<%BANNER%>

Manifestations of One-Dimensional Electronic Correlations in Higher-Dimensional Systems

xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101118_AAAADE INGEST_TIME 2010-11-18T23:30:01Z PACKAGE UFE0015302_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 1016085 DFID F20101118_AACBHA ORIGIN DEPOSITOR PATH saha_r_Page_115.jp2 GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
793001d6cd9963e47b9c3ffaa49ced3d
SHA-1
130ddfb67cd761cdaec78dc1160a6dc6e3d41d7e
759381 F20101118_AACBGM saha_r_Page_081.jp2
ab4987f7c456122cd8eaf95003adc1ad
c40dcae7130439d13d2cf7ac15a458372b9b523f
783827 F20101118_AACBFX saha_r_Page_059.jp2
a8d8dd0f6cc757181169525da4a1bd58
23948eead89aa6c7151466c0fe39121daed545f4
877639 F20101118_AACBHB saha_r_Page_116.jp2
6aeb67fc6568a83959129f080b122ec2
386d40787c1b6593a2979481d253ab58dbdb4c1c
695516 F20101118_AACBGN saha_r_Page_082.jp2
b14b82990b34d25614ead822cd90e0ff
256fc13f7595e3bf7d5b2281f8bb80a6871cb6e0
998917 F20101118_AACBFY saha_r_Page_061.jp2
36b217753f557f8852ec703afefc4e42
b3013ca8cb398396271136f4e55c414b30c21219
931795 F20101118_AACBHC saha_r_Page_118.jp2
cbfa24bc053f76204bf89e3581a8fc04
2dbb5569a7feba7123789bfe95688c9a64524acd
946254 F20101118_AACBGO saha_r_Page_085.jp2
689d9286c31e15bb06dd4ddd41d853b6
abb0f797adcdc15e141c4f5446c0e556a6c082d9
724005 F20101118_AACBFZ saha_r_Page_062.jp2
5cb965d1cc3425860aba6d8048ce5cb3
02d543184444e70bf7bc391af39ebf120eab6542
930667 F20101118_AACBHD saha_r_Page_119.jp2
53906ef60f0bd4b9a802d246531e77a7
9c019e0bacc5812afa8fcc3052333e90bbdc88f7
995018 F20101118_AACBGP saha_r_Page_086.jp2
d90ca90a8bed55e5f87bcdf4ba0a9cec
254cb773a54d2f1a591a35f021ef684beea818c4
884261 F20101118_AACBHE saha_r_Page_120.jp2
4ace65737f535a74c36a83cf8f826342
9d4cc9174386bbbea6361eca401877c1ed8d7d7c
768779 F20101118_AACBGQ saha_r_Page_087.jp2
81460c3cf5ed7343f481b13527a1b6ba
85e1d683c698be1502567c24c7a1186299be7b7e
786538 F20101118_AACBHF saha_r_Page_123.jp2
b4d3e672a962425993dfb800f4dbd211
74a334204c56db927163a75d869d6b85142b370f
918686 F20101118_AACBGR saha_r_Page_089.jp2
80f115474c39b78ebd04aed97597449b
d1e0abe9c0e4a060b383c16a5ba0f315fae7c25c
1051939 F20101118_AACBHG saha_r_Page_124.jp2
8614d6c885546020419865697b7f9d71
b2908722dc6d5ff091106362ad9b92c9d981eb6d
960403 F20101118_AACBGS saha_r_Page_092.jp2
40980ea180c89f733598ef78eb3e7a0c
1692961e562c63fbea2252cad7786a96a30cfcbf
671037 F20101118_AACBHH saha_r_Page_125.jp2
bb173435769eb0f9f54bf5849bf6293e
2987961a6468b135feded4407553285ad2cd3a4e
1051956 F20101118_AACBGT saha_r_Page_095.jp2
9c21814b82dc2ccdc9f4b3c25ebef9d9
36617c9ce2209b14dc6c504c4e28ad08d6cbed15
F20101118_AACBHI saha_r_Page_128.jp2
8c11f6ced081244dcef83d8ad8f16e22
c7dbcda6cba5be1b27ef441db6825f453b3c3111
1051981 F20101118_AACBGU saha_r_Page_096.jp2
344db2dadddaf3428a5ac9ea8f8cb465
1dc79347168b1680fb3d3f374eaac18eeec3365e
778281 F20101118_AACBHJ saha_r_Page_129.jp2
69f76a416888d3c7c10f72dc8287fceb
22d24ec229b4d78b2d1c49f43ce0b0671634d262
1051974 F20101118_AACBGV saha_r_Page_102.jp2
d702d8aec9b7f35070e586143538d59f
79ab1eb2fe3c2e3edc8d6945210f1369ac94065e
763301 F20101118_AACBHK saha_r_Page_130.jp2
b5897b0f9ebf46837d2ba4cbd5873637
ceeaa7a4c6166a0ffa4c52bd136742e5f5e363d6
87071 F20101118_AACBGW saha_r_Page_105.jp2
deb6f8922bca8db30db00e614f3b5ce8
ea66e1d0f2e6ae224daefa898b9d91856a0e9955
1008310 F20101118_AACBHL saha_r_Page_133.jp2
e4d9290d31bdbbc617ee2201ff214694
226c151aee3eab8ad648639827fa42d622cc0c80
1051980 F20101118_AACBGX saha_r_Page_107.jp2
4a3dfdc619b6f0ad39d48f27e7033534
e58b702a95e29afc5ea268ff9098d45673266ddb
961913 F20101118_AACBIA saha_r_Page_153.jp2
18fe6a8128d111733771b6c5321fd642
170dcf755a1187e1402c8b76bcda17f991254913
827791 F20101118_AACBHM saha_r_Page_134.jp2
250e052c25ed93463f691171f10f19ed
69bdae13444d9592cbc8701a6fd45e92d4c5cdd5
1051946 F20101118_AACBGY saha_r_Page_108.jp2
b6c12cd65db749fb2527a37722991ae9
36f607c5668014260d28329581a0a0efa3c4ed59
646937 F20101118_AACBIB saha_r_Page_154.jp2
199b94b1ce326a29aa9f74e2739723c1
34e7d0d4393f6971a256f567ee0e75ea3d7f3805
1051976 F20101118_AACBHN saha_r_Page_135.jp2
5cf9c22adeb2beed8c417d24c2b897ec
7313c68aaffc801d4cac477f007ecea5084c6e1c
1051986 F20101118_AACBGZ saha_r_Page_111.jp2
4135e317c0aaa16818e09842cbe61867
9d5d2e8bf4b6c4acd223f785316693e9b27af950
455268 F20101118_AACBIC saha_r_Page_155.jp2
5b7ed40612a93ee822f10842d4908367
1c82aafcca190db7ae9c1e186b84b66850ed661c
665597 F20101118_AACBHO saha_r_Page_136.jp2
b19b98619b97b70a4129fd608228644f
e68148d128ea3f4e7cb30bb625cf36b8f3a1d592
849937 F20101118_AACBID saha_r_Page_156.jp2
984a89c1b823c03edead154742b4b8a3
c42c385ba6a5d17703dd9cc256b129f41d19da72
982755 F20101118_AACBHP saha_r_Page_137.jp2
03af35b746eefeac790c68dd87983684
6146e6bf6f96642208bab3426b70fb699997c0e7
799981 F20101118_AACBIE saha_r_Page_159.jp2
d11ba8e82e0ccc940c61d1549fb17857
5a44011c52fda3d8d271d02ab8b38d38e5181904
855935 F20101118_AACBHQ saha_r_Page_138.jp2
dc7fc160e1ac7d5741ec37fade03429c
538fda676da464695f751de900dfa87e2fc80217
609456 F20101118_AACBIF saha_r_Page_161.jp2
b7c17691c143ff657248e37cba32f4fb
8513471e1b980e6714e107ff54b0257c365a5ff4
F20101118_AACBHR saha_r_Page_142.jp2
ea0303564d0f68bc260cf7e1c37ffc23
6207b1364152a592fe4cfd2db814570e9598eb07
73589 F20101118_AACBIG saha_r_Page_162.jp2
479b0b8c46df4f5088aa7e5de918a441
1eb57450b428ed8188f8f870b2096b7dd05a38bc
621785 F20101118_AACBHS saha_r_Page_143.jp2
e497b80e74b35bc90154e6bd1bdeb6f6
edfa276014440fd81f3bddea5edd02171938e491
683632 F20101118_AACBIH saha_r_Page_163.jp2
e36f0ba82e2e9b753a51710f88408ffa
2bb6aa8365b1d0ca135c7d74ef0a9e89923499ef
70998 F20101118_AACBII saha_r_Page_165.jp2
9ce1a851ab262f8ad3a8570d2a79dd5e
7c97e6b2f9ae12122becfaa88ba9c10291ea3aa3
877768 F20101118_AACBHT saha_r_Page_144.jp2
7f31c59f8d9b821447871d378c8a7ecd
2d6698afce5f96bfb050c2887c93836c6d97ca2f
739995 F20101118_AACBIJ saha_r_Page_169.jp2
0d564cb04a0e2dc4335a8ba748d054af
f6ad1bb96f3082b6acac85a063d09058e9c77e0f
906320 F20101118_AACBHU saha_r_Page_145.jp2
6afdd23e11cb6dba63c2a3c7a4811163
47321cb62db89d16ed8079daffc46e3523f0661f
62859 F20101118_AACBIK saha_r_Page_171.jp2
1eaf887c737d43fffc0f29bab0331bf5
df2add088b7cee9ee83e2053b2beba9ab1087566
1051943 F20101118_AACBHV saha_r_Page_147.jp2
a80f1ed91a98b390dd9affeb3ce8089f
fa613c5b2885527544aad01b843c7ef46b20cb0f
744113 F20101118_AACBIL saha_r_Page_172.jp2
6ad212625bf6c95d1daffd978cee5f67
03af704a5b7831fc1c37af86139199459a0b4592
84382 F20101118_AACBHW saha_r_Page_148.jp2
de943414d864bbaca49f4c8fd9910801
ea436db042d9f2b122da9ebdc2e98d3c9718a5d1
25271604 F20101118_AACBJA saha_r_Page_020.tif
db0c31a592aa7b613d93fe30b3ccef94
a20c912d19a54fc161a7e13dc146c15658cfe0b5
712890 F20101118_AACBIM saha_r_Page_173.jp2
8331b2374dbc82bec78bd35181715bfd
4dec52276c2d61c89d98d2dae40721d246d21f44
F20101118_AACBHX saha_r_Page_150.jp2
85279c764121875b806b5a5cbccae1cc
a9311db4789d7aa8487753756c7e32b05a5e31a9
F20101118_AACBJB saha_r_Page_021.tif
55f4cb5fd38eb738895bc074a7bc52d4
d0721616b7572c07582a33b846662b2dded5974a
858577 F20101118_AACBIN saha_r_Page_175.jp2
509f16f8ea00d665a9fa8ce53ea0401b
78e9e6a14634ee6b251a13b749fbda98f3a315d6
1051968 F20101118_AACBHY saha_r_Page_151.jp2
6e322d07a8cc2b0c840512173d4daa6e
ebd73e791147ecec1dadf069d36aac985b954788
F20101118_AACBJC saha_r_Page_022.tif
84b8c8fd17ad0ec751e169b290138c91
272434cf5690d3a26974da7caf93d00443785b9e
95678 F20101118_AACBIO saha_r_Page_177.jp2
1f788f8f0e78acf621afb06138902347
3b33d1b3669f6d5929a2a20e08a82e6e02ed94e1
1051910 F20101118_AACBHZ saha_r_Page_152.jp2
3426ec7abdbb99e39ff80192303ed65c
db99e417d9c38ba02878447b16a56f02a14b6655
F20101118_AACBJD saha_r_Page_025.tif
0914114dbde401525b1c4a72d9aefd9a
a47c3cc8cf561ec08724e18a41c793adcb963c23
114662 F20101118_AACBIP saha_r_Page_178.jp2
05765fdc66eae24aab07f10c31ba0c17
223e30b13872b7075d005aa71292c05437149efa
F20101118_AACBJE saha_r_Page_026.tif
3ab5334d61f3e550815c9e867c008c1f
bc16dd69af2291c61fff2fdc49204707254ad82f
106835 F20101118_AACBIQ saha_r_Page_180.jp2
77fc279bc084c47cd275d5cc7a3e65ed
1b22a6b35d7595ee992bfacf32e1b8207763e6a5
F20101118_AACBJF saha_r_Page_029.tif
9c4a61ca0df071782211e6a319483caa
b79353443148b49feff7cd54aeb60816e2421d6a
99310 F20101118_AACBIR saha_r_Page_181.jp2
ad5cfad088be360e2da6a56304d375bd
461b202b5c3a3ab080f3f9c8a989897d8c1f60c8
870 F20101118_AACAGE saha_r_Page_155.txt
89a7a7fd0f3996e8ea2ab48efd765bc5
2d23d9545112ced7418f93e068bc6f6d256fc169
1053954 F20101118_AACBJG saha_r_Page_031.tif
1ddda44d2bbc381588ee428651a5ebf0
b41b5a58cf1f204afa4d5c839deb1db4ca569f3f
97085 F20101118_AACBIS saha_r_Page_182.jp2
be6876690dc25c357c7c3c1df045cfdc
f1cd679e8a88cd30141bfb46d11d0e93b8c4c917
1537 F20101118_AACAGF saha_r_Page_103.txt
0e917f4047b7c35832fbdf66180eb67d
773740938749ca39f6bae0cffae2448c2666c522
F20101118_AACBJH saha_r_Page_032.tif
3c8e838557d9771c4617f45aefb1980f
5298a7f59576131829375bca92065b62d5f8d443
56956 F20101118_AACBIT saha_r_Page_185.jp2
907867559a70c93b84a91c7080dfa10a
765c121c0f1f9829066931eb7e9cd547b8fef2b5
54837 F20101118_AACAGG saha_r_Page_049.jpg
0be16bf3af521b7e36b4c87f7432e6c5
153749bd7c8cc2c7801b6287d7a18a8d113a863b
F20101118_AACBJI saha_r_Page_033.tif
78ac8f8426d3ace33627d701342aac05
bf8e4a18d485badef93a79dbf03ac05803a7cac6
81 F20101118_AACAGH saha_r_Page_003.txt
aba3f1506c44523d2029cb4c32de0e2f
aecbf60a0cab0c1eb1fbc33fe6da1bf558725e32
F20101118_AACBJJ saha_r_Page_037.tif
dbbe16d2254d4717ea57428a9d878b17
33c8a3dc199bc806978652541ae75109ea64d066
F20101118_AACBIU saha_r_Page_004.tif
bfbc71dc77fd7f1ca22aed4fda0d2b06
90e87f4cf3379c58791a19e788c65a126c86fca5
6034 F20101118_AACAGI saha_r_Page_147thm.jpg
9eb816e9e92e3d229dc96e3a8bd34b70
1a5bc228d4de8cb016ee83b29df3f291c9580c1b
F20101118_AACBJK saha_r_Page_038.tif
788787064577234dc526d22eaf2e07dd
ba739145af36fc2b8f3da21f6c0f3af5d11e8d6f
F20101118_AACBIV saha_r_Page_008.tif
8f51464cc579b4e74b393767f08cd5f4
2e2c014edc55e0552df6f15ccfb088c1937ff04c
F20101118_AACAGJ saha_r_Page_007.tif
cb81a67eeed13290727bcf52a25a8fe7
55382708a584c96fd71a8635a0c00c3ee67e06f9
F20101118_AACBJL saha_r_Page_039.tif
c6e1184a1acdf6dffea784eddad3b5b6
d961a98ad26be39180f8a0033512412751b073d2
F20101118_AACBIW saha_r_Page_009.tif
fbb02d75c6d3a5f7de35d31dfce37e8c
21965e0d50e33d15a189b1c051d3b02a460ed11a
F20101118_AACAGK saha_r_Page_130.tif
e5f923802980403e80322e9aedeff1db
c73d2adc202867677bcd6cb33d717aa9a8fefa19
F20101118_AACBJM saha_r_Page_041.tif
8303ea0a1e350b27ffb2ede7932521ac
b324a161ac7211c7941e38b24f0e6d9a6183bf20
F20101118_AACBIX saha_r_Page_013.tif
0d6c588e2ab022bd1206678992d2a2a8
1dcaf20c35e6e63fec6c40438b5c53ee79351e62
F20101118_AACBKA saha_r_Page_061.tif
56fda523d890310db9cd4940f63b20fa
3010f41351ff29876df8ee2d8e784b3e9d99e373
22572 F20101118_AACAGL saha_r_Page_069.QC.jpg
a63ca5d4f047b32d4dc72d5ba1df9aa9
3a6f007df212090ccb6d3636c1285e69af93768b
F20101118_AACBJN saha_r_Page_042.tif
318e2be91b6f4c434b115882ef811865
8f195d90cedfb76a292218ee558a33ccbc88939e
F20101118_AACBIY saha_r_Page_017.tif
bdf968ce7114e1c6ba77af8acbe5ee87
2b5c9f381fa9575fbbd196f5ccdf3297a7bbb5c8
F20101118_AACBKB saha_r_Page_062.tif
d471fcf4626adbbf6d3f3c691fabfe2b
001179661af3c9d7dda81024c47abfdfed8b442e
933608 F20101118_AACAGM saha_r_Page_083.jp2
00e7ba004e38626df41232d471e2f80d
824d4dd7eef89993694c5b72f4851e0e70299f3d
F20101118_AACBJO saha_r_Page_043.tif
04f72d8ea4f6a54e89a30ad5dac1e5ff
1b8d11580328111184ae4d19e1347aa8b1e925d0
F20101118_AACBIZ saha_r_Page_018.tif
2ac0cfedcf4c06c246ad4dac2e24fd02
57ed5b5d541d088a2ff8dbdb23c078dce5d127f4
13432 F20101118_AACAHA saha_r_Page_171.QC.jpg
3b0f0dc07bf1f58c2fe929a0b8b878ad
3c2e585e818b3b1049c4bb98ff99f2cdf9d415c7
F20101118_AACBKC saha_r_Page_065.tif
b02f3ae632f13b80af9baed015a24049
972272a8111a860b7f28044deb3f9ac6698eec56
783090 F20101118_AACAGN saha_r_Page_091.jp2
f1b2cf255edf57112ba9ea59a7542644
8b7429c771bf9fd4a18b474dd1c47357dc14480a
F20101118_AACBJP saha_r_Page_045.tif
ec4ce8e0cbfcc76cb2a7bc482fb26c11
3d81b519ffab7dd602c266780bb810e81809502d
597 F20101118_AACAHB saha_r_Page_003.pro
6ec907603e3843cbe1d93f860e8c5d1a
38c067b81a02c319b465f367740fc635a1376760
F20101118_AACBKD saha_r_Page_066.tif
fcd70988a0751e12da7f3baca6a342f2
aef2611753066e1f550f0218fd1c6dbb785245d2
5880 F20101118_AACAGO saha_r_Page_020thm.jpg
0e48c9a1dd3e5095b872f591db24e151
45476cc6a10b1597f33429c22cc74406e832bee9
F20101118_AACBJQ saha_r_Page_046.tif
7e358acf37666e12144ef9e73768ca91
447499ed4ad72d6b4c349af05e36c8127b487ae6
959240 F20101118_AACAHC saha_r_Page_121.jp2
fc49001f99c43698ff68eebc0f14ea34
928cafb9cf10a08be748442930736abe4df841a8
F20101118_AACBKE saha_r_Page_069.tif
010f1492f975883e69590a3b57dd2fd3
d9d501ebd5d10f4588e0779ece1c2a52b63f4f26
67980 F20101118_AACAGP saha_r_Page_126.jpg
5ca7e5e50225f7f02d78995c50c4ebb0
39adfbf443668a60c4c1aa29e2673cdfb656d3f1
F20101118_AACBJR saha_r_Page_048.tif
d9178aad7afa7ebc779e668114d99fb8
d8ae989efb91d8a713c905dff82e8e64b7f888a4
F20101118_AACAHD saha_r_Page_079.tif
438784a8e482d51c481293c5bb154027
85603c2e4af96238df0cdbfbe6cdd5ae872aec07
F20101118_AACBKF saha_r_Page_070.tif
4ecbade412ea37e94dda6c5f48ab604c
90d2614b6b11fd33d9433edc01d94cb4f8420e67
1552 F20101118_AACAGQ saha_r_Page_163.txt
55190a600930a53c05c478f715b06385
1d052541bb29a1c7ad01d44373c057249a4e02bf
F20101118_AACBJS saha_r_Page_049.tif
f9cc04551842cf37eec8f9adda3902d4
d185a9d9114ecf2765eb2b16b763996c88e9e2bb
5498 F20101118_AACAHE saha_r_Page_158thm.jpg
7c2b5f45d65b5e2fcd04dcaa8fcf46db
9ab22bd52f0db8f027b4eb240db2a70b70180b1c
F20101118_AACBKG saha_r_Page_072.tif
30997f84e0042efe58be1313026fede3
fdc836d4850c6f28f8c642bca5d78ee8dc8b9471
69146 F20101118_AACAGR saha_r_Page_112.jpg
fcdb61ce082fd272a55596dd6d9bcc42
a01b0cb46b6591971755f5b3b6ee9e602373ac84
F20101118_AACBJT saha_r_Page_050.tif
d870a1338e5e775813ed0ab2bc747906
0153b4f37bf39aa6b376bbba5e605b9542cfbbd4
923533 F20101118_AACAHF saha_r_Page_168.jp2
72d1bfe70450d367ea3614b72bd2c8d1
da3ec583ad62c1f04e2852ac9fe8334a5938d078
F20101118_AACBKH saha_r_Page_074.tif
94de0fdee8ca753e8bcd1045b598abfb
a9ee9382681dc1880edecd317f465676c2437d40
77019 F20101118_AACAGS saha_r_Page_028.jpg
d4bb85a95d14c5985d8dda6bcaf557e6
d3987c4eaf47c142be83f21918cc3c31a192689f
F20101118_AACBJU saha_r_Page_051.tif
36433c5f0e23be433bee088ac58f1225
2dd7b490f9367b02c4d2f1c032bf06b72f6c5f63
42911 F20101118_AACAHG saha_r_Page_133.pro
72aabff923a912922103a63e7146c179
3f6c4c55308e3d7637d928cc4486681da5c0d100
F20101118_AACBKI saha_r_Page_075.tif
94832c96b5d8d09d68f0ff30fabee754
f1250cecb882d0bc2c3d6039b8ca0dcba60319b8
18546 F20101118_AACAHH saha_r_Page_179.QC.jpg
91fd1cba704c862832f78e0071847a00
ed311e87c5b1e3c46dd3171437c9d68042c36675
F20101118_AACBKJ saha_r_Page_076.tif
77f252544321647f2a81aa0f6cf5b12d
414e2c57fdb1bee7c1515879596f43d054e73a43
F20101118_AACBKK saha_r_Page_077.tif
9cd0bf68507900224a3d2256d5fbbbb8
110c2721842707f93e65d35fad430e2e8a96f4de
2132 F20101118_AACAGT saha_r_Page_141.txt
fabffe45c3215231fd5969df02173ac4
8987389bbf2f10e508a64deaf8b63fd4c49f96a9
F20101118_AACBJV saha_r_Page_052.tif
928d17036e6494bc65dfc3ee440e49e6
8781fe919547a22bb950844a91e620051ab5024c
1352 F20101118_AACAHI saha_r_Page_067.txt
abea07eb90ce7abfcb480063738fd4a2
5ac5d78754420663f68d3a88ddb740cbc68d9dbf
F20101118_AACBKL saha_r_Page_081.tif
6317e1c5e60105fba760f46650f9009b
b28737969626c3ea004ecae93e126c578a0aff97
F20101118_AACBJW saha_r_Page_054.tif
289eed64fe95a31a18a9cfdb4c6bcfae
5f8eac71ed369b96820557a3517f0ca7545daf9d
5534 F20101118_AACAHJ saha_r_Page_025thm.jpg
919f935094bf3b2107de884e77b75a53
40e5a511bee6a51a8bb67f653492ea319316e0a5
49413 F20101118_AACAGU saha_r_Page_160.jpg
f71fe26546fa8822b19967252ba13144
03935c26bb54533ad919324eec2441505b634548
F20101118_AACBLA saha_r_Page_106.tif
496ea2c79b4faf1161f1d77315997374
827188e63c339c000f7aece8f72efbfa02c2785b
F20101118_AACBKM saha_r_Page_084.tif
8c672664eb16652b8524e8fcc7a6e89b
63f82081743cff99446375f33e05d45baaa3d2f6
F20101118_AACBJX saha_r_Page_056.tif
6ccf9a986dbf3f7578253d229e4dd5bd
31c1ad96d95c07b1991859389b225d41d1a8be0b
F20101118_AACAHK saha_r_Page_092.tif
7bea02ea66d1e6e1458655f6baa63dc0
3c60638bebe65bac6a086c145a638e0731cf7436
63694 F20101118_AACAGV saha_r_Page_182.jpg
672f9c4e131c216895c501537b0dfebe
3d05498871eca434d2172171cd1de1851c05ca1c
F20101118_AACBLB saha_r_Page_107.tif
f6f61c907d07b67c79238e2601bd06f1
bf09ddf29138c485682a69cb151a81bbaf64325c
F20101118_AACBKN saha_r_Page_085.tif
2695dfdeeb004a4bbd050fa91979c997
f55941ac68ed72646768c042b9de3d450852e2cd
F20101118_AACBJY saha_r_Page_058.tif
646f0db6e9b14092dff03dd830829eba
4eb3603d4c1e7261b03a5a03a3505e2d26d62bab
6347 F20101118_AACAHL saha_r_Page_095thm.jpg
dd62fbdac542201730970b2e1ec7182d
991fc382721e10caf16a3d36e7ce1e560eb5c16c
1051961 F20101118_AACAGW saha_r_Page_084.jp2
399e92ba19484320668f8697ff9b8456
7fc4958060004d924e7f5c9b470df60c79cea343
F20101118_AACBLC saha_r_Page_108.tif
373250bec5a4a2782f17983d32465a8c
4183b3fad0ea5c0b8ca766c6f70524c40ea24ae4
F20101118_AACBKO saha_r_Page_089.tif
9403ef01fff815124dbf90040dc69808
f79e76259c02dba4422dfab3d755fcfc7c948b3d
F20101118_AACBJZ saha_r_Page_060.tif
10cca1926a3c95de1f7c8841be16273c
28fcf8c4be3730d9f51b51385e94eb4995f756d4
4849 F20101118_AACAIA saha_r_Page_037thm.jpg
547fb8584651b40f65fb529471076961
92ab665694f684ddbec97eddb88549ef9818b6a0
42638 F20101118_AACAHM saha_r_Page_131.pro
9b6bf7d5113d6ed38e85827b963501f5
7b364cf559a6e6c58fbd34c0fb0352958e7d8619
63807 F20101118_AACAGX saha_r_Page_083.jpg
57063dfed9418bce3580e6c7e64ad02f
43fe01cc84f1ee90941abab8b6b55318ececb2d8
F20101118_AACBLD saha_r_Page_109.tif
9a6998b1f82ca24a3faf43f547bc82c9
f7a85a015888223bc2b368a560fb5b4c95bb3293
F20101118_AACBKP saha_r_Page_091.tif
655693b83ea9b6a7ca1b62adb7880c65
799c3d431e3a2670d2df5a4e7120f2ef1cf47856
18863 F20101118_AACAIB saha_r_Page_156.QC.jpg
5a10ace1604b06ec12bf8dc33b1065f2
9df35cc2cfe07bda706e49f72fb8179c5b886986
F20101118_AACAHN saha_r_Page_114.tif
5bf3acd278c8f894f2f39205e8c164f5
3dec6473c66717c5f977d1c686b3bd0fdc6e1369
63854 F20101118_AACAGY saha_r_Page_068.jpg
df6ff4f89d4d983fa41965117f81990b
8fbff50d5cfb1d517c88b30acd439ce7e4d6aaf2
F20101118_AACBLE saha_r_Page_111.tif
b2aba3d6d38814f7f379430602e0d24d
a050cb56da6de54523499a4458d5f3c247a09403
F20101118_AACBKQ saha_r_Page_093.tif
860809cfc09da06b04664c02f2a67553
4bf8852ba7f81e32a039d115b9063e0841cfa0e9
1967 F20101118_AACAIC saha_r_Page_145.txt
8c26c7f841e6ac65d3b9089e918deefc
b1c05291de74d707c709d423895cfbf4c1ddd5f9
72867 F20101118_AACAHO saha_r_Page_115.jpg
a3d3d29e971b10a1c47666b7a81f986f
9788f02111f2d680f35cb4b6f5b9c24f344003b6
F20101118_AACAGZ saha_r_Page_152.tif
98bbad4f436f23f98cc45e6cc2a11ca2
5b8e2a339cea73d912556c00e37a97cbc00426be
F20101118_AACBLF saha_r_Page_113.tif
701ce1a3a8f914bd47db75e6ea6e708f
22836547f32b8506f6be4b89b61905b3821a1262
F20101118_AACBKR saha_r_Page_095.tif
7418306ef2f6fa286f2d5b8135926eb4
57bebad1c28e5978c1391db03bace9fc6138b2bb
916091 F20101118_AACAID saha_r_Page_036.jp2
8e0745bb0432b0e7ba3119dc8e4ab555
9f87485d12df6185562fc27fa19a089910b2f525
5454 F20101118_AACAHP saha_r_Page_080thm.jpg
a4b56f4dbc7fa85a8c640472a4da9583
a94665a61eba948cbc8f3d3428774a3f4b936a1a
F20101118_AACBLG saha_r_Page_115.tif
c7b080b3f215d9061c18c3d39c3cdccb
33533cde7a60b555073a191633bb29415bf18a55
F20101118_AACBKS saha_r_Page_096.tif
d924c73de6e63e5b07682adb42e0abbe
259b354c45eae4cbcd12c20f121dd236e0ff46f6
F20101118_AACAIE saha_r_Page_163.tif
428368e2515561ef3e8b58286e9d723c
00e246608df92ea31df4267e3f73eed4b416f962
6035 F20101118_AACAHQ saha_r_Page_054thm.jpg
6658db2ba0386b3167102d475ddf2461
5665aac305bb308b2563f38d6c115c904d990fa7
F20101118_AACBLH saha_r_Page_116.tif
055fd667e6a1ebc2247eb95b49029ab8
73cffc1720660b72d116dcdfd4b2ab40f0b2bb68
F20101118_AACBKT saha_r_Page_097.tif
c8dd01710de64fd141b16c00b16637a0
e505733050f0d34b16bbb02047a76e2d83d105fc
19466 F20101118_AACAIF saha_r_Page_083.QC.jpg
8cbe1889959cf94d9a377f91b856cd66
839eae97fb240f6bdc533b78a5224d19f591b705
19743 F20101118_AACAHR saha_r_Page_072.QC.jpg
52a4d384d03f0275e749b8fe94e8b092
9991cf644cdde560257b9b86e643681e19205788
F20101118_AACBLI saha_r_Page_118.tif
c07367661eccc9a3c6070c5727305024
9eab4d5f3a9ad9363495e72ed9c425107d2590ea
F20101118_AACBKU saha_r_Page_098.tif
3aa7d09054a3e88bf188d6bbd9c234eb
31177ff27eafda1c23caed6e4941cd19a6096118
19465 F20101118_AACAIG saha_r_Page_006.QC.jpg
570b35383d2dce3c6600edea72d214c8
d3c6e5e71eb2bd7470389cee8580c5139326c2e3
52188 F20101118_AACAHS saha_r_Page_023.jpg
cf17efa1b7b53f5cf273b0b8daf37c2d
e1ebf66b4d2ca77219718facf4691c40eac5db31
F20101118_AACBLJ saha_r_Page_120.tif
7ee43e10bf4d96a1b718a2487d3f10f1
640190b9ddfeed38b4728b078aa780f91f106cb9
F20101118_AACBKV saha_r_Page_099.tif
42aab854db29bdbc6a0d6281d5cb0375
41ba021278efc9edeb82aff4195dab94cffbb108
F20101118_AACAIH saha_r_Page_124.tif
f7afb81b9e7f2a52f5002867f8e2d1b4
c5c9f0a01b7c58814b078976c5e9ae46c44a5881
5729 F20101118_AACAHT saha_r_Page_180thm.jpg
b204da083be6d2b81bcb5028d8b9a9e0
25f90acbec6a971411686f21707088590c5db92e
F20101118_AACBLK saha_r_Page_121.tif
58cda0f5835bbc8af78c24b9f4bf0872
3c12e991ac4c3896ccb0c21fb59fcfe0c9e58266
28226 F20101118_AACAII saha_r_Page_067.pro
6f165eeb42c3a5a7e2b436cc1b92d7d6
baea8bb3f4ea5f70e30ad106497170860f3cc478
F20101118_AACBLL saha_r_Page_125.tif
e56683edb68d88dee6c0745d0f015edd
86da63b7f643c4883e9ee0e4e95bdd50ba54c76b
F20101118_AACBKW saha_r_Page_100.tif
540c1cb38aae6a76ca2ad064bfa8cdc4
d9439c791eb7a045727309e20c115d78763cfb42
4989 F20101118_AACAIJ saha_r_Page_050thm.jpg
45c004640cfbfab77afea1775ef1382d
4139e6e4ab1ee2baeb7aa1436482464ccdac674b
5468 F20101118_AACAHU saha_r_Page_120thm.jpg
945a2bae0dc0ac5ef9922865f50b40c4
31e2eb77445514ffa8942a46c1b7e006775a41eb
F20101118_AACBMA saha_r_Page_155.tif
2713043ea03367ce67b39d9e4a43a6e3
bc9758b8b4aef9f41aa13e214043d5b4982120bc
F20101118_AACBLM saha_r_Page_126.tif
277c90b9e26ceaf6a608f5ca813190e9
d947e80967b532a1401c2a791989ec0f692ee08e
F20101118_AACBKX saha_r_Page_101.tif
2501d74a8a6a71b674f762d8f7d19258
04155ce4a4c2b1c109687ed2527aaf18f84cee81
27052 F20101118_AACAIK saha_r_Page_108.QC.jpg
470b881eaea75a1d24f725812c271e96
6ea03438c28d85739676cf837f14ee2fe9df831b
20734 F20101118_AACAHV saha_r_Page_065.QC.jpg
8bb2d534ce0cb7074a841f13a857ff56
92bc1d68cce1c19d140e313dc8b96415f7e43213
F20101118_AACBMB saha_r_Page_156.tif
25da2e6134be1a6d61804bb27b60f3a4
b2240f08d40ac3f25c886a522e1a54b920764dc4
F20101118_AACBLN saha_r_Page_128.tif
7980ce6ff3db4308fc24e1fb53bf110d
7dce27d202e37b77be3d7eb8e5fb66fe326ec8ee
F20101118_AACBKY saha_r_Page_103.tif
712e0e39367a4ac0fa21f2526e248f64
317257c181b714834b831bda3e1849d90a6cfb67
74928 F20101118_AACAIL saha_r_Page_046.jpg
87579807da067ef4b56cf0ddee9f8f51
ae0ee7cb376a3072f4829d68e3aa9575701ff63e
20009 F20101118_AACAHW saha_r_Page_116.QC.jpg
9fe43e91a70d9645313ab0ef453fd696
e7054b29b75b23d16687e5ac44d2614b1e3cd099
F20101118_AACBMC saha_r_Page_159.tif
eab9016e4205824f7cdf8df2b0f4e822
66138ae5be7bb0c9e85650cb21be58278ee4c518
F20101118_AACBLO saha_r_Page_131.tif
64f86999d0504bcbf02dad6851f55a00
441f6824d84b4dc1af725ca86377ecac4b8257b4
F20101118_AACBKZ saha_r_Page_105.tif
8e6154a45a482d874f249f3f325c5e41
9bbc5f3130bfb78140857eab219a374c21161cc6
1026383 F20101118_AACAIM saha_r_Page_113.jp2
bff7495991c58a4b663f7127882428a6
3ebcb56173bd88e892d684f72a89f89ebc1e9559
21846 F20101118_AACAHX saha_r_Page_075.QC.jpg
7307f4f5ba9d50f1c2fe346cfabfc2b3
35bc5263b853f607f5f710d3540f859cc263e86e
F20101118_AACAJA saha_r_Page_022.jp2
be7d6180fba1f7af9260a47c7ff29c68
91b7701ab58d3781b6ed0762ff7c40ae406a591d
F20101118_AACBMD saha_r_Page_160.tif
b98084f6dbfc13157bb18fd36d5e0d6c
4e15dfd0a5a7f9bdd8103ead5c522e7c9d6554ee
F20101118_AACBLP saha_r_Page_133.tif
4c04191093f7016a2ddf9bc821c22c10
e675b5217ea962e82991fb9deadc36aa2e8d0d82
2094 F20101118_AACAIN saha_r_Page_175.txt
b0f9dce33839808a188cb1cb5764c821
004091c1f0c5e418caca89e91263c9ac3536e026
921363 F20101118_AACAHY saha_r_Page_078.jp2
d68182a1a08b85e013ee6c1e2fd71488
bfec6657e099a1af75e29b48cfbb2b0dd133dfa5
37605 F20101118_AACAJB saha_r_Page_097.pro
b6978c13c5e8fc44877201d39a3f3ab6
f1b98358bc71de629e4a60f3e07baaa7c165ef8d
F20101118_AACBME saha_r_Page_162.tif
5e76adc52409f3900c1b72a275a4c5d6
4dd4c269af2c94d9c32fa3cc9ad258b08dda3420
F20101118_AACBLQ saha_r_Page_138.tif
28d3d07950d9cac852a91850a9ce9bde
53d45081a3cd70f78eedb59360d96c0aeff7c915
F20101118_AACAIO saha_r_Page_090.tif
174f969f84563d2d1978ce3c03c24731
a6df795e47e95a378e2cb95d4d8a44cae4893d2e
1747 F20101118_AACAHZ saha_r_Page_138.txt
b6ec2835bfbfa2a22a44269c42c52eb0
62d066f0b6f1cdddf9ebe72a595a566e8a0aeac9
4695 F20101118_AACAJC saha_r_Page_105thm.jpg
79e4007dded21f55041f6419b5768fc7
5b8866f14544f9668acf81125d138df594896629
F20101118_AACBMF saha_r_Page_164.tif
fb6de5d62d74457eaf9b523ffeea68c0
3a3dd61f0b94897873f2349aaf0a1c6cb99ec97b
F20101118_AACBLR saha_r_Page_139.tif
8a6d76e2a20209c1ceefad770f4f0c59
060d625a9626468b013c945c4c8be15ff3ac6ad9
73884 F20101118_AACAIP saha_r_Page_178.jpg
cbe6ce021b55289acfa33cd18f414527
299a61bb3f77bf72834b92318f76fcc82d7d0a3f
2097 F20101118_AACAJD saha_r_Page_117.txt
670e7013c9fb32fad44162885d01d850
2531cd7656400cd4cf3762b178f5d65e48934d52
F20101118_AACBMG saha_r_Page_165.tif
b686ae87de2855e1ebef2d85c908f296
d06b529d2c61b03ec697241a16c3ba131df7682d
F20101118_AACBLS saha_r_Page_140.tif
d33d3f7f7037775d8af6f5cd61222be2
41ee7cca46dde40b8f502d7bda4ffe510433a58f
57291 F20101118_AACAIQ saha_r_Page_103.jpg
39407f39e8611c2626aa04ec37f4011a
9adc19447ebe0644c5f1198ac9fe048f71186ac6
39885 F20101118_AACAJE saha_r_Page_098.pro
ca3cb8eaf70fc0d1b862be9d8104d31d
7765db60198202c8aa614de967d20a141baefb30
F20101118_AACBMH saha_r_Page_167.tif
9e26a2cec3591eb46abb9ae9fc7b45ff
6ecbc7d00d0cae60da4e1dd618816110b0f38b5d
F20101118_AACBLT saha_r_Page_143.tif
49b90ffbdeb422160172b171daaf0b50
7d9548c6fba86c3d9ef31ec9e6ef10d6be4d9a32
1051972 F20101118_AACAIR saha_r_Page_094.jp2
dcee23dcdd1a4580fc15dfd1ff8a9f1d
49d63a386943dbbb38815901e550501a5fabd970
53124 F20101118_AACAJF saha_r_Page_013.pro
e8e399373b3f7ea79ccfb01e448f3684
96a7492be8ed9656892dac51399215de545a1cf7
F20101118_AACBMI saha_r_Page_169.tif
8f01b985dd74f0cdf806c875240bbad9
39f0753be04b54336b23f44569b0f40d016b274e
F20101118_AACBLU saha_r_Page_144.tif
ac1f91c7d347240f69c6094c7d88ca2c
5ddb34b03428656897b33c729602863c0e4d32d4
15632 F20101118_AACAIS saha_r_Page_074.QC.jpg
3d76ee58763cfdade8bda7e1b1ab8d29
aaf2b0841f668f5a4290a764425b88af380936da
79204 F20101118_AACAJG saha_r_Page_110.jpg
9a4fd60781a69d9dc74e4a673a9a6d8e
4fe471e7a998c68eff841f0b85c67e8c525af4bd
F20101118_AACBMJ saha_r_Page_170.tif
2863252d3b6014a529199ec211c83a52
b58e00a089399364397af182385445dca8603926
F20101118_AACBLV saha_r_Page_145.tif
80f9689b0625fd83951f3454139dc65c
53f0458d4f9aa8cea9f0e1009e4a081f339e2f95
53584 F20101118_AACAIT saha_r_Page_110.pro
54748172b751e2fa08ef61af9b8d4e6d
e1cd7df11c334d46c9b082db4a70a24d219e01e4
25030 F20101118_AACAJH saha_r_Page_150.QC.jpg
6b23ad31652c702ab6af30b25e884b16
d024806e67b9f0adc4947af634fe233048fff66a
F20101118_AACBMK saha_r_Page_172.tif
cf5f9cc6b0f9f00fd3eddf97b6afe7a8
dfb7842e77b3779f063d4dd66d9aab9d8d5277c9
F20101118_AACBLW saha_r_Page_146.tif
a9b187af319f681a15c60777ca057fbd
d1fc07c5bddb96d3ad7fce187e396c84eec7153d
741847 F20101118_AACAIU saha_r_Page_164.jp2
e1db95bbe9e2e24dc5b135c324138f6e
9d9984a4a3cc68823cf7c0019e0c41f13bb44946
1363 F20101118_AACAJI saha_r_Page_090.txt
31acb01103801f14928046aad7bf640d
4e1c453d09de268b2b7162d4dc5fa43866583b79
F20101118_AACBML saha_r_Page_173.tif
f0138ffcb11cbcffca182aa0e527c128
d684b26bacbb572f64c2a19fdcae594d31ea48f5
20587 F20101118_AACAJJ saha_r_Page_126.QC.jpg
1769e313e1dce5fa0ff286490dc539d5
75b340374519a55fdcb2179fdac73ef949de9a91
71090 F20101118_AACBNA saha_r_Page_008.pro
f07de28ed4d5f7862a8cd859264b488e
d12eec6e09a72198d32b24508d13a676983cb863
F20101118_AACBMM saha_r_Page_174.tif
685d55f7dac87256dea31112aeb459ba
6896856c32e8980c88c41b9a9bcab31a11c53ad0
F20101118_AACBLX saha_r_Page_147.tif
42463ba8c7bc8651ffd415abfdb66786
082afba90235651876f2a981950cdeacff4ebce1
1675 F20101118_AACAIV saha_r_Page_063.txt
1ed9de95aa31a1a5ceb2381b70927f2a
f89111903560f295d03d07ebe27bd4ece788842b
74767 F20101118_AACAJK saha_r_Page_139.jp2
c254ac137513840897199873f4ed9298
d83f296294072f52748e1bea0b70fd3b90db03eb
42930 F20101118_AACBNB saha_r_Page_010.pro
3cbc16af7c98b62f5499a4b235734196
0e9f3782018be7811c967b50aa2c5a59128d801e
F20101118_AACBMN saha_r_Page_176.tif
c3c70a3bd0c25f500d989537da532037
483a6d891fa24c95c3ce0ece206ffcdcef546457
F20101118_AACBLY saha_r_Page_148.tif
64f5b697c8e5fb2d05db39761c61ad7a
0ea119a386804d60136f601872162c08d09e46b8
F20101118_AACAIW saha_r_Page_151.tif
0e7d0296e224fc44833132eae4f44af8
ac11c92b8fb661b3caca24fdb50aca1bd321f7d0
5864 F20101118_AACAJL saha_r_Page_183thm.jpg
3a634649cc19644a34be6969e3d8dba2
491747eccd559679451a6a19fafada803303c29c
52543 F20101118_AACBNC saha_r_Page_014.pro
0210e1ea7064bff956badbb2a376d9eb
16eac53a9396abacf76ac7821ba2a387358cf8df
F20101118_AACBMO saha_r_Page_177.tif
379b2c93402e83bb35620cdbb0fbfd64
99a1b4b9065182294e7562dcaa2ebe2038bdbd1c
F20101118_AACBLZ saha_r_Page_154.tif
b389ff8214b3eb974f5f6287bb4c4fc3
562df818289f87d2398b59c041dc0c7652bca0fb
1944 F20101118_AACAIX saha_r_Page_068.txt
224664d9cd8ef620b224f0651591f8fd
9ac8a29f15471aa7a9add852ee5152c002435af2
F20101118_AACAKA saha_r_Page_047.tif
f8459ec002178cd538984b54510e50fb
5deb0dde6e155db26717b36ec6885d5795fc2289
41805 F20101118_AACAJM saha_r_Page_083.pro
c47abd16d44e71d500c80bc29fb54137
6443c7c7b8e86d1ab3341051e548889b30499c12
52251 F20101118_AACBND saha_r_Page_015.pro
6193de878fffce2cbae4d116128a01b2
a61252d6501612c696e5cb8429a74bea6a841f49
F20101118_AACBMP saha_r_Page_179.tif
a6084af4cd34dd1135558a2634ea04b5
08ad8ba7504d96d6052652fe45c5aed2f45581f5
1905 F20101118_AACAIY saha_r_Page_104.txt
1e76fe60c5a8745be68dc039a994cf35
3d1629c88853869ef174db20a32f13aec87cc6e6
86094 F20101118_AACAKB saha_r_Page_151.jpg
cf3af4f9158e4b4ea8ab0dfa9cc400a7
14685d15a1cc4e10496a83079b892052779b44ce
894286 F20101118_AACAJN saha_r_Page_068.jp2
e4499a2c64ed3479e23e79755c3204f1
101034eb9e4d3e353d692f4b08e434c71a77ae30
34541 F20101118_AACBNE saha_r_Page_016.pro
4e2c3ad63b15bb5093aa0fefbc0e1672
14e63a24649284a63c9c04ac7887af29f533b7f4
F20101118_AACBMQ saha_r_Page_180.tif
9ac511efe2b1f99c8db2bafc0d801594
9f1133d28451b9c84a4678913e5dae2d4e0a1d18
959967 F20101118_AACAIZ saha_r_Page_126.jp2
93324344794442a7f7be94cbbf4dd328
2fde4d54ee9414757d0f5ceb0fa9c46ee7f3ae7b
1051935 F20101118_AACAKC saha_r_Page_015.jp2
d0fac298fe7f155ef9e216c887c2e496
1c3a183722e387a6f1de1f1671881b3d0cd13604
2139 F20101118_AACAJO saha_r_Page_111.txt
9d4f1436d30667c8319eb87190874d37
e8e33a4f15247e01b56e6f41de951cf6f96574a8
46142 F20101118_AACBNF saha_r_Page_019.pro
74122215686544c0a605d0e9d364da49
2281c3b698c75facc44ad9ab70978b5350a398aa
F20101118_AACBMR saha_r_Page_181.tif
b4b48edc4984ff4d6fcfdc3063527d55
7d08d00de5e50ab55d334b1480444366a41efe99
1402 F20101118_AACAKD saha_r_Page_101.txt
d79cbe7bbc0665297086ddaaf2bb8e1d
337b6ef9a615a985906b023401124823c44bab1c
4885 F20101118_AACAJP saha_r_Page_173thm.jpg
071f3d44f218820025a09a53881914bf
7bf4fe0f5053e24b9f8cd123328cb4ddbc1f5059
48773 F20101118_AACBNG saha_r_Page_020.pro
64d34b845d9156f657b7fcab20ff816e
01b48f0eead2e9511b1dc32146e1289799ca8929
F20101118_AACBMS saha_r_Page_182.tif
dc4e71542f61c82d74cf50f0b3c989ce
aefd2aa513dcfe9d0034fefed0a27e854ccce7c8
F20101118_AACAKE saha_r_Page_019.tif
9d9157c0aead43aac8af65d0a77c829e
2c9e84b47a4368c108e0d0479faeb256a1bace92
793847 F20101118_AACAJQ saha_r_Page_103.jp2
69ba6d88b939cce2dff6e3a8011feafe
3e6050828db875e5f1880dd3ce96ce65864b5cb1
55273 F20101118_AACBNH saha_r_Page_022.pro
df8c67991e6e46756816c0b91a3d5359
8006c5b956198091a18a8cfa86ced0cfeed8efaa
F20101118_AACBMT saha_r_Page_183.tif
ae9f17dd88d7acce0594091d298dbddb
1c72ee4e6191d056d74fe1726c01a6bc58c1ab78
1970 F20101118_AACAKF saha_r_Page_046.txt
817ce2b325b8e9e2a642ab85fd6cd367
d04c66f10ba3e156caaa42e35bdcf43b778ddb3e
1833 F20101118_AACAJR saha_r_Page_182.txt
1f083dd49d801e907de4c3334c8a364d
535f5b739b3ac69143567f4d4d9c696e785a0ac6
32185 F20101118_AACBNI saha_r_Page_023.pro
545775f1b4c8234ff3c81494eff389d4
46e1da5099a2a7ba623147b4dff55f437566db54
F20101118_AACBMU saha_r_Page_184.tif
5f4593f906f5469406cf6e0784207d64
d6a06418a92548792b227095ba15f5a85b808822
40257 F20101118_AACAKG saha_r_Page_078.pro
9c4f853f65bd20a5aac79ab952cd7447
862979842d941d76a76c337d404952c92538b358
F20101118_AACAJS saha_r_Page_055.tif
fd4ff9a88f437d0670221452db145012
311951e99ff19830b04c5d14a6e6755710fc5204
46413 F20101118_AACBNJ saha_r_Page_024.pro
2d32eb60a26e171957af07784981b8b3
83754102900d7d9e0d3a81ae5aad39bcec701615
8514 F20101118_AACBMV saha_r_Page_001.pro
b607b0016677e98a8c94a1c4d686ac07
11654a463d375b1f5532fed92d2195f6d3a3fc82
2541 F20101118_AACAKH saha_r_Page_011thm.jpg
8e6727534b449e12348f82ed186eca13
7c7af664ef179b1be820166bfc1542f1d4d87da6
56763 F20101118_AACAJT saha_r_Page_042.jpg
8f5286e4082a81a80079c091ebdbe78e
5d762864dd7295467112db6f2721710052a2e0b7
41333 F20101118_AACBNK saha_r_Page_025.pro
88bd63051b77051b4c7ff9f79d43fdea
3cfc9009f61b1d76d31175dbddfe2c8327f9ca79
1095 F20101118_AACBMW saha_r_Page_002.pro
70baafbf60652e4f4bf233d28f4ea1cc
fcff9fcecabc73235b041a89f71cd3e973848829
F20101118_AACAKI saha_r_Page_053.tif
3b0c883628d7d003b9079eaf845d3c81
6ee75ecc1aaa45fab0d5706437ec952001d91401
F20101118_AACAJU saha_r_Page_059.tif
a59b168566e77356479ac4a6df55fe8c
5e7e7d531b1bc9061bfb9e54c6e6844e213314a9
39364 F20101118_AACBNL saha_r_Page_027.pro
7a3d8754b83f9f05f6a1275b3a50e227
62a19a74aac80c6edff78b63210bc34b6dde1a5e
53714 F20101118_AACBMX saha_r_Page_005.pro
22e74e9db646e37c81bfc154b497a235
2441e2a2aa8ddb752f87540ca46e799a53c36bf7
1585 F20101118_AACAKJ saha_r_Page_160.txt
e69f1cec423cd341075546ec8d0d32b9
38a9d20e52acce342738ddc8da30c4cf2b82be90
6657 F20101118_AACAJV saha_r_Page_108thm.jpg
52fde2e223f563c86af984901b5b7b15
c1d48f39ea507ec92a8628a9c011098f83c5b8ed
31581 F20101118_AACBOA saha_r_Page_049.pro
e172194ad22ecec0bfab2e941eb44ae8
85649c128347cb95938d31718f3be53e8b157a08
50327 F20101118_AACBNM saha_r_Page_028.pro
38efbcf7fd42ad5f8121b9e47d93d44a
17c5bd30bf443f4ef34ddf3883468ab5ab61be2a
F20101118_AACAKK saha_r_Page_117.tif
5308a4ba25bddf7d59e3e4b15088c4eb
cefd524dda14a952955dbcddb0ce37954e572535
56492 F20101118_AACBOB saha_r_Page_051.pro
575fabffb684564bc51b1361faf12bff
a1a19d9a1ba379a9b3781b8064e47a4f40bc39aa
56226 F20101118_AACBNN saha_r_Page_029.pro
7d3924c81a863192fe3aa1ae5a2bfb03
2795f04101d387c3bf3add50c82bd57c290c7ad3
51586 F20101118_AACBMY saha_r_Page_006.pro
8fbd8ba24afb8344cb6d0036ae4284ea
426564955db5716a968d0a4d93504df72900a124
F20101118_AACAKL saha_r_Page_127.tif
fb624708d1aa39a466bcecf457e25c44
74892279a3b6a9ce672f242c7628d7bce88e5846
81186 F20101118_AACAJW saha_r_Page_055.jpg
d83d7987de03c323acd153e6d7409072
b2ce6a4549b9e22b9916ded13c61d5383d4611ca
51232 F20101118_AACBOC saha_r_Page_052.pro
56902021e76e9c258db546136a5dbeee
45904bfab72eaf51bb07996f29a2b4d6850741d9
50764 F20101118_AACBNO saha_r_Page_030.pro
8fb6eaf8266a2b44c8ef0652fa7b6a4b
b31e5dccb4084aaac2f712a379d397ce81134aae
50407 F20101118_AACBMZ saha_r_Page_007.pro
e39c6d859fefa3fe1d733c17563d9bf6
9be4df7c051ee72e4c5331a76bf14f6aef7c7ee0
73327 F20101118_AACALA saha_r_Page_147.jpg
bb79a261a9fa79bfd7416bdcdfa1f61c
53e96e177f2fba1ababf5a84565004dc0e01fc3e
F20101118_AACAKM saha_r_Page_137.tif
f93a7773b948264f67240d25f0292b0e
2986df5f4dce549d400b5c75e021dd9624b010c7
6451 F20101118_AACAJX saha_r_Page_151thm.jpg
1a1b4aff190b4f2dc2dc7e242fb4d910
99c417d4c1ac396ccdbc2e70f61857feb2e3895a
7566 F20101118_AACBOD saha_r_Page_053.pro
cb6e6a3a31359518e4e2125983499c7e
46ed66474516b7641dcbae75b8a63b2c0cccd80d
50324 F20101118_AACBNP saha_r_Page_032.pro
ceec609a8bcfe9309af36c7043ec2c4b
c53e4f9e1c706734c8dc532cd9ce77f9990ab0c0
55367 F20101118_AACALB saha_r_Page_055.pro
695ea6c2bae304d33b07719a9df7c1f0
b12716b8e012fc3225be7d2d65f0330a02f0fc0f
19146 F20101118_AACAKN saha_r_Page_144.QC.jpg
2554989cb726e407b9722da19722d779
ad561e9b912c36c8e2bc0bc276ecebe9ae8f84d6
46139 F20101118_AACAJY saha_r_Page_069.pro
7af026ac3d461bcdf8f0c4e8624dcb94
e1958c99ad5eedabe7add89350023a3a3d0f309d
36003 F20101118_AACBOE saha_r_Page_058.pro
14b09cf591a4c578744f007a5494a1fe
239c7b33eaaeca18a3cee09e23ca4eaba1f3fc0e
35299 F20101118_AACBNQ saha_r_Page_034.pro
367183cfa8e53ba4b9cccda7ed766b61
391dbb6371abfecaa9ee9dd6a89083e2414bc11a
24363 F20101118_AACALC saha_r_Page_014.QC.jpg
8396e4557c80f77510de30d422ae25dc
c1ca94dfc16b3de41348319dfe7e1fd04a3ee2fa
5327 F20101118_AACAKO saha_r_Page_031thm.jpg
00e8e7a9b4d7ddc766802de970b92dad
7a7f63de67032ad7987c8e1f3e3fb3af9d77a01a
40833 F20101118_AACAJZ saha_r_Page_009.pro
20cb253f56ffc8fcd14a40ffeae81ba6
9109b1467eff0d83c0db57217fbdf16c2450e52f
34513 F20101118_AACBOF saha_r_Page_059.pro
778d4c45191a2f2b47d6fa598369f44f
b35a261feca86c42c09cc8fbd4ff334013952b2e
42028 F20101118_AACBNR saha_r_Page_035.pro
4723b37742ed645e45ba06b2e4c5f50b
03840d08b1014c1392e7fa2ccecc7b4d93b29d7a
51488 F20101118_AACALD saha_r_Page_140.jpg
50e2463579ed89c3bbfb15cd06ac7c85
ac0aabf5694ac22a6ee4a1d6dee93d7ebb374ff3
41382 F20101118_AACAKP saha_r_Page_105.pro
8b7452f356c46c48923e503dbb7cc3e2
40f0d0ba6ed397218eabecbef32468873835a461
46313 F20101118_AACBOG saha_r_Page_061.pro
f9baa35100f61c11ba7c782f7d26940a
80564a8299f61b1fbbcb4c79cdd80a4cc5868427
35095 F20101118_AACBNS saha_r_Page_037.pro
de66aed1e167d3a4deb769f6807126b4
156d9b26839be4d9be3ca56ab42dc7b2a5c12956
5383 F20101118_AACALE saha_r_Page_078thm.jpg
9f29c0ec8d9c4dc84eeccc6de87094d5
97cd6ce3a539070108b28d346ceb027de7d2a1fa
6014 F20101118_AACAKQ saha_r_Page_127thm.jpg
69491fb52f55482ef86aa4ff34901a10
ad993dd53ee8f1a759edf218ec5f816d5a26bc64
36155 F20101118_AACBOH saha_r_Page_062.pro
548388dbaa5fe9c34548d4e4a0156b5a
d4f72eb26eca2ba668fd9279ccfaf842fe6f2d5b
47670 F20101118_AACBNT saha_r_Page_039.pro
faf593b2b768f7439f754a3e3506b8e7
366c2f579f49aed6931b29d905f78d20f7123e8e
1736 F20101118_AACALF saha_r_Page_087.txt
5b995238ff07676f5471ab77bfe3e40f
4e9fd99e71383104c7818281391865ae74bc18dc
23321 F20101118_AACAKR saha_r_Page_020.QC.jpg
2663af27c9f0dcf6b0137d578546e1c9
955731819aa54d6ee41d22793a9df0cce4fdb9c6
42460 F20101118_AACBOI saha_r_Page_066.pro
e98800595ca953d6d87cd452b88e02de
f55fd02b06c72479a66381053407f1245aeacf31
48471 F20101118_AACBNU saha_r_Page_040.pro
86d781034d96ed7e0a4646290f92936b
2cf01fef2420f0c33633535ce6f1d931529655bd
F20101118_AACALG saha_r_Page_104.tif
ec6e4c250012fbaf048c9a0e273a6976
d8cc639a615c476d2380cfb8acaf57a5e4562247
19100 F20101118_AACAKS saha_r_Page_053.jp2
d850922be5135fc42f13e70f4af8c723
e48c86245e658744db14dff8c6bb827082d226a5
41452 F20101118_AACBOJ saha_r_Page_068.pro
a9d6013086a3fdc67ed5da6f0699c91c
2e320a0fb1cf33d8fd5b813fb8e4cd68ca9e79ee
55848 F20101118_AACBNV saha_r_Page_041.pro
9eb73ea847a2fc360ffe12342c818aca
f2d4b9f32a2fabb08be28b95c068c150d718a49b
F20101118_AACALH saha_r_Page_016.tif
6f33233360836a09bcf959dafa6538ce
d50a490b21c04322a69573a9f21310d984849f3c
35335 F20101118_AACAKT saha_r_Page_103.pro
669cfae1bb0076bd1e023d07d92b7e88
344f361a8740bb8c3b27a785bde2ba761e99f083
42828 F20101118_AACBOK saha_r_Page_071.pro
ec6a986792bb1e6bd812ed62dec8ffb1
6abb6e91d27cf7db167baaaba75dd066552bebe8
45423 F20101118_AACBNW saha_r_Page_043.pro
9560e233cb4b80656ac962d119d2e383
f1e288a87212d0b4fdbabba21c522dbdb95de709
50021 F20101118_AACALI saha_r_Page_094.pro
472e3388d473dc1f98aa44bb9f6c13ab
2314f34a378a52ad9c47921ab50484e7dbff28b4
F20101118_AACAKU saha_r_Page_150.tif
c486982deb1cfd470ac42b544354fa88
2fe22b5aa062abdd11285f753c378b1cfd0a5a86
40133 F20101118_AACBOL saha_r_Page_072.pro
fa5c59d75543e838fc3c5e4c690c9429
0aca570b59d4d4c179e92e013d25c8754ed744eb
48929 F20101118_AACBNX saha_r_Page_046.pro
612182208fa1a2fb5eed07899f3b14fb
10151e054edf51737422d60a97678eea559d37d1
73399 F20101118_AACALJ saha_r_Page_017.jpg
61a4e316a807cef25d97711241d6e620
d23f419137262dfef3b1feb52efc35d69521d8a8
43036 F20101118_AACAKV saha_r_Page_070.pro
d48510fe1837885b214b0248d36c6b7b
4e6223de452b1b2c35fd33115548a467c2a42f8c
40288 F20101118_AACBPA saha_r_Page_099.pro
4cf25e30465fa7ca6b0b5f5a1b3eceb3
bd836394c5484d7fce8e1ae0ecca216a449b3ea3
F20101118_AACBOM saha_r_Page_075.pro
2d32b03b36110cc47a99dc717c1ea1c7
47a7cc68120f4622af93d9d8b03499b39199493c
53023 F20101118_AACBNY saha_r_Page_047.pro
9586c339a47055398db3f63e7fc00efa
533bc69d0391f5f364245d610d2abb67798f93e6
42734 F20101118_AACALK saha_r_Page_033.pro
273cf2d7b0e33fa63ac5fb6140ecbf48
53743d570a113cde8b80979c3ad9d062f0c7b7bc
F20101118_AACAKW saha_r_Page_135.tif
583684c3bbfb9bdf822cbb4b3da1afd2
a83133b02a59b201d034ebbab51449ac0769d6c1
35194 F20101118_AACBPB saha_r_Page_100.pro
059b7c9e9ad522290a41291aeffa958c
ea94f255ce31fa73bac48af90182fad2369eaaa3
35926 F20101118_AACBON saha_r_Page_076.pro
7f4fa50e8acfc4571af021f6399a6711
58a1b72cc8d650150f95c749ae619da2aef54e22
70393 F20101118_AACALL saha_r_Page_075.jpg
d54e02aa4d15506991347754ae3b31cc
3d2bfd8ad6e091a0c1bcd22a4df88e31f0b592a9
49775 F20101118_AACBPC saha_r_Page_102.pro
89a83ee2c57b3cdd67b6b9a7e5346091
362af8617e0147a14ca918e5eb872dee8673c35f
51103 F20101118_AACBOO saha_r_Page_077.pro
09cfe192f0ee877f249310c45958eb55
b353c2d8a572bd9414e1e4bbb59735fa4686c1b0
27376 F20101118_AACBNZ saha_r_Page_048.pro
62ede39dfa7abaa1e764b5f638d597af
b591970e3ea958b426f919e455f5ace95fe8ea36
2144 F20101118_AACALM saha_r_Page_021.txt
8141d5a3253d77cf3e1a42a39b8900e1
2c2802aa7086fdde819d3720e2d2ea991f398e59
69527 F20101118_AACAKX saha_r_Page_127.jpg
b91a7b713d3123f5bf5a9b62a975e4ad
f54062f6c9b7624d53b37340c3a2ace01c5f4e2d
5556 F20101118_AACAMA saha_r_Page_002.jp2
df087a47c332d57115803eb63dd20bb6
61599edd613975f8bec957cec5397b3e9c72bb61
47760 F20101118_AACBPD saha_r_Page_106.pro
4a0830f16f765f2b4dbedeccb75650e6
5a2c004da5b88c1976b4443d1f0ea4fc635852e3
41136 F20101118_AACBOP saha_r_Page_079.pro
47194ab096cc49ea75acedd9dcd6e772
092796c0e45a3c4f2af13cacfebd83b2b0f6fde7
67531 F20101118_AACALN saha_r_Page_099.jpg
68513d7f7978ab30a3589d769b77d001
8515e1a1be7076a556b0adb74bba082e518bb8af
84732 F20101118_AACAKY saha_r_Page_045.jpg
9b6cdb60d4b02d15ff66f76e1b496845
2ce29610baf74af5ebc1778838c320dc90c29862
97576 F20101118_AACAMB saha_r_Page_031.jp2
5538ad517820888daf56adb386cabb29
4c84663f8d8f4aaa0f40db2c05011179ebde81fa
55378 F20101118_AACBPE saha_r_Page_107.pro
d20c69a3c57786f3cb189aaec6d1d917
170043be6158c9aa865f5a777d0b1cc34afc61bf
37329 F20101118_AACBOQ saha_r_Page_080.pro
1232d1b052d0c62a1f52c079a9471970
34dad63a068ba65e89d6972d5011b917cf0f9557
61060 F20101118_AACALO saha_r_Page_010.jpg
36eee2c06f05e74680142ef4cc5fa128
9537218a3ebb9ad70d053c3883bc237ae82198c1
33099 F20101118_AACAKZ saha_r_Page_167.pro
1502ba690ee310a8c812666dfaf89460
d4d1ab783b7c9caa47cbcfb1ea2ac3c0542b38c6
859360 F20101118_AACAMC saha_r_Page_146.jp2
15b04edd5dc31ccb173721e84966395e
6e45aa7228750da285aa3d9524419f62a487227c
58450 F20101118_AACBPF saha_r_Page_108.pro
721a26a18b1f11e23ec7e341f1d4c9c9
4fc7797be5e71104d722bec31239cb7f8a7e93c2
35051 F20101118_AACBOR saha_r_Page_081.pro
bf9b87d5b30b45cca63e99b32584f53e
0bfad7ee489cfac9614c8cc0de5677c591a5d8c7
5705 F20101118_AACALP saha_r_Page_070thm.jpg
10447ac3a2e243f1ae11a7a4fc99bf8f
9de3feb4d8de0422bdc7ec6cdd8e0e35b326ea1a
5795 F20101118_AACAMD saha_r_Page_075thm.jpg
3f9d2fb6ef0f1d13812add8de780a4d9
f8872c914bd14854cbe449200c6c284cb383caa8
44382 F20101118_AACBPG saha_r_Page_112.pro
c79f91da7191f9169682043488422d4f
76f45a0cb7a792e6923e432682ea66ff58c25cb8
46847 F20101118_AACBOS saha_r_Page_084.pro
dde8c22db5da9dde9221640e8598a76f
4e93577d87024feb3b6047cc170e4561cecb6306
1499 F20101118_AACALQ saha_r_Page_100.txt
bda80039f6caf083afa9c70fe40c8364
9a3887c461b82f2b212a1966d7f25901fe07a925
65268 F20101118_AACAME saha_r_Page_168.jpg
cfa5e6809dc4ecbf4406dca18c9c40f3
1e0e3003af8e5d5988a9863501c782e34a027b47
47628 F20101118_AACBPH saha_r_Page_113.pro
a67d68f62dca42e9be5f124922705935
88fee81e2d805ea3e3c7a010435d3f09d9d8c0c8
36126 F20101118_AACBOT saha_r_Page_087.pro
fc7800ea6216d5252f8945550c3c31df
b346e7f1bfec0bbc0121ecee8213a5b54c5319bd
17010 F20101118_AACALR saha_r_Page_129.QC.jpg
18fd9b4cdd1777381e3d0ab8438cb1f7
35a8bc73e51f3ca2e8ac50322298db132870b36c
F20101118_AACAMF saha_r_Page_028.tif
600231128d59fb6edb0b58093dc73644
7f31297218ac33eae845d6c9d2b8899e47f5b692
50900 F20101118_AACBPI saha_r_Page_117.pro
3eee97fca91453526727cd880af6f46e
fe7d9a6d9f5ce0281c43f5ca2c647ec605141a28
39224 F20101118_AACBOU saha_r_Page_088.pro
8b5ff6ee316c315ca356990105d6d41b
61fb563d94d8220d4bd205f1223de445dd32b643
102606 F20101118_AACAMG saha_r_Page_008.jpg
5d04746ced5bf6c3c0e3e67a585ba635
78f7778e9aca40b770bef34ecb692fc2c823f335
54331 F20101118_AACALS saha_r_Page_111.pro
c25b5ebe1c3e4d82d8b05d8390c72704
363b6eb88a1bbae4f2a89ac69679df3c3ed86105
39774 F20101118_AACBPJ saha_r_Page_118.pro
0be0aed1fb985c8b44a45656cc12268c
6ad9d738871e8f0fb17091f74408160e2c30bdff
41940 F20101118_AACBOV saha_r_Page_089.pro
29f19383e3b4b33366042f6f69340766
93051784e06b8a7a791efa673268733e88cb4952
44298 F20101118_AACAMH saha_r_Page_065.pro
72bd260c5b5bbbf5fc490cc37acba5ef
889fac255155fce39f863f93350179ef4a73891b
5345 F20101118_AACALT saha_r_Page_033thm.jpg
6dd3b88913790dfd5be79332125ffe7f
813a9a2a6ced224829a619add46cf3164e75ceaf
41812 F20101118_AACBPK saha_r_Page_119.pro
5aa64c2ba154113631f44dfed0b6bf9e
a6d97ca794aef6d5872f447c9aff683f704377ab
28204 F20101118_AACBOW saha_r_Page_090.pro
fedee92e5582dd93f496292ac114fd1f
c821b27dabedd1c0022ccbffec92e0f5c915aa0b
F20101118_AACAMI saha_r_Page_088.tif
f430ddeba6a283621fdab82084ff5b49
d1945c9603fb2ee1e443be7dce51814153e6446d
883211 F20101118_AACALU saha_r_Page_158.jp2
a1d0bc8342a1f8944fb8790edb0b5806
8ba96cf52d85bc6739395621c12952b3cde3c145
44122 F20101118_AACBPL saha_r_Page_121.pro
8b93f8a6a3a366f68268651c07f16230
f73febb5dcf9efb2cd155d1dfb44d718372d799d
36093 F20101118_AACBOX saha_r_Page_091.pro
9b2d1db3b07f90cf0b8cd94691c94ae4
2329f6011797f8f200f0a790644058f67c43713f
65425 F20101118_AACAMJ saha_r_Page_145.jpg
5dd94c9450d7d5dfa21f933ca8bc7cb4
2ea1691515b984391b8db65436aa9dabe9fc852e
21420 F20101118_AACALV saha_r_Page_153.QC.jpg
6fce6d361ca22368df4e9853d185916b
9b102f803bb9d31ed9f3c8bf7b45553798b2cb12
13102 F20101118_AACBQA saha_r_Page_143.pro
e148d6fb5f9e0b33af7700c465f43222
4d471f2020796f61da40b4a81ce22e0a040db23b
40813 F20101118_AACBPM saha_r_Page_122.pro
7a5e9e035caab9502416b8b36b8d8dc4
fabd44b9062ad5812664dfbc3e5008937ba01b7a
44624 F20101118_AACBOY saha_r_Page_092.pro
228b8cf83bd398683b329edbe64e440e
2301493b79ad7a5c0922991e4201e5577f54b1c2
28616 F20101118_AACAMK saha_r_Page_093.pro
75628e1acd846540952dd03f53d06815
30ffbd3637e9156f67f96fd568c3cbe4b5bde288
61522 F20101118_AACALW saha_r_Page_080.jpg
b5d765b2c7bd7223e174d7b21cc82b1c
7b90c290ef3ab3007ea1b672a0b5afe69ee7bbaf
39443 F20101118_AACBQB saha_r_Page_144.pro
f66be6a2a5a43c5778b66d6505ae063f
2d905906a375e3f25d53a5ec9a20ce73f1a46a2f
48806 F20101118_AACBPN saha_r_Page_124.pro
580deadbace39af552ea89011059fe55
4db5243bff08087c6c163e7fcc6169bc0df06f90
52614 F20101118_AACBOZ saha_r_Page_096.pro
6517ca597ce14752825f3f72581d1e31
2a349a73ad8575bd1830c2588b99a0a2a11e4e88
2175 F20101118_AACAML saha_r_Page_013.txt
7c84a10f4fd0c25efcbfa017ebbf1461
cd21ecd5a3ff16e223d4e99b5d8d795b81751e3c
54936 F20101118_AACALX saha_r_Page_045.pro
4e964c765c390bbb98498efded1821d3
883f4b840f382322ea548dd8b57be849c39a8581
40515 F20101118_AACBQC saha_r_Page_145.pro
65d17c0af006d825faf99d317fc6535a
c4c2f12fda763e6160f3f0581afb157afd9f8a2b
42987 F20101118_AACBPO saha_r_Page_126.pro
8ee12f39716a7ee2adbc2ec3c9c080dd
72082bf320a4484e8a8b4a0a0fd7317e9ddb1ea1
13576 F20101118_AACANA saha_r_Page_184.jp2
42f74cace7e1dd895fc917dd8a486864
9eb773dcff5e6d02b91bbb78becc05f9246ab977
768940 F20101118_AACAMM saha_r_Page_132.jp2
23a91e07677b9a51580bd4cd497ed1b5
23a6b61afc490a3979ad5721ca1fb6cfaefca3aa
39243 F20101118_AACBQD saha_r_Page_146.pro
f3cb964029e49f8995761785981f90df
94639e2d58085e9ef40a3f19a743a12dca05dd4a
45532 F20101118_AACBPP saha_r_Page_127.pro
d98f899fc10d3195b76fbf94935ccbc4
bbe52ea128b47d4a144a91eb5451008d10be23b7
28024 F20101118_AACANB saha_r_Page_073.pro
106984921a6007bb0d1619dc8083621c
18b2fb117207dadb940804297d15546332e8fde7
918520 F20101118_AACAMN saha_r_Page_097.jp2
85d3d3af9e9a6735b6bf316bae9e9da1
7f14d00b94498b8250fc3b8c578579c90f80d9f6
F20101118_AACALY saha_r_Page_011.tif
8097d1fdb724264fdc048c26573bde09
dbcda0ac68e189ec98a9efb9c986f4b8048d8055
39986 F20101118_AACBQE saha_r_Page_148.pro
95774007bb6c9aabb620a236c8287ee8
d3ec689963cc95a462488cf81a529cfb15e034d0
49038 F20101118_AACBPQ saha_r_Page_128.pro
b66b01629323ec3d901c922e7b04e171
2b5a4bd98465da786bacc47cd43505e62742c1aa
64871 F20101118_AACANC saha_r_Page_116.jpg
2c118ed0a5d49795cfdd8bad22bd5bc2
f9fd001177a94c3886c6d4f46e189f3d551178a6
5589 F20101118_AACAMO saha_r_Page_092thm.jpg
7c58b38c674a54163d843665028f3ae3
dd477e36e6f0f382c9527e04e25c58dd409da3ce
6649 F20101118_AACALZ saha_r_Page_021thm.jpg
8a7b8227df43c51e402a7e047471fec3
90f52b201d0bfd790f02734906e69d6623daee22
48543 F20101118_AACBQF saha_r_Page_149.pro
84d5fcd391ca5f26f6d7fe66a7ea1bae
77fd84080bb947bdc7e6ebb17c82c0a3d8c1e071
33886 F20101118_AACBPR saha_r_Page_129.pro
ead41fd1b2d183b71f183bae358bf8c8
d81592db23c0906033abae956084257854ebaffc
54506 F20101118_AACAND saha_r_Page_095.pro
8efdd3eb491b44d9261e513d04f47ef6
02e1841392114b0761b4b01d422dda625bc28b73
4653253 F20101118_AACAMP saha_r.pdf
eb7bd08a4ccc3499a0865ffc7e74fa44
2e6b6fee07f0958cdc72e5c59ad9fc4c38540f43
53388 F20101118_AACBQG saha_r_Page_150.pro
d335cc57759c9f76470a44c147e258e4
0fb138f2f9ec762f2f638db917123bbdb25b96e1
33571 F20101118_AACBPS saha_r_Page_130.pro
ee30eae4d2b0f0963aca350d1a8bad6d
cbb6f433b31c6a40a19fef710658f5047ba99e89
8515 F20101118_AACANE saha_r_Page_011.QC.jpg
7cbfd90cc5fd2275a7399ed71e23a675
8c679691cd3be2a3193203a9c42847a7b163c7d5
F20101118_AACAMQ saha_r_Page_010.tif
04d8a32942a84d1425bb237d1bda6ee6
7c5645bc1dea73df7fc3b0a21fe1d0703629eff9
50173 F20101118_AACBQH saha_r_Page_152.pro
e30e716dd724f8a818f15e15baf8cb36
bb1575753c2692046731c4359bf6e09e677781ab
33724 F20101118_AACBPT saha_r_Page_132.pro
80c91a421f84027394dde5812f57b48a
e9d2d7cbf05bde580fcad2e3e0393ca5bdba9850
39813 F20101118_AACANF saha_r_Page_120.pro
21e08bfbbc92cde8acb6269be9e61e01
93b6c6c4cecf4b36a257708356ce4577f6b54c69
14417 F20101118_AACAMR saha_r_Page_063.QC.jpg
9ceae43e9ecd365f0efd2bc1f4f9d4cd
4dd3e7a615b447ff817a4a63c50a4a00fc822c39
38443 F20101118_AACBQI saha_r_Page_156.pro
e575eb21a4a39ede3b59ceac3d8965a6
164d91fce424a013086f16effda31fe4ee3410a6
49650 F20101118_AACBPU saha_r_Page_135.pro
55d88842111f3f67d1a06dfdc99767b0
c1e48ea1b365f0094724a8a199ee36a5a94ee8e4
20259 F20101118_AACANG saha_r_Page_119.QC.jpg
9c8d6cc0b8f4eb7e0a181e5398a76cd7
a8c49a74c2d677b8d110d91a1b043a1c637b7705
1784 F20101118_AACAMS saha_r_Page_057.txt
14e5547d6f78aea1f12d97c19db28426
f383e924151b8c7a2bf11a4ab70fb62690a47083
41803 F20101118_AACBQJ saha_r_Page_158.pro
2a8c876d13853f363cf9983d1e0ad02b
da50b36c442c251b02f7c7157fee031d5dcdc5c1
26624 F20101118_AACBPV saha_r_Page_136.pro
8e8bf9edc8801d846aa7df1d1c0e92e9
4f13a86b62e49a217696c85095559bec732c7307
4419 F20101118_AACANH saha_r_Page_073thm.jpg
95c0186f4152733c6a01b2a7a36c1e15
c2c7bcaba09e34ce1e61fc53242db72174cca8cc
26430 F20101118_AACAMT saha_r_Page_101.pro
26695c294fed374c8539af981bcf9958
e71e7f61086bdb1b94dfea998c269d3e2bd27016
34016 F20101118_AACBQK saha_r_Page_159.pro
89d4eb872517c3d47cf091e150a1e41c
43b82779db502f5443816b2026a80f9daa9175c0
34447 F20101118_AACBPW saha_r_Page_139.pro
12689f578227789d98deb3b4f1ec1c1d
7d6255a92ba80009676c774145748515ddff19d4
41168 F20101118_AACANI saha_r_Page_116.pro
12eed168d3713c690280a132d2284813
590508681b7f67b2d0b40fb292a840da73b7fd9a
1902 F20101118_AACAMU saha_r_Page_181.txt
cc5b697046d1216bc91a4fafd8096c03
695fd71ccd4828cad4ebf557ed9e232da6615010
29914 F20101118_AACBQL saha_r_Page_160.pro
a82c40bc3e537f4b89add86cdd3904cd
0c7c24397c30b5b764fb87cbf650f301974ed91e
32153 F20101118_AACBPX saha_r_Page_140.pro
67036e64d108f428b049f33b17b203e5
0b95dd0224866679d0eb58536dedbc0f88a21880
31122 F20101118_AACANJ saha_r_Page_074.pro
9599c441f40e10b50d4737c55a077570
c9ce1bf2f1e530c650431caccb51e4f637dcf960
74075 F20101118_AACAMV saha_r_Page_084.jpg
b69fd6ead0f1591d2a93a6fe3d0c0b82
fc8c170d9b2951e1bfd71ef879f188ce9a7812b7
41418 F20101118_AACBRA saha_r_Page_179.pro
2264fbe27e75a1f2641321a1bea96c47
4fd9118d61a6d11a27dd567ef260177f008e85a9
26279 F20101118_AACBQM saha_r_Page_161.pro
219cabe436e636ceae6717015aabeed3
3d30c435f9b43fc0d10a8ac49e89ab0b1b57eab2
50430 F20101118_AACBPY saha_r_Page_141.pro
fe18ae182009559a0ed303f36aa79d60
8fb85c7ed019c123d91ba8300ba7c0af083f07c9
25347 F20101118_AACANK saha_r_Page_001.jp2
428b7a9fe5cdcb3cca8dc5f9df36fabd
c10a5c303d6625087c7987d5d0f824e899b7ab90
43903 F20101118_AACAMW saha_r_Page_093.jpg
a4545178c7f082cd222697374344a6b1
48e88d05ea0d30064fbd4cdcee66785153b07684
50836 F20101118_AACBRB saha_r_Page_180.pro
9754a0d8a19674d909ccb8515e8a3bef
cdb36fac1ce810435bb7a48a7ad8f666baf69327
35592 F20101118_AACBQN saha_r_Page_162.pro
2d1ca48e0c6657a7d8e91ee7d17f1dbb
8624162edab80ba3b14976badc8370f7e9ce36e1
50681 F20101118_AACBPZ saha_r_Page_142.pro
b617bdd2e7be4c51f10dc124c04b37d1
69d361b719a26690111df224414aaa07d15b2354
19553 F20101118_AACANL saha_r_Page_146.QC.jpg
14bfcae3204559e46e3cd67e4cc0f639
b1f62236a353730943362390c32010588ee741d0
4193 F20101118_AACAMX saha_r_Page_004thm.jpg
20fcc8e6006f39bcb7ee36d8b6498027
521c3e9c5ed427bbe512225abeab68e35d8bc732
47785 F20101118_AACBRC saha_r_Page_181.pro
bf3e83ad4d3f9a2f80eb5ced395f6a5e
6720fee0f3e14ebe8a99a2f024cc65882544930e
29555 F20101118_AACBQO saha_r_Page_163.pro
d08bac80d336ebbb81cbe3104f95c70e
f8e88c946d3591ea89a93a8ad337295b2cf96b23
23123 F20101118_AACANM saha_r_Page_032.QC.jpg
cf73d6ead879b8ae58a9e7984c1f4be6
51d4e90d57bb23c2a02a090ea1878a01dba9205c
F20101118_AACAMY saha_r_Page_134.tif
e621a5e931bab4557288419053d88b7a
a363cfead559af8b01db8df9a8bf65d139bb0789
1926 F20101118_AACAOA saha_r_Page_099.txt
4506c5b639f8d1075bd58e1dad112163
bfb0145a71b0957679b5ef1c1b564ee284b0f2ff
5183 F20101118_AACBRD saha_r_Page_184.pro
90dc8cdf4058ab45d70592dc91b135ad
c31dc4b85b6cfe1c7b0b770942a9ca318c07300f
33956 F20101118_AACBQP saha_r_Page_164.pro
dd0a0ae2b532ce7b4e2eec335d1a2d3f
bd2ae3f0f3eefb88c75a4f576a45b9617d0dbfe9
4517 F20101118_AACANN saha_r_Page_161thm.jpg
13f833ed9e4dbf1862b0aff38d72c680
0ec68776417371ec3cbdca1e99eb3927e8bdd8bf
18428 F20101118_AACAOB saha_r_Page_159.QC.jpg
14c35ad2c0bb8fb795cb68fa208ed190
8f4123820e51d9f0dccf5ea5e68db998ac7e8315
25493 F20101118_AACBRE saha_r_Page_185.pro
7aab9d9198e6dba4b55a495fc1020d1e
d85760fe27e7f98e1d783c7e82d991e62f698c2b
34662 F20101118_AACBQQ saha_r_Page_165.pro
bf2ef5885f9287dd14fec748e548013c
0e17647b18fea18fd821ecbb7f16edff91cab7d0
4643 F20101118_AACANO saha_r_Page_023thm.jpg
26a197951bcd7776891481dd5d61fcd5
ca57e713b8e1d7488ac9d94231cc3e96f35cb334
5281 F20101118_AACAMZ saha_r_Page_097thm.jpg
1006647c2fdf4d246790575b19f0230f
c7793944ef73002498778f492a5eb17b641fa3d1
2086 F20101118_AACAOC saha_r_Page_128.txt
b6a8be64b7e67436cbe779a93c4f5abc
be199ed2b0a9580c6bbca6e75199f7a69bf24989
470 F20101118_AACBRF saha_r_Page_001.txt
a1f33048299fe4eabb75f0cd5f385e35
adcc9cccbcd3e217add646b50a8199987ee6198c
34230 F20101118_AACBQR saha_r_Page_166.pro
3eb5ab5342cdedf1f00fde20fc52d899
1fba1fa7d18262ae16aeb74946fcae595e3c5e45
2085 F20101118_AACANP saha_r_Page_083.txt
ed12ff2eb1d376413fea8c45e55e3a2e
a0a1462219f13b293469bba24a98fb4aebdf2dfa
21012 F20101118_AACAOD saha_r_Page_121.QC.jpg
d2647bf9aacadc7aab7f63ded4c41e6e
cbc3b6edc993e3bf75bccdcb6f04808374e29a43
1358 F20101118_AACBRG saha_r_Page_004.txt
5ff49a28c9640c94eb5535bcd46fc9e8
5f82b6b4259724c6941c857be25ab2519455b243
43336 F20101118_AACBQS saha_r_Page_168.pro
919d9df1d3995f01f7a35db729a876c7
11e4904137f64aec8991e6c06a456ded35796ff7
46785 F20101118_AACANQ saha_r_Page_038.pro
211e7639980056334df97e609794fab1
c3833f7c8cc2ede40988b0e97a719272b792fb5e
834904 F20101118_AACAOE saha_r_Page_100.jp2
d65f5b9240489f2b64b5f1604bf53083
db913f2a503d3a9bd49c889c24e3eb6492024d07
2447 F20101118_AACBRH saha_r_Page_005.txt
797b9486c2c36a63e6a7d53404220440
ef9ebd4ac36db588a097612e00b1798b6d762447
32246 F20101118_AACBQT saha_r_Page_169.pro
bed106336c9331967f28a5367f7b91a4
6c19edd445633833374f30c32fc0e52e61578b01
19336 F20101118_AACANR saha_r_Page_033.QC.jpg
1e79f18bd38afd98e9108a723e91652c
8386c9df1558f4e040f954746529f353c612f7aa
5573 F20101118_AACAOF saha_r_Page_089thm.jpg
943e5836bca0b047cef2c14cadb321ae
40310cc688268ddbb4a45caae2146a244efbf005
2128 F20101118_AACBRI saha_r_Page_007.txt
79053c347308c2cc719219f7d756e42b
95a053b0bfeb7831896b32d060d843ebc5397a47
45116 F20101118_AACBQU saha_r_Page_170.pro
db321dd9a8bfa2e6fc177b6d7fb9ce51
b91902eff69e5b553efa72274e8b3764061ce78a
F20101118_AACANS saha_r_Page_142.tif
55f39b958101ec6ea3955dcd9412c81c
d5c0c507b259fa2e7a8acc8d8da5b46698e5a140
815910 F20101118_AACAOG saha_r_Page_157.jp2
5025fc0fa6106d151ab89f7f3c09b967
420d2937733d56d0372d96239574eabb19776975
618 F20101118_AACBRJ saha_r_Page_011.txt
08d541ce09073544e0440a354da26f3b
035ecba086935020dd327d93854647f2d381fe61
28702 F20101118_AACBQV saha_r_Page_171.pro
6144336d843f7520421e2d821c4c38b9
ea151c48d7f10d9e782f2bcd08fd4b5fb4e6da0a
916425 F20101118_AACANT saha_r_Page_114.jp2
77bc0f33cb12b2f8974f6f753343a5ee
4fdd3a22cdaeeeda46d4f1ce58ed13d899762d1f
81016 F20101118_AACAOH saha_r_Page_150.jpg
7b96db8c6259352f1ffdfa84dd791243
8a838e15e0386d7a27fe203c2e350c90cd03e65c
1817 F20101118_AACBRK saha_r_Page_012.txt
f5356dfc4ef3e11ec8395ae12af018e9
3464c5808b135fe3824c21538f175870d569fb38
33884 F20101118_AACBQW saha_r_Page_172.pro
61b4899e55d6bd04a9db3c20d27d86bd
ed9fabc934a52665ecd9eadb3353759776f3f632
2199 F20101118_AACANU saha_r_Page_041.txt
1abd3d2a09ef4fce4a810d48ed1a7776
608922f36264f65301b0045bd0e055d854b3e206
1952 F20101118_AACAOI saha_r_Page_149.txt
9d536ed4d6e11ad238d5df7253f35898
62c67691b8abe228954f8eceae71dae2ab408b41
2108 F20101118_AACBRL saha_r_Page_014.txt
9c342b55503c85d4ea2265f2d64fe3b5
6cbdffb3f3e102450948fd51821fe7c11477fa2c
29278 F20101118_AACBQX saha_r_Page_173.pro
d6fc5aa5a2ee1652fb06f2ebb4f721d0
80c83f8dc4fd9c21f1f76b860128ffb21f072aa5
F20101118_AACANV saha_r_Page_064.tif
5afcdcf7dae53c604381416602629549
a6449c568a9f1e28095536f3ac42cadfe7f14b4a
57073 F20101118_AACAOJ saha_r_Page_151.pro
de187b359394fa7f2a64865c13432831
1007b68c59aa73a81c7271f46c4bb6505d200f3e
1907 F20101118_AACBSA saha_r_Page_035.txt
365bf6200d7ca854d34e785434e58045
c9d0484b70a8a8a6d218d11eaeb0e26b158d20f5
2069 F20101118_AACBRM saha_r_Page_015.txt
158207e4096fa3f8157e7da4a6b1a75e
e7da733fe9e1b2cad1b43506407036edba353e45
16031 F20101118_AACBQY saha_r_Page_176.pro
ad517bc4a120b4680c0e6592a077e2d9
a3f077869ee0efa228fb513ebf218952519ed7be
1051983 F20101118_AACANW saha_r_Page_110.jp2
e6332289203b93435111dd3257b1973c
0cf456a1c457426fb9197f9be45e7b0e0c580a8a
6282 F20101118_AACAOK saha_r_Page_013thm.jpg
eefe02517382bdfd8108064f9147ca24
2c4f1752847e92dd97dc922a92f81533569aa29d
1640 F20101118_AACBSB saha_r_Page_036.txt
9dfda5ca20020d8dff89d55f98ecf14b
aa928183984877b3eae98006dffdaae263420d62
2047 F20101118_AACBRN saha_r_Page_017.txt
098dbffba0ba93f5ee1952ae81b5cc75
d0d68655f4c97989f3c6b2fcffde1186af673ebc
56138 F20101118_AACBQZ saha_r_Page_178.pro
071bb9f4fe617f06e49ab6ed8f05f8d7
59d97a327560107e0d7928b737cccedc8ca19fb5
752995 F20101118_AACANX saha_r_Page_076.jp2
1204325396f3ca1bd4ca165070e6a1ac
cb20a89ead364a70a51973bce87bdf3a99b8c5e3
69058 F20101118_AACAOL saha_r_Page_121.jpg
4dd7e79d6eb0ae1a7591f0dcfadb1925
3053d774e223aa59b09ec9da678cc3f82b6f5adf
1571 F20101118_AACBSC saha_r_Page_037.txt
0f867b20c3075bee73f333e1084acbad
965e0760a435ca14b64e74a86ad409d3817f742e
1205 F20101118_AACBRO saha_r_Page_018.txt
2bea47fe5aeb5ad8a3fb0394d3a1379b
3d8210521c8483238ab51f9f7e78beac5ac2b74b
44697 F20101118_AACANY saha_r_Page_104.pro
2a8f7b52c287d96b93458be017fff864
e9e59f77caf5f80835b124b83e3981697f753e28
43984 F20101118_AACAPA saha_r_Page_026.pro
2ca88c19e4088e882d92c0f60f7bcb44
61f5acf81fca06c4f0dca44970a4e38ee64635c9
F20101118_AACAOM saha_r_Page_036.tif
28071eaf6cba11ca1cd47a0425f91772
7a2d241d3d889e8004751d1abed516182d535ebb
F20101118_AACBSD saha_r_Page_038.txt
c4602e1d83f87290391c68586ae0a26a
0f7af2649f56df533e9d0966df69fa2125505fc3
2012 F20101118_AACBRP saha_r_Page_020.txt
d9c4f0842c4ba5fa1cd200e1863cea0c
d119a92efb3746c9267b25404dc3c286970252c7
49173 F20101118_AACANZ saha_r_Page_067.jpg
4b4d519fe372dcfb19e215229201fa44
36137a7fe6e39729ef58ee723ef0ac1fe51c849e
2106 F20101118_AACAPB saha_r_Page_150.txt
a73db54dcd0f997c3a857e2be4ba3158
15a5111d79dffae8d44fd42b7dcc48a61d089c83
15550 F20101118_AACAON saha_r_Page_073.QC.jpg
73717687924cb7f676fb2d80800b9ada
8d0f9b41eef0c0de421b5ba24f3dd80a5b5f0437
F20101118_AACBSE saha_r_Page_039.txt
aaeca37454a85cf847f02bcb108367f8
59065f3b29199ccac2d034e27cbdf0a53d6959a8
1403 F20101118_AACBRQ saha_r_Page_023.txt
4f1d75f25f4bf16c71a824df450c8c35
b6c3099f176a3c721fc624b372119bc7b24e2908
F20101118_AACAPC saha_r_Page_071.tif
59e39cbea871e4f3b809f9cdf8687dd0
7a361ab9a9809a6af5c01c24a3a39ceb64e77869
53370 F20101118_AACAOO saha_r_Page_167.jpg
8ddf1044dd1a55a4092ded244aa1557b
e3e28c1684b47e12b41bb3c9766099e55c4bdc72
2234 F20101118_AACBSF saha_r_Page_040.txt
3698d2940ba8410a96b5f2659b928ba0
61d503b8c070e6238cbf1839467979b762f6aaf9
2181 F20101118_AACBRR saha_r_Page_024.txt
dbdfb6e80c90b06e48d1438a08e1061f
34212e0f1658da6331ff01c275235000bc549413
F20101118_AACAPD saha_r_Page_119.tif
d7fe1f024f6110df740f11ef9d046661
2ccfcb2d485daf8b7bf1560ee0606ba05e9d4115
6889 F20101118_AACAOP saha_r_Page_001.QC.jpg
6535071311ade3c9c377e8fdc65a0cd8
552d6d41b42a9783ffdc5339841d6eef4ffb2fba
1244 F20101118_AACBSG saha_r_Page_042.txt
8377d3bc15035901f8107e700de2719f
05ed6049a115e67bb007b27f593de8eb6c17e4cd
1858 F20101118_AACBRS saha_r_Page_025.txt
a53ff83313b86ffb17a17490a57a2066
24eb9f7d374372e5fbad2e98f3b9eccaaffb8add
74244 F20101118_AACAPE saha_r_Page_149.jpg
c7012fe71c746b4e2589ee0134c1994a
9f9037ac470c0ab2cda38a63bd2767281d80a2ad
23730 F20101118_AACAOQ saha_r_Page_141.QC.jpg
0c1fa7c30e5ef669dc14d2c1b3c51fe0
3e707e3c21cda9f348340ad3cb5234032468227d
2240 F20101118_AACBSH saha_r_Page_044.txt
e0d56163b2a1ce812ac8b7be8f88b1a2
2f0bc7bf98fc42023f42ff95aa0e4a86cee09dbb
2071 F20101118_AACBRT saha_r_Page_026.txt
35f740b12c5a9fd2a6763e342b89e78f
d84ae32cc91639ba84303550ee61557ddb96d862
20713 F20101118_AACAPF saha_r_Page_182.QC.jpg
125c47173dc3a76e0c45a70008830c9f
5d468d011351151d46bac713a0cd8a166f060932
60226 F20101118_AACAOR saha_r_Page_134.jpg
49fc942efa3eb83bba415dda88942672
1bd79b40d7542eea2d5ff4929e33835a90112796
2216 F20101118_AACBSI saha_r_Page_047.txt
70442b1b117c9dcc2e4339d763dce415
781de3448de8457823e2f394649b14337769ca12
1766 F20101118_AACBRU saha_r_Page_027.txt
fee17b92d1e7146f7999f94082dcbfd4
7300b95750a5888ec6c914151646247d3a710264
30034 F20101118_AACAPG saha_r_Page_082.pro
68c71d960b288848f1d7b37503cfd9b5
05ab0abb57ebcc46c52f0ed2c5605e1b651c7be0
F20101118_AACAOS saha_r_Page_132.tif
ef9b779fd06ecdf25f36c8552eff141c
ac001624f953c680f9034555938047c079b300b0
1203 F20101118_AACBSJ saha_r_Page_048.txt
7f43a2d4a2a7cd0228dcbc6e2cb2d49d
431c0096df50457e9052e5f0008c40497812d873
2135 F20101118_AACBRV saha_r_Page_028.txt
5593a9dbb2e9990a0f291e7782f484b8
f8d28086b4442b841e4a1edc3a0f57795e020f96
F20101118_AACAPH saha_r_Page_078.tif
26d74f2681f390dee5aafc421b9ce1aa
949a48f107950673569cef9bd3e13c362f74c9b9
1479 F20101118_AACAOT saha_r_Page_093.txt
e188aecea89232aee1b0642fb67ea5c1
1eaeba1366198c08f353382ec0ceaee68771692b
1621 F20101118_AACBSK saha_r_Page_049.txt
7b614a5052041f3e26202290448af935
c4d04c49ed28d38883551862ba1bb89385dd90e1
2122 F20101118_AACBRW saha_r_Page_030.txt
d220ce8845e8dd3c293ec722ce51a954
34c3883423665345c9cbdad65953f32682fa4fbe
F20101118_AACAPI saha_r_Page_030.tif
5c872b692a8750f9ada5fb7027c8e19f
e6f7e8f50c4e9dbc3cab327121bf4de39d5c64f6
17531 F20101118_AACAOU saha_r_Page_062.QC.jpg
666a081bd48969ec8278e9d2e9f96404
927020a1591e8c127c903e1af761d06cc63764e0
2217 F20101118_AACBSL saha_r_Page_051.txt
16dd55535f71baa14a10d7fe4c4f1753
dd52d7ce97561e43b71fd2d341d38b373d77816b
1949 F20101118_AACBRX saha_r_Page_031.txt
90251134e7e0fa091828d73d314f1c72
9fd636fe1b7b382ef635925a7157c760f75c876f
4451 F20101118_AACAPJ saha_r_Page_143thm.jpg
51463ab232c970e6e08a9eaeba1fc743
577037b8baedf2bbd3e16052b8042c9d347bae7a
56639 F20101118_AACAOV saha_r_Page_109.pro
691f2efc9b1cd384dccc86a6cb7768e3
68a62a8529cc82f58f53625eb97d5c7e3b03737f
2136 F20101118_AACBTA saha_r_Page_076.txt
9ef0e2e937af431c11abcb645f40423d
5c774899cc2cb60130191290b12d3fcb4c33938f
2131 F20101118_AACBSM saha_r_Page_052.txt
4665bb636703a2784054b2733a72944f
f6e227eb627932fadd39605a63c21fbfe9f04813
2173 F20101118_AACBRY saha_r_Page_032.txt
135a0bda805db5a0365ddb01fdc5e07a
ced638d961d3d0e2118737ae475021eff4724a7b
2011 F20101118_AACAPK saha_r_Page_147.txt
1f5bab1c1dca33d1ce9c9091abb837d6
4d26b1e947ccda63a9f15a64efdf9014acf1675d
1876 F20101118_AACAOW saha_r_Page_043.txt
b574dc8e30d5d973edee34e3b46ac65c
8285797ab7dbdcc7c71e663443b1fcba40e24301
1726 F20101118_AACBTB saha_r_Page_078.txt
a9d17b107b9169c8139477315cdf218c
1d60ad46dd1d2aab8676ca8dd9245a3bc365adb9
343 F20101118_AACBSN saha_r_Page_053.txt
5a6bd7d7d27b3fd8de3fd427c8d6736b
69b6733509db911140f96e98a0883e4f1b901de6
1957 F20101118_AACBRZ saha_r_Page_033.txt
ddd5502da53485049f6ca8cb99d153df
f1474f134476eaa85b8b191f7c4c61378cd3f37d
41760 F20101118_AACAPL saha_r_Page_137.pro
01167c29c276a40b0a57ec37005ef37b
0273c1d8aee2087d98c2efc279b23587e4ed0c15
F20101118_AACAOX saha_r_Page_014.tif
d1a4150e906b966db73237bc87fd8fe9
c5d7d59573b9425d9b323346b413a405b3c070f1
1918 F20101118_AACBTC saha_r_Page_079.txt
bcc77d5374e4cfbea0b4f708dde3f855
d16e1a1977b922c913259a66bc052fd3adba70e4
2024 F20101118_AACBSO saha_r_Page_054.txt
bf52ff54be2a675713d8482e266f5a20
06aa2de5176948522b49073a023af6363b0e6a16
58197 F20101118_AACAQA saha_r_Page_123.jpg
d2f56c7a51d601bda8e85390c32ba3ce
e454ed0f2da725b173ca3c44b73ac69198986241
F20101118_AACAPM saha_r_Page_068.tif
e423f7099721d6594c88a112636d5738
5f3e906125384f3e61710291313c404abc3f136f
19028 F20101118_AACAOY saha_r_Page_050.QC.jpg
f8e6a2b57ee553719ebb9a95cfd9b5cb
2e99aee527e4a41f37d0b3f2289988fa51fdedae
1825 F20101118_AACBTD saha_r_Page_080.txt
30ac751a85dbba38f8e4c68a5f144e5a
c0d1acea8090205ebf9e6320d0e711555570c3da
1476 F20101118_AACBSP saha_r_Page_058.txt
1ab33affe09f1c68eec14d4071e8a7f2
0001196bd9ebb218186cb4a958a9d3b95336d4ca
5432 F20101118_AACAQB saha_r_Page_118thm.jpg
a4038f0ce2dc001bc7ffabd70aff0422
8ecede5b7cb9e7c50180702626b879a62cfae375
F20101118_AACAPN saha_r_Page_080.tif
af0ecb27ba430289ab92721f89d0b2a7
3b2d79a129ff776800bed7b6bdf3426fb61605b5
29830 F20101118_AACAOZ saha_r_Page_125.pro
226fcb3459773fa0d4a10488ab8a6680
6e7899974acbc2e952d5da1014ca8020ba42acc3
1834 F20101118_AACBTE saha_r_Page_081.txt
45aca61710eda492932aafbc7217a5c6
3585e695786911bbada41a50515797058e8837ce
1617 F20101118_AACBSQ saha_r_Page_060.txt
f7e631c77e1667ac43782960c324fef6
a597af3d6a97199208612bf7796b2b2a9d33ee01
5612 F20101118_AACAQC saha_r_Page_066thm.jpg
bc887ec0239a5b25e680aca8234bed84
1b3a11b6431e5634d8469cf0fc101c2a32f52e36
48582 F20101118_AACAPO saha_r_Page_136.jpg
aa180b6f566d822082385cd15f36cd8a
484d84d4b9419ef47b262dc722da799f8176cdd9
1514 F20101118_AACBTF saha_r_Page_082.txt
62b287e0447d8bcafc87f7caf52a7997
3021bd00e80ba8b1cbb81785f616aee33d11fd02
1973 F20101118_AACBSR saha_r_Page_061.txt
21b55951d61559801452e30b4ae89a11
06b031a7b1007c45012e414ba98be8555f94f1b5
24970 F20101118_AACAQD saha_r_Page_095.QC.jpg
c62e88315a3e8f62c66f5fb2e043b986
9ffb5da4db25afc6709c0470fe0b1681e2ceb606
1471 F20101118_AACAPP saha_r_Page_034.txt
0263d56ea99e954537de9a45c972c330
28b61d865906699e131695bb1425bee108bed8c6
2018 F20101118_AACBTG saha_r_Page_084.txt
e9f82be0c4417f1f5d87f3c5e1c8f032
6a330a34d4f750a699f6206e2b677ddb56f53905
1709 F20101118_AACBSS saha_r_Page_062.txt
99eba5862374e430a01c6bc9ffa6704e
d838c399ad1414f560709eede83348e238832321
1687 F20101118_AACAQE saha_r_Page_086.txt
e4bd6b2912667c6823d1b685a6682dee
d3f4ab8d3b67a8469800fdc8ef7c57dd42bc063d
5629 F20101118_AACAPQ saha_r_Page_168thm.jpg
15c40bcc973b7d0a9c657fd8d65ca9dc
1e77f5c7bd49635889f22a81d64a9074495009b6
1888 F20101118_AACBTH saha_r_Page_088.txt
b817eab45ca02074fd70485e82e73536
377a4e6c4bf147d59f2a78c2c0f7c91bf9a909fd
1962 F20101118_AACBST saha_r_Page_064.txt
082c1f73b2873f8663840978d083bafa
019942f06d6bba87073eab4e7914738f1b6aae51
F20101118_AACAQF saha_r_Page_083.tif
d552734b8ec4868e6d865b4c62146d79
1d933e37a9db6402c509571c130d5ebd8ac50770
49589 F20101118_AACAPR saha_r_Page_054.pro
3ddc56f9cc09e14bb0fe580ebe65b600
9c58a9929e6ddd4c8d5e169b4459ed967b4c5ed6
F20101118_AACBTI saha_r_Page_089.txt
f53984f90d12323cf2d5c5cac3395334
44e950057deef959264bdeb3fb6f58bc58e340cb
2142 F20101118_AACBSU saha_r_Page_065.txt
d1c0c4f6b1b7e7b22ec98c9fbb59c208
8c43a8f4b5d3af8135cf68e7d5179a2566a41969
944420 F20101118_AACAQG saha_r_Page_072.jp2
0cef6144c32e4d686bf44db5d52957f5
d798fcbb402e604d2c2af1baaf504857118d70b5
15470 F20101118_AACAPS saha_r_Page_067.QC.jpg
c06a33b3ef65aab43c8e9e55d4906aab
e9bb04768d49ffe2f5446305ca67eefb844640a9
1647 F20101118_AACBTJ saha_r_Page_091.txt
f91152ab53d204e6b1b12b3a5e975e76
eed138ece03d86144b4580acafe69a59c51efbfd
1989 F20101118_AACBSV saha_r_Page_070.txt
dc6a457afad38d6115f40ad634f71b10
963ba6754fdd5dd74b338f26140dc44658d9f2b0
108 F20101118_AACAQH saha_r_Page_002.txt
628bc37a8cbadd30854975ee76e81266
56b7339352c505406fc56e0515bfbba98bef28a6
F20101118_AACAPT saha_r_Page_019.txt
5ee8ba665aab53ef338e713ba19e74d1
b9e22d1b4158687b299ab25fdc1f13302c6fcfa3
1963 F20101118_AACBTK saha_r_Page_092.txt
5bec63a43c8831c64d522596fe192791
163c5c1f6844e550701da50337c1fd447bed43e5
2095 F20101118_AACBSW saha_r_Page_071.txt
f8c8ed8124b8b2d623ed4a151a216435
7b5287dad9fd80fefda8c893d5de8fd42f53ecc4
2421 F20101118_AACAQI saha_r_Page_077.txt
77fe8d665700c4120685c5603faff6a3
bcf8c61bb7df96520fa66ef6fe150c85a9c7b51d
66587 F20101118_AACAPU saha_r_Page_131.jpg
ae55c043185c4810817bf245e2276e25
8a2123de733a7d5a4921c4453a0bb1f9eeb423ad
2119 F20101118_AACBTL saha_r_Page_094.txt
46439be014956c9285318deda25e0cf2
72d5c6e830ae762890c055c8afa9f82079dfd2be
1718 F20101118_AACBSX saha_r_Page_072.txt
03f050876397ee21640ae64b32cbda31
7debd2320cf88ffbd51eff86a478c27848bc7912
624825 F20101118_AACAQJ saha_r_Page_093.jp2
ddf70da5f390a355f95e2d6fae85f50c
7c362d9f3fe7c7999ed88c62c0b13d9fe00d82b0
19704 F20101118_AACAPV saha_r_Page_031.QC.jpg
d272ef7942a2f82968613d3146bfecdf
83af2ac64d881426f4953ec96e8b054f948abf9b
1725 F20101118_AACBUA saha_r_Page_119.txt
96ce0f454ebc6c3e4625b019ede87469
b91bfaffba58f89637d64b68e6c4874377e67e22
2077 F20101118_AACBTM saha_r_Page_096.txt
d0fb5323a71b38e4904e803f80553ba8
b01094d2dcdd937118f0e63c86e772c9b982449d
1308 F20101118_AACBSY saha_r_Page_073.txt
cc2c441a9bfbb7ae5f8922b352900db7
a20c759d7c9c603a463bd2a9eeb63c22d0a48f55
31592 F20101118_AACAQK saha_r_Page_060.pro
4de6b2ab2f767c29cd05890f8695877b
ef2318d530e1927c64e70993866f506840c738f9
29426 F20101118_AACAPW saha_r_Page_154.pro
3b7e6f011d8421b57cb2cc8c2c7502c6
8b4d7a16c88647aa8c3703956bfbaec01c55112a
1670 F20101118_AACBUB saha_r_Page_120.txt
5841811654baa24c805349438486f310
9161d128e0155e4ec97eada371805a1b43553818
1872 F20101118_AACBTN saha_r_Page_097.txt
4ddc61d995f4d233e54b3bdf8ae59874
c7188c4af56ec0c7850416b8230d703091fbeb56
1553 F20101118_AACBSZ saha_r_Page_074.txt
7aaddf165e136e1906a1913e1837e1ed
645ff34ffdf18461846afeeb412c75ec1e0e4b9c
1051977 F20101118_AACAQL saha_r_Page_141.jp2
57b2e292ecdb7f305142cc3fcd00797d
22d60614dbb41a16d5252e51683b207762eae6d4
6091 F20101118_AACAPX saha_r_Page_142thm.jpg
427deb2749efa6343fc604f00da0061a
df5755457fed1494f97c0c012aa069215029ef32
2008 F20101118_AACBUC saha_r_Page_121.txt
70759aebf85e8005dd2e1efe3941c8f6
1a5e0de891fb53fd8c95fa738a5fce970f452b69
1749 F20101118_AACBTO saha_r_Page_098.txt
c21d8d48a9fc9161f0488207cec78f28
c3971e964fa689f0a619dbe7db0ece844430c219
911068 F20101118_AACAQM saha_r_Page_035.jp2
a7b1be3fe2e9ad69212102ec1583c596
ec637f7449b75c5267322189273e2cd8f19314c2
991362 F20101118_AACAPY saha_r_Page_127.jp2
c02c235739239c6f1f7de79cf8e69aa0
2e7ee8e04bee32492cbb85b6ce8b6af160a5b986
11024 F20101118_AACARA saha_r_Page_155.QC.jpg
ed082bdc7f35d6631dc0e40b1c5af782
1689c60ba8efa3f4e8ad174ab8ab21b3ae92df8b
1770 F20101118_AACBUD saha_r_Page_122.txt
2421c0a8adc4856bafd9d07d52fab7fe
2bf9de3248d377f1e973c71b539741d0773c53b7
1644 F20101118_AACBTP saha_r_Page_105.txt
f84f4292627e4cb23266d1762b7d03b3
95ea5ccb73749242b88fd181db0cfdb97d6a29a1
1051913 F20101118_AACAQN saha_r_Page_090.jp2
ebee3757a7e2ad96d031ed1a0a0ea1fd
2698d37146b3d29e95da3e3d66c3f34d170bc8da
18003 F20101118_AACAPZ saha_r_Page_155.pro
6bb883895588186bd255c118f48af6a7
71db0fb98d728744ad9ba5473ba39e98671ea171
68762 F20101118_AACARB saha_r_Page_038.jpg
f497843bca8f91a22f66fa2e4c575862
3f50a24667d26260a597e8a118dbb219dddfef07
1508 F20101118_AACBUE saha_r_Page_123.txt
3ffc631816f93ed52df10dee6015bb79
d5c800fc5aeaabfd3bae9ce8e16d5536bdc17f64
1935 F20101118_AACBTQ saha_r_Page_106.txt
a5f8d5f863ea0f0aa6a827f93523d4eb
96bd944505daad6fd3ebf5c978c2b6d231b16d16
70734 F20101118_AACAQO saha_r_Page_007.jpg
6534042ae93fa5eb2743ad7b2fe9173a
36492e7741763fd1d3e1da8ca1435d025b8a189f
6164 F20101118_AACARC saha_r_Page_141thm.jpg
e6efee8a7723c7ea3a75eed68f350590
3a59a4b57255a52fddaba4fb144ddf3515506431
5050 F20101118_AACCAA saha_r_Page_005thm.jpg
eddd59cc41c0250fbf6a3f1b196b4b34
96f0e569551e2fd6001f64138da6becbfade0182
1931 F20101118_AACBUF saha_r_Page_124.txt
d99590bccde49a939389fa559950808e
b9d9f685ab60b670d4c460c2924fa4d3c66e4d00
2186 F20101118_AACBTR saha_r_Page_107.txt
fd5348be2cda23e4f145f7fa84e92a92
acf83c506b98abaea4e01a3d1aadec69fd6f10b3
744426 F20101118_AACARD saha_r_Page_074.jp2
4ad38b1f98d231ec6eb51fff8be597e4
e013f528efe9630b239b208a16d7b2e1c73bbf5f
826516 F20101118_AACAQP saha_r_Page_088.jp2
5d335119d530a45fcb989f6d97408e96
e6d650625f96651121b775e304d02709958d264e
6203 F20101118_AACCAB saha_r_Page_128thm.jpg
31aba86d7fe4bc9b49d0e3aa3ecc5fea
ba1be771700ae897d628aa65f5bec1665af64cc3
1487 F20101118_AACBUG saha_r_Page_125.txt
b45b33c1a207b84a839af7f4ed75a551
a3a3f58298faab99d323a848b94735051244a223
2291 F20101118_AACBTS saha_r_Page_108.txt
1bd59c8fd847fc8fd9d96ca6922357bf
6539b2a048455a319fa3b5753c6fa9b373e06fa1
F20101118_AACARE saha_r_Page_129.tif
5d77a80a9f4020cefe24f0b4c1c3e7e3
2fb28d08fdbb66aaef269ab5240a56cf6b98198d
40552 F20101118_AACAQQ saha_r_Page_174.pro
7867ae71441c8d1cfd419e5ec3c3586e
fd219f6853338140b8de15883b0d75fd9379681e
20964 F20101118_AACCAC saha_r_Page_118.QC.jpg
b80321de4147dec4a79d0e47f4804ab9
7c18237924a75b597f73693085aa86d8d8a54504
1904 F20101118_AACBUH saha_r_Page_127.txt
820f5ef83459b8039522271fdc2c1a99
accc7e756da3ddfa6ad86b6848bf685449cd1db2
2221 F20101118_AACBTT saha_r_Page_109.txt
5b8a6b262765369bb29fb80203eec54c
2cfbd04779ad8aa4fe845fb791a3ea0ded2a0c8e
16479 F20101118_AACARF saha_r_Page_130.QC.jpg
8fe429af5c6b6977a99cf124e21475b6
6a5167237988b16fee9761d29480445acc80ec7f
550 F20101118_AACAQR saha_r_Page_143.txt
8f53f6e75a678b63936a21720c991bf0
10f668b6ab1353fa56f95b509990683c939a4deb
16476 F20101118_AACCAD saha_r_Page_163.QC.jpg
43bc20700462cfd8cdc8d48381526f2b
cc8e2560ebef0d97d8f27b0f8ccd296b44d568a0
1364 F20101118_AACBUI saha_r_Page_130.txt
74ccb1f25401b96b3f89436d120ad98d
7e544f800bd4c96f759a317ba57feecf6d7c202a
F20101118_AACBTU saha_r_Page_110.txt
7579338335c03d78ed2d1820ef31958d
d1e5060b3290892eb5933a66dfb19c04bad93c19
F20101118_AACARG saha_r_Page_171.tif
e546b32e3bcf69dad97c4e3bdaefed49
228526772daaac1748f69fddfbb543dbe22ca3ce
48074 F20101118_AACAQS saha_r_Page_183.pro
ecb813f85c449188331268560db8b958
43bce69b2479a80ddd7fa51e43dd41feeb4ff1b1
5885 F20101118_AACCAE saha_r_Page_170thm.jpg
c5acf938173c9639d6fe1ac049de988a
1dddd93c42d5cdfdc91b17760e1d944982080aeb
1819 F20101118_AACBUJ saha_r_Page_131.txt
1bdd09ae5559db7191e9882bfff32ba9
26e997627ea1c023b342c27f313fb64369455009
1867 F20101118_AACBTV saha_r_Page_112.txt
47611ccedd137dc8a327c4fd176e6ac6
67e9e189ea4d7f4bbe3d110c110f2a58ee84a1ce
F20101118_AACARH saha_r_Page_136.tif
a7e979c957b1d431c1701e4c58c1a73a
8e87b791bc0e14c40d416f4cdfac31aef854b0db
2041 F20101118_AACAQT saha_r_Page_102.txt
7fb5c2de1e448a0bfd838665012a4825
e9daa397f7459da28a3d9f847e221dd06a07fce9
25267 F20101118_AACCAF saha_r_Page_110.QC.jpg
f5aab6fbebfd72233579b4514269247a
1c0a31b09576db4e207af94503d10c558b4fd65c
1389 F20101118_AACBUK saha_r_Page_132.txt
aca58206447f067d2af5634284ff379a
1a1aa32d2dc25e158d6f60205d9ed7b4b0dc749f
1998 F20101118_AACBTW saha_r_Page_113.txt
1581f3ffc53a6b1c78d3351557722da9
844c64880af9f6fe37a88d2b4c28ba911e2e7b84
45981 F20101118_AACARI saha_r_Page_177.pro
7420db19815acdb340d7c4143eae59a0
fb319e5a1945b743331fb3c98cb455ac5c2cc18d
F20101118_AACAQU saha_r_Page_165.txt
641bd4a4e1b9010a508172a9e9acc083
1bacdd56d956c38a8d860aa2e1f2d7c642fbf575
4660 F20101118_AACCAG saha_r_Page_139thm.jpg
7e178ebe7b9d9b4e80e5e8893f9cb5a3
fb70344237a9788d175c70f7b7aba5232925db1b
2148 F20101118_AACBVA saha_r_Page_157.txt
eb5411b8f4b32290df14dd745483776e
4b6bd456280c8c490e517aaeba3b0bf400de81b8
1840 F20101118_AACBUL saha_r_Page_133.txt
8b1c2bcf357516b43d63d459b93afb4b
3d3280b5d6e9ca9336ed3225a789545dbb5c7649
2056 F20101118_AACBTX saha_r_Page_114.txt
c608d9771ce00c694e84e1501affd69a
c2f1c5c5dd91bf3c9fda7ac3cfd445dc1a24a68c
91051 F20101118_AACARJ saha_r_Page_010.jp2
a4199a0016d3648bf2032ff19ef564d6
703fa7d0626c6ec648f1eb040bbc41fc59819836
1929 F20101118_AACAQV saha_r_Page_168.txt
133cffd9aa7763da0fc88ad6ac9a22b7
31f37b4fe1ba34ae698f96c1fbb6fc8c30b4f1a3
23634 F20101118_AACCAH saha_r_Page_135.QC.jpg
f14d5f566646f63b14346e3d3300c092
0437af2b41c41d50f0adaed725f912e6b7d356b5
1516 F20101118_AACBUM saha_r_Page_134.txt
fa008e54e730e273064486a66d0fa5f2
e9435b8659f6910f04a35e463e1b14f341c059cf
2129 F20101118_AACBTY saha_r_Page_115.txt
39790cc27413a6d0008e543afa44c744
906e66c34f6deecb53e8df3662d9bf62bce28b5c
1702 F20101118_AACARK saha_r_Page_085.txt
24b4c06bf5766dfb3c8591835c6893e8
ebcaad68f640581c6f9a18b3b87b13f930066331
45052 F20101118_AACAQW saha_r_Page_050.pro
761a85d45f8c4c2acec7e582d3e97916
2607e8821d0021f9d24c715578d5e0c55d923c4c
6219 F20101118_AACCAI saha_r_Page_111thm.jpg
50e98381f69f51b9f2e5b5570743fb03
05b766dbcc93ec8677ecefc53115137c2c977d7f
2003 F20101118_AACBVB saha_r_Page_158.txt
7cc3bd0bbe22d07c1a9b3857f3befb9f
cae2d0b961b257ca38fa523fa4566ecbdd278509
1983 F20101118_AACBUN saha_r_Page_135.txt
1ddccc9f70a111d254c273a1b18e0cf7
fe9618fd5e1dcd823a6fda98b28ececc8d2a6642
2023 F20101118_AACBTZ saha_r_Page_116.txt
e7b4a66263d8ef7bf82c7776c83a49f1
e6ae3e64259f74cf72bb81868a7b8c70cfb6c326
2026 F20101118_AACARL saha_r_Page_050.txt
d1d03636787fbaf3dc8a216bd86f3fdd
8ff9c77d19e7d02ab9ed9cd9192b51f0e10fee6e
F20101118_AACAQX saha_r_Page_122.tif
92c3857f061a984f88420309426130d2
dd6a59e28bfbd45ff2bde4fb775bc774f26188e0
16544 F20101118_AACCAJ saha_r_Page_132.QC.jpg
2d8e8e392b9d51d02486b4f552943914
b1f1b5ec25111c6bf49ef86ab7d934f72a318b19
1506 F20101118_AACBVC saha_r_Page_161.txt
7de8f357d206366d551bc3280a58109c
6c173e36749fa33bd37299117535267621812be7
1420 F20101118_AACBUO saha_r_Page_136.txt
3c2d1d4024270aaaf9706cb949d454a4
4d708f8406e9070d7c9bd5a710e3871681ec719f
F20101118_AACASA saha_r_Page_055.txt
909a8af697e1d2b0f66e498a4a2b1f8c
058499d77d557e6529b59df7c41edf4f16467c4d
2205 F20101118_AACARM saha_r_Page_178.txt
b207083a948b03e3e063f484b0190bf2
de0834f1e57277a0fa3f6971a33e8537f39a522e
2177 F20101118_AACAQY saha_r_Page_045.txt
b343fc2b48f978f0cd5df245a537e20f
94e47a980ef837b69807570053b124a94889a7a0
5653 F20101118_AACCAK saha_r_Page_181thm.jpg
86d93d98ff5d11304652622fa80eea21
b33809ecbb3b13d3c81a098f947c92cefce7c92a
2016 F20101118_AACBVD saha_r_Page_162.txt
e77003b176333041966548c24bc742ad
fd89c71f36032dc4e279e1c06e6f759f25665331
1862 F20101118_AACBUP saha_r_Page_137.txt
71b1f9afa3b15a7989d728ee55d124da
3797107bca364cb323c9d6ec5ee6c4e56faf2236
35619 F20101118_AACASB saha_r_Page_134.pro
68f6ac6fe68ffb59ae278018d55c23f7
a648cc3c8eaee460a8de83110978de7f191d6dbc
76647 F20101118_AACARN saha_r_Page_102.jpg
dce5a603d76580ddb507d62edebf842e
3669be64d0b9e694e6bd592cd6b5dddbab0a9aae
2037 F20101118_AACAQZ saha_r_Page_180.txt
25c85dfefa0e5ca90e7df91f80ee340e
ecd22a4f1f01011a26d4d17c62c7dbc626b3f213
1642 F20101118_AACBVE saha_r_Page_167.txt
9d1664e8e0f2bda798c57dc95abd6783
e339ebec7541e972dee59f0a321472cedf1e515f
1584 F20101118_AACBUQ saha_r_Page_139.txt
184d4af07f2ede93d946887058275cda
28c138b8ccee98974f53a4121f8cb29725085698
1051984 F20101118_AACASC saha_r_Page_029.jp2
07aad0838615e3d395e72829af598b6d
2bd4576f88bfeecb29dc690a12931b9d4b2778d3
4681 F20101118_AACARO saha_r_Page_067thm.jpg
87c841e0122adf9347f8ec01e2b3df51
bf4b520f4a0a3f9f0cf03ac3ba46f89dc24661cf
17864 F20101118_AACCBA saha_r_Page_091.QC.jpg
b4d1beb8a1faf997022af94181ccd173
f6cd9d405ec42694a4a7ac5ddcda9b09f2b7858f
5810 F20101118_AACCAL saha_r_Page_064thm.jpg
03bd36b9dc17b48891745cefaae199b7
0020dc766dd4ff1fdca23f65688bc5d07635fa5e
1455 F20101118_AACBVF saha_r_Page_169.txt
ea4e21acfb6b17a57b16bd684467e7a4
a8e1d1cc68c7ec824cd05518b689f3e614423b53
F20101118_AACBUR saha_r_Page_140.txt
c73d0d6403f57c3200e8e291f1d6af36
5970416956672e94038caf62f848e046f10fb72f
22708 F20101118_AACASD saha_r_Page_106.QC.jpg
94591a88498d9981c807bbbc022616a9
5c908d91e759f22f6a080b0336881d9683a15f57
20378 F20101118_AACARP saha_r_Page_066.QC.jpg
f682f14d965dc2fbc78584058245706a
3c7a5dc92d11a13f71bd0bb99b664265d8142fdc
5350 F20101118_AACCBB saha_r_Page_079thm.jpg
6b9951bf12a2cb9ac2bac1be1bc07bb9
d8035f196f83cacbbb53437e206bc3a68f7c9489
6059 F20101118_AACCAM saha_r_Page_115thm.jpg
adc9f0e1d6a410861f6b3972951d4067
306ee8274c43bad656593238a6485cd274a460c5
2107 F20101118_AACBVG saha_r_Page_170.txt
52f921317adec16d4e990015ec52b843
d2eeb236927c0730f4ad178d0ac2a60fb671bfa9
2152 F20101118_AACBUS saha_r_Page_142.txt
63f9c142d569c1bac3b26980a8851088
ca52fc569a9b8712826be5dc9c6305e6c4c85f77
24085 F20101118_AACASE saha_r_Page_152.QC.jpg
ab0b837d795858ca6bbf395060488630
9f6af16bfb77906ea9ca127ca33135f162c4b8c3
1679 F20101118_AACARQ saha_r_Page_118.txt
2d94e7b01b1c433c75b287a962daff74
1bab5131065041378461a590bcadaf7ca7180b62
273235 F20101118_AACCBC UFE0015302_00001.xml FULL
d2145a509366abe7c33c4681dea1dc9d
33f0a9488120e79ba42e633b7d06378075cd0e15
20334 F20101118_AACCAN saha_r_Page_089.QC.jpg
77a1982f485b9e097048361372bfff77
fd24ea6b123b55ec12d17007b6eef88e82473269
1485 F20101118_AACBVH saha_r_Page_173.txt
a6081c478850329917433d62e8916b76
fc5a37bb260908c9ffafd7c117888982ccef0911
1857 F20101118_AACBUT saha_r_Page_144.txt
122d572f32e3e3a159f2ae4d1ca939d6
1214cc6ba17c4e5df064fecc109c4a8dfe808ec2
5240 F20101118_AACASF saha_r_Page_027thm.jpg
1093ac6a71a36edf7873756d96467d6e
96f5ad06b46ea7862847fc4d847a1c29e2ccf56f
F20101118_AACARR saha_r_Page_003.tif
9ae190759fcdcf29a4f6c639b7436462
7faeb2267098feba5425dfbcfa80a902c7d5f998
15401 F20101118_AACCBD saha_r_Page_004.QC.jpg
185f395882970e377f94fd3a5185f498
6e225b630bb0ec579ef52e980e19e0bb76b43f5d
5891 F20101118_AACCAO saha_r_Page_133thm.jpg
fb2b1405b1e4d9f661e8ea6ef28f9abe
a2e5e1332585b36a98db0ef2722e7f787a59c0c5
F20101118_AACBVI saha_r_Page_179.txt
76c81d1af68f5c6159763e39481b422c
7d28d3955d328950860c1078b75af5928d220930
1624 F20101118_AACBUU saha_r_Page_148.txt
84bf87068ef8f17fc8e38e200d3684bb
6f8dd9a2adbfe5e9727dadc7c3aec15f40c1cb95
57577 F20101118_AACASG saha_r_Page_056.pro
49a9cfc014ad92b96a668db295ef2b36
a1ab6c8e3727fdf39356e9e226cc5858fec6ae5a
67547 F20101118_AACARS saha_r_Page_153.jpg
cad1a83281a775726c4ea5146b3b8195
1c56ed3f0504092d0670399bb3b739d4b3df15c9
4752 F20101118_AACCBE saha_r_Page_009thm.jpg
d8a06dac338d9be9661ffb27a29dc993
de6b138335c7ef243ca47550ff0ca73e73ae3ed7
5076 F20101118_AACCAP saha_r_Page_059thm.jpg
1068f332106ffa6d6b5a403477c7a018
b34dfb5ac2d64571dac33cdedb432e19856bfc6d
1932 F20101118_AACBVJ saha_r_Page_183.txt
30fc6405cff9d378b1fd9e64e5d40a07
ef6c9c4e0c42ed52f396a15b9b6e13a2a8615ef8
F20101118_AACBUV saha_r_Page_151.txt
1bde9bc42e92a028e04b7fbdcaaa0d13
4c7a0068853ab82a9700462c904769fc81b85845
18472 F20101118_AACASH saha_r_Page_134.QC.jpg
8a80e96026c39beddfff1c6eb5ec5622
e6698baf120ccfe313c8e6c990c86738b2a1c901
1871 F20101118_AACART saha_r_Page_075.txt
88dd4d5d7bbdf07daf02aee94dd31f00
5d779e5c2e40161fbff768d87aae7481c49ebfb6
6437 F20101118_AACCBF saha_r_Page_015thm.jpg
183a9c6f4516ababde0b29f6f1ad852b
beeadb49bb53e9e9b2d2dbb89c92b913f6cf3b0d
21802 F20101118_AACCAQ saha_r_Page_038.QC.jpg
6afaea16adb7d38893f7664c3668220e
8a79cab85d0a7cbe6b77795ac2d94d248bd2b29a
247 F20101118_AACBVK saha_r_Page_184.txt
cd28f22f45596694a7490efd1026ff6f
1a6dbb92173f1a47b143fae6ebfa39da13b36ae3
2080 F20101118_AACBUW saha_r_Page_152.txt
33af690e906808062d97417479a2878c
aadf95eb344041405adeecf75cce9f7e156d245d
1051971 F20101118_AACASI saha_r_Page_106.jp2
7d3a68f706d26481ee438222d99d402f
a2f40bad90a2a6b2b64c6b7102f08bfcb4e13d51
4217 F20101118_AACARU saha_r_Page_060thm.jpg
ffbbc9c0487b6d49a0b731091cfde3a3
f89bcc18ce5cbbba52fdfc893678607175ee7833
6074 F20101118_AACCBG saha_r_Page_017thm.jpg
3019dbcc61fa29c22370ba0eb7702c62
a981f78a1376d1f8af62de1ceaa2608fbcb1d2dd
17618 F20101118_AACCAR saha_r_Page_087.QC.jpg
7e02edf548c4281ecf2216006f203828
ed616dccbca812747b93f8d318b37825ae338132
1359 F20101118_AACBWA saha_r_Page_002thm.jpg
0fa00fe78ba840f056cbab49e6c901c0
205d4ec8ea4d3f6368b128f8703369bf7d147dbc
2353 F20101118_AACBVL saha_r_Page_001thm.jpg
514d30eab5d22cc6425ab7db97d52657
c0f7e88c24a80269d059a66061202fe7ace18746
1853 F20101118_AACBUX saha_r_Page_153.txt
5be15240dfc6317cde0a3a3f70037b1a
90ad20be26281e370b7bb22084f59652dfb4d61e
720645 F20101118_AACASJ saha_r_Page_166.jp2
fbc8126cf051f1c3e5ff4ef55ab3402f
56a960071e1e61ea4d0b656c256644b5b7d86730
F20101118_AACARV saha_r_Page_112.tif
db41d746962496db2d18737bbd779240
408feca436af120a920997d9f6889d46f4210162
15192 F20101118_AACCBH saha_r_Page_018.QC.jpg
e07b254d333fe52de8ccd53cf08acf45
5c33ebc411ac5f8540adfc956734dd258d5740e8
23227 F20101118_AACCAS saha_r_Page_077.QC.jpg
370b564bb9af0d5b754edc669fbeed45
a42f13e45ddf7cab2e64a90b727161cd790cd4b0
20770 F20101118_AACBWB saha_r_Page_131.QC.jpg
e06541d8bc33037e55320531f41439de
6c286c33583f3b781377dea92c33e985418c798c
6406 F20101118_AACBVM saha_r_Page_052thm.jpg
51de104d296392c4e808586ec07023b2
13d6a39de3cb06adb15f226d98a07615dccaaabc
1600 F20101118_AACBUY saha_r_Page_154.txt
81f0bd5af1866e3339ada3c23f22eff7
b5dbbd4163b2679676548e9fb0e59a2042cbb11e
1589 F20101118_AACASK saha_r_Page_172.txt
73792812b17c77388eb54483a99ef676
7dc46322b706cafe0c4e49ecfcd9a25da2655327
27145 F20101118_AACARW saha_r_Page_018.pro
ecdbae4627f9c148f7d4fc0bb7f76bb0
938051fd2fbcb11084e4492d8c48d6de4fe228b4
25455 F20101118_AACCBI saha_r_Page_022.QC.jpg
705da86c07d6c1cd3fea0c1acff0d442
cebef2e75d9089c8d0c448b535756d3af4beb882
4569 F20101118_AACCAT saha_r_Page_165thm.jpg
0bfc3285c6bb26e8118b811bb3ca7757
6850af8794e3f22a74023966667bd07bde65e2a8
5263 F20101118_AACBVN saha_r_Page_156thm.jpg
ddc7d68dffb49e333a8a5616f4ba0e55
0d8481b097a17027c861a491142ca3d51d852784
1938 F20101118_AACBUZ saha_r_Page_156.txt
dc7e47eb0e7f89e03e4b43bb9b5407d2
74c910a3c5cdcb02f283cfef36411a43c739a0e7
1489 F20101118_AACASL saha_r_Page_129.txt
3f4ea01e35286ea4780f845b87e02f54
8fc56d270fb968db57490d2fc5aa409aa37f457c
16183 F20101118_AACARX saha_r_Page_139.QC.jpg
a6dd448037d111fff87e0f82c709d79a
e845d2861bbc313b3d51415bce3daa29e14e43d3
16595 F20101118_AACCBJ saha_r_Page_023.QC.jpg
e44411e6b50b667de36460b9881dbc9d
9c3e82d9ffbf807af29011cce8e98997dfee869a
5831 F20101118_AACCAU saha_r_Page_126thm.jpg
03fe3cb1995b2e843d097b07b0614626
f72b2059ffdc5496b7d5ce335803f161e9194175
17795 F20101118_AACBWC saha_r_Page_123.QC.jpg
088c7fabaccc3ad66e5ce81aee1f2d80
4344c4d2df65516855d2bf18602bcdf4872e9f0c
6396 F20101118_AACBVO saha_r_Page_110thm.jpg
764fe63336034b04a5b7083c206ce6d2
6bbcb7863cd0a875f8d1676865e1f17c719565d0
25735 F20101118_AACASM saha_r_Page_107.QC.jpg
593f69ae8eb44903e88e24c2af7eae6a
c1d567719be849b6547ccca28f9b9285b2594490
40686 F20101118_AACARY saha_r_Page_175.pro
998e09ab7dd88922aa5e4e6238fe93e7
c2c401ed15821f66862898379a91b16c7d3d4f59
52834 F20101118_AACATA saha_r_Page_044.pro
a6a519a6514b45164f6a04d62382b53f
aa3c30149a9e49417aae35a6bfd348cd6cf7b62e
21111 F20101118_AACCBK saha_r_Page_024.QC.jpg
79a02fe0dff0a2c3d641a17a03da6d76
10815fd87654988b8d1e4da992bfeb7d0d6839a0
16124 F20101118_AACCAV saha_r_Page_162.QC.jpg
3e45630495bea10c9aaf0a6bc1f56949
4859b8ca6e3c754e7795910b64328cbe82a7b864
19109 F20101118_AACBWD saha_r_Page_080.QC.jpg
4b4636eb650b9e826ed65abc43c62947
2c757bcdfdff698649f2ca300ea89a0bf85509ed
4998 F20101118_AACBVP saha_r_Page_130thm.jpg
c935337105e9927354982adc56c8be25
a90a19b30e5cc8661a76ffded669e0afaa262a07
F20101118_AACASN saha_r_Page_012.tif
ff524ead8a9b99f33cf8e1aacb7d04b1
6c2b566f91fbbc3fd8b2c3989dfc25c75fb82f83
5244 F20101118_AACARZ saha_r_Page_072thm.jpg
65eb32d4c82e1f1408d19e038aaaeb3e
a66079437938b0e13dcb2be578c78cea36e1abdf
64449 F20101118_AACATB saha_r_Page_158.jpg
7425776795744c5add9fe97a4f067352
ee8d75cde5f066d16622b877cb913d65c94a6c9f
20188 F20101118_AACCBL saha_r_Page_026.QC.jpg
522cf4b99aed20edc8c4b53bf3c9ef4d
6dddd27fa10a95ebaa6f95fdb5ace442a528234a
27121 F20101118_AACCAW saha_r_Page_008.QC.jpg
57e124e4c3e86b6cca7cdaf29e7a5fdd
668f4ed5f8f1d37efd957df4a3a408de23ab3149
19582 F20101118_AACBWE saha_r_Page_088.QC.jpg
0cc454cf03eb7872d7aab09298c103fd
e63855a4e43efda7958ae27b4591552fd312a724
5793 F20101118_AACBVQ saha_r_Page_113thm.jpg
f312a9784196c8a93d4afad7a16f7075
4e6af9b8d80e81798e3c1fad703cc25f0e28662a
22604 F20101118_AACASO saha_r_Page_047.QC.jpg
24966baf8901e0387499af2645df0222
06635a76822b7a93b9c88732c41056a3845fa7c8
63274 F20101118_AACATC saha_r_Page_031.jpg
cf78f0cfecc4b225b1e90e05cb44a47d
94e471fdf1df667b36f391d6de8b570acc9e1c33
27006 F20101118_AACCCA saha_r_Page_056.QC.jpg
79e50580bdd03f7ea7aaa87bde62d2fc
f7c8bc616845c9ff7604755da93fd407fa6b2ad7
5966 F20101118_AACCAX saha_r_Page_069thm.jpg
c02ba336324dce3bd23bc221dd8f02c7
63103359ee46b485bb4d40154c62052919852a01
23777 F20101118_AACBWF saha_r_Page_128.QC.jpg
d940834bae825a1d72c49ea7af51802d
c1845fb7f659b1d1afecbdfd0e89eb61cf1749fc
5869 F20101118_AACBVR saha_r_Page_178thm.jpg
a7a38092bf4d4dec9143fac2ea9abb60
6ee9808a3472f0df8b76dad459e9d9ff5c33a475
F20101118_AACASP saha_r_Page_006.txt
9cd2bb9e57c38608c89514d2bf0d4d7a
93173b0e175346eae1d769f1dfede6ddd5d632a5
F20101118_AACATD saha_r_Page_076thm.jpg
6ccdcd205063cf1f6c64ff851041bf4f
b3b33bdfe4a5e758d13be97c26640ce543fd8b0c
20547 F20101118_AACCCB saha_r_Page_057.QC.jpg
798fa434240c59777087d76fd3e6fe40
e10a55a83e47239537a1f56067419ff75a1f1be0
5732 F20101118_AACCBM saha_r_Page_026thm.jpg
199b64fec9ebbcc5410119989372dfd6
e3e65e6ed4a1f99b40b142d92e6c1a693cf213f4
4185 F20101118_AACCAY saha_r_Page_048thm.jpg
8391ebe0fbee46bd65c7f36bb9a6a9e1
50d556969bfe43808b2526c20bcb734810ed1f1f
17882 F20101118_AACBWG saha_r_Page_105.QC.jpg
c3e53685f88f62170795670e0188803b
9c4fc2e444d4ca5e6cd90ddc57466dba0165aecd
17331 F20101118_AACBVS saha_r_Page_164.QC.jpg
6892eb5e9f3ce9630bcea5a83c1ed84d
9f1bf6e51fc9030a0136c1e5e4d611f175e4de81
23437 F20101118_AACASQ saha_r_Page_102.QC.jpg
a06931b038b73ab1a141e9c495569fba
d261c820603c699207d8efa690c6104329851120
F20101118_AACATE saha_r_Page_014.jp2
ca6e2337e7f2256ff4f416e96743e5ab
671c569972890273d0373aea3e807cfbb4ea2129
5351 F20101118_AACCCC saha_r_Page_058thm.jpg
68a6f6737a5fef11a8519b24a490332f
ed2fb07d61ea28bdfae54a9b497dbd80f45ef052
19329 F20101118_AACCBN saha_r_Page_027.QC.jpg
151b81107b418cbf467f31d82b62c3f2
22fb3b36187546ca7c267802d416ee9df66866fd
5507 F20101118_AACCAZ saha_r_Page_122thm.jpg
889218677c8cf0843ec4cad0ec4df5a8
a7498fa759c08b8d161c762683f36bc5e72740cc
6718 F20101118_AACBWH saha_r_Page_056thm.jpg
55323750941f227186c98726b754b9b8
ec5eb01efa885a1ed1b1b4652a087348c7405e11
18814 F20101118_AACBVT saha_r_Page_016.QC.jpg
a917a6409ebc7278cc40e189b6c4d57b
c53733db37a70874fc53fe7625d7edacdf7668f8
F20101118_AACASR saha_r_Page_185.tif
7a526935101ae93e1f7483a095434c2d
295286364de38bc27163b04e87fb6fa687204965
4834 F20101118_AACATF saha_r_Page_006thm.jpg
0ee78f8b34893c3506e423e6d352cfe3
6547f623ca646431bf9f1ce3c603f4c88be15a89
14646 F20101118_AACCCD saha_r_Page_060.QC.jpg
8b91aefafaac817654d17cef94f90b4d
9cee9fbd332ebb79345de367de9b63fe6e4f6b37
6028 F20101118_AACCBO saha_r_Page_028thm.jpg
9df0fccf47d0e55b3aadd27483394756
937425b99362136fe74a105e981ae632b0915dbf
25085 F20101118_AACBWI saha_r_Page_013.QC.jpg
45e21fd4597246aceb38dc485cf46bd9
ee0bd2080769598cac72c036400893d6a8cc4e5d
4219 F20101118_AACBVU saha_r_Page_093thm.jpg
c44c5a042e4ae5bb26acf1d742342b6e
535105e6907c1aa5f9fd20cd3237b5a41511fb71
863642 F20101118_AACASS saha_r_Page_027.jp2
68c77d4cd6b9e9d3d3868d049c8e98dc
203f248e8a1d7977da7f64123a2da19c67172d90
F20101118_AACATG saha_r_Page_035.tif
092232cce867b0db14baa63fbb0a2e9e
2ae519cba82f22718dd5a876e051a310fbf12839
19836 F20101118_AACCCE saha_r_Page_064.QC.jpg
128eec03a6f90621f13c5f8876f6c7c5
9b78d90a541ce788c5313c39a73e1796d9e6c202
6512 F20101118_AACCBP saha_r_Page_029thm.jpg
8835c550f613461e7508219acb1a11d4
a38390bee16d9b6a7958e1fb5cfeb6a85e12c41d
13983 F20101118_AACBWJ saha_r_Page_048.QC.jpg
bf784acb054fb5c54391ca2945e83c8b
661833706b7bc41bc2e5b4673e4327a0aed49c97
23675 F20101118_AACBVV saha_r_Page_142.QC.jpg
9742796c39ba783204d5c22d28135eab
8d28cda7977ba3325fe7516f0649c60ba8f01877
5404 F20101118_AACAST saha_r_Page_131thm.jpg
d4bf79bc9c346662234b16a8f7e77f17
2a9dc44f55c828fa751faa3d3ee65ba8b88da193
83698 F20101118_AACATH saha_r_Page_029.jpg
cfb81d87f4f9c759a712e469310007a0
fc524090e540805aca81c00987f6e4a818034f1c
5658 F20101118_AACCCF saha_r_Page_065thm.jpg
0280206eb6dd2be299d37cfdeaded227
afe2609b8a6cb8f004d6c122e4ccd8e051231459
19325 F20101118_AACCBQ saha_r_Page_034.QC.jpg
32bde30cd97e777fae6deaeb1fca3510
424748175a6fbfa76e87ee1244bf8bea9f99f42d
20042 F20101118_AACBWK saha_r_Page_068.QC.jpg
e032d2b94769d858cb18954d712570f8
f9e6f4b2aa422907716ea6bcc9b53814b8351148
5282 F20101118_AACBVW saha_r_Page_016thm.jpg
6297338c1c79b0aea5027ad02903e061
cf2a682ca8735ce5e8908e4f344da56878538aab
F20101118_AACASU saha_r_Page_149.jp2
3a92b9fa9ae8b2301e5bf5b8f2a22859
23b94542192a0dde50281f3d3e11476ea6e5ba4c
46809 F20101118_AACATI saha_r_Page_031.pro
9d3bacd9f43fb9c724f7b22b3acd81a3
ca044212f188bab7387cda99c5c030ef14d465b5
20957 F20101118_AACCCG saha_r_Page_070.QC.jpg
39f721ad591eb9713800a94efc0036f5
fd04e7489c745b742547506db899f7c09c88aab9
5101 F20101118_AACCBR saha_r_Page_034thm.jpg
ab480c69e2b16af20acb9a095cf96ce2
2b7e4ac0ce5edcbe8e20ad1856101c42a424f4d6
6046 F20101118_AACBXA saha_r_Page_077thm.jpg
2737356348b955f5feb7844c10b35d23
abb1bde04597ea340b2d8f1214b3bd752d4e1cb5
25322 F20101118_AACBWL saha_r_Page_021.QC.jpg
01cb44ad6460633b28cbe86d2a7d9e51
e7d05892bc6160efa5bccc26561f5192c3dff7e7
5344 F20101118_AACBVX saha_r_Page_144thm.jpg
3ad235aa95559e31ecf5ea5a96a4168e
16575f9b27a17eccdeede372238a57528e22248a
73964 F20101118_AACASV saha_r_Page_069.jpg
a84578c81295aaae95ade03862cda1b2
66e9f2d0859a83dee92a57b52bba4e5ad94c5c1a
38514 F20101118_AACATJ saha_r_Page_057.pro
ac9b996abcfc0ba45585471599995069
3f345b7a1bf49b9417c9b3fe19203b4fd37dd91f
5472 F20101118_AACCCH saha_r_Page_071thm.jpg
9997687207db58437e9e9e2e096542c8
dbb1a18800217cf435e1c35b5fbf3ecd127244bb
19969 F20101118_AACCBS saha_r_Page_035.QC.jpg
db12fa3fcc31037dc7b1ecbda31ed0ec
81ac8d3d37919f54743d26e60325c34310ddce36
6229 F20101118_AACBXB saha_r_Page_014thm.jpg
9a4d860d23eaf8085b3e478412a3e6a9
a37ec438a6e66d2f8f3b898ab396dc833501051f
5256 F20101118_AACBWM saha_r_Page_138thm.jpg
d6544ebe9d90936da40db0359ae1ebfc
403ff529ac92f1457895471137f1b70ff0f3fbd0
5316 F20101118_AACBVY saha_r_Page_043thm.jpg
de9321ace3fd44ef562379fb10b50d91
eadd74bc9877508e5c167595acedc93d0ef36444
80623 F20101118_AACASW saha_r_Page_044.jpg
495395b155a7fb6de053561d4b90ffae
ecfb85cea43a353627771ee2cea1b8f4d4a55dcf
992696 F20101118_AACATK saha_r_Page_024.jp2
26399b074602b3b7b2746d763a161dc7
cb27a612d7a162df75ea5f1471061515e33e83a7
17242 F20101118_AACCCI saha_r_Page_076.QC.jpg
ea293ca3f1893296926251a8e8d187a0
3637191bd41442c56a361c864c4689af34ad08de
5681 F20101118_AACCBT saha_r_Page_035thm.jpg
14fda22cbab006a293f2626ab00ef0ae
e63148739d8c2909119cf6db38f20506a1651984
6615 F20101118_AACBXC saha_r_Page_051thm.jpg
2e1e0581912e083178c37b3bbf5a5da4
e2b932c45669902b5cf267c521a99ac6c4e7635e
5396 F20101118_AACBWN saha_r_Page_068thm.jpg
5d4ec701ea8bafaec0e7783940a2b2a4
4376f5bba84eadf43a3fc2c5d207c27cce2267af
22912 F20101118_AACBVZ saha_r_Page_149.QC.jpg
124ed1ae2312279310ed3af14b1e8b25
6faa86da2043a9b931c9b70985b07f028da56b36
21612 F20101118_AACASX saha_r_Page_112.QC.jpg
ef0d0d7e75106e4aa3837d732aa574ac
bc503bf11bd6cff92d7f22e369ad0f4be1ec4b8c
19202 F20101118_AACAUA saha_r_Page_157.QC.jpg
b10086249fd028ce87a2034c7e07bd7e
798d6d4a5351760edfc86340cc5ed9a17b28cf4c
F20101118_AACATL saha_r_Page_141.tif
138cb1305320a3bf24cb0cdbf4427694
9bd26e29d5c4a5d6d1c809a14ae6570b7ab6ec4d
20733 F20101118_AACCCJ saha_r_Page_078.QC.jpg
c5b09018e83ca48217b2cb28389dc633
45b929c9ab5e43f044fd53ccff253e747cf84b2e
6032 F20101118_AACCBU saha_r_Page_038thm.jpg
64f3de39ccdb92bdc90e56273228c5b2
de41397876d9f90732c3b4bd364f83f7c8199803
14859 F20101118_AACBWO saha_r_Page_165.QC.jpg
8b30e780c8d709c365b78eaefc267c6e
0ee8ca2624cbf279950cb1fa87412c18f839507b
20797 F20101118_AACASY saha_r_Page_025.QC.jpg
2a7c503a1c828c08282beb9ddcee9818
c8672a8092580927dd55e8fb085a19513701677f
1894 F20101118_AACATM saha_r_Page_177.txt
6f11c7b505f1d40f137f244539adf7f8
754d6d54ccffe0380619d4ad4589fe4d95cffb5e
17040 F20101118_AACCCK saha_r_Page_081.QC.jpg
7d454870135d82535e0bb1761b3b15c7
b86459f9ce1f048b7b431adcd8766cc0c6fad948
17688 F20101118_AACCBV saha_r_Page_042.QC.jpg
6f1c7f28fc8fd12dd4966d231bce05cd
765acdff08cca00118adb6b02d3ea9febc2ad902
4555 F20101118_AACBXD saha_r_Page_063thm.jpg
0e952ce340e42d3faef3567cfd15bd2b
bd21981b9bda79a6a956970d1d64c5d84ca08afe
19390 F20101118_AACBWP saha_r_Page_138.QC.jpg
e398f138817af51153a5bef860264c83
a035271740a29b511c835550e855093279978d2d
5139 F20101118_AACASZ saha_r_Page_088thm.jpg
d7b7ba94bf89c1223c612213a20dca98
a7d75c00c6dbc3e072bca730cceb4d35d6b1908c
70721 F20101118_AACAUB saha_r_Page_170.jpg
c578711dc2bc86c8c7613ceab913bacf
b680404b70fcdd7ac4231b5e1076d4715afb8f95
6228 F20101118_AACATN saha_r_Page_022thm.jpg
bad9d8f5237c54efe803cd8b8fb19614
efdd7a55af3c332e64b69dd0f4a47759e5439458
4606 F20101118_AACCCL saha_r_Page_082thm.jpg
02881544a4f40b67acc870b8d5cfc53d
87d563a834dcecf8caf9bf0cd283264ff1bc28b4
25770 F20101118_AACCBW saha_r_Page_045.QC.jpg
c97c6ca72cf3d5c7c251b18426369cda
f704f96db79e7b9c64ba8215050d54de813bd307
5802 F20101118_AACBXE saha_r_Page_104thm.jpg
2f767409519626b916e9a8ddffdcd713
aa1ff24876fd5018eacd6a171d73569893061316
5058 F20101118_AACBWQ saha_r_Page_164thm.jpg
1fe383143312ae17111ceb43671f6eb6
d04bfb9fd72e26e2865c86e924e14247768d38bd
F20101118_AACAUC saha_r_Page_027.tif
fbdfd4dcfa6704c6a665ec51bd1c75c1
7e0508fd30c918d8c75a99692e97fbfee6cf27dd
38706 F20101118_AACATO saha_r_Page_036.pro
2756f0d5f6d7a309d336db525af5dfd2
bafb3e20f1b3fbe1e8dfd9473dc42d703b92f01a
6443 F20101118_AACCDA saha_r_Page_109thm.jpg
6b1702afb352ac08d6478df538f50fe0
37a9964c774f5ef1bd753ce0d7915d4d0cad3dd0
5542 F20101118_AACCCM saha_r_Page_083thm.jpg
79d2d5a1d9a45416d24ee23e97ad21ed
07ecac39ff3b7e47c13a24c33fe32f7027afb3e0
23836 F20101118_AACCBX saha_r_Page_046.QC.jpg
584729dd11f6fcb236741b5f87958b2f
f870934479fefcd498d5ce823658e812a4ae9131
20502 F20101118_AACBXF saha_r_Page_085.QC.jpg
245e7b381aa88bf23afaeb4310e0b694
1f8a6f808561b1f77c0ac3958ddc2f346f765834
6287 F20101118_AACBWR saha_r_Page_055thm.jpg
00d43a65975b86400dd5f81d13346456
62ab3df2ddf64f13a8822b4a3fd898dc2f12828c
41050 F20101118_AACAUD saha_r_Page_064.pro
0d005e0c650117f41f50e96c45a4e88d
72adef3e6843fd571308d710a11fbf73999eca63
F20101118_AACATP saha_r_Page_034.tif
60f6917c6751a2287bfc9ee597b98dd8
92b1f9ba15c6002b2b1cff84ae0a5a370008ed2c
20296 F20101118_AACCDB saha_r_Page_114.QC.jpg
c0b994499f666608ef83b8f61f7aa026
577abb6222db836e32445e28563a9c67d5df9e30
6058 F20101118_AACCBY saha_r_Page_046thm.jpg
cdff78a84021cea41e589fa56be9cf54
ed8da97564631aa3ce50daef3394ea769c9e5785
5090 F20101118_AACBXG saha_r_Page_049thm.jpg
1db13f00c631724c227bee9df35fdbf3
a0b8182b5c850319f6f34799a1239044f226efb9
19410 F20101118_AACBWS saha_r_Page_007.QC.jpg
20a8b6cb731433388a739ae8fe0bb09a
2fccf8f3fc24e26342732f1f4d4826166a2c574d
16177 F20101118_AACAUE saha_r_Page_053.jpg
fedb875883b9da1420122623a825d328
5e230d032a473a682c383431aa8ad934f0305bf9
17074 F20101118_AACATQ saha_r_Page_049.QC.jpg
b6dc229dc999c8b22f7e2f711a5c197d
13e3c3065bcfbea2b577ff4c0b4ad307f2a9e5e5
F20101118_AACCDC saha_r_Page_114thm.jpg
2d2488708a845db2114e4bb26c9ced5d
e4bdc77bcb92c59fc16d07ade2721eebb928c152
5523 F20101118_AACCCN saha_r_Page_086thm.jpg
f37135af9788e7e873e771375a7508d2
aab18bcbedb199d8b22792803680bd5d452c5122
5394 F20101118_AACCBZ saha_r_Page_053.QC.jpg
f0ffbf0c8b7bd4b83ce12bed30c03067
9532b1b57ae2297473b679253bf887128c000410
23930 F20101118_AACBXH saha_r_Page_124.QC.jpg
456e06d136ca640bd6269dca8372cbf5
d1336f4917d8ff2d7f55200a49218c1300f7c754
5797 F20101118_AACBWT saha_r_Page_106thm.jpg
f3a5a8cdf8a70aa91b1c1cd7c47a9286
173050ae0ff3803293f6c557b53052b5bdc6236d
22989 F20101118_AACAUF saha_r_Page_054.QC.jpg
56cd325879977cab6fc7ff954aae5162
fb8e8d9e5c14f06235aa7ee95be763066a8a2c3c
F20101118_AACATR saha_r_Page_110.tif
85895aa5607638c6ace8f3fb3e69a847
fb2a7cc85c8476e085494c3805e5f399c84c539f
F20101118_AACBAA saha_r_Page_023.tif
b73e7e3cda565a058472f1aeeb21e26d
b9c6ea61d236ce9287f574c8db1c01ae6bdbc254
22445 F20101118_AACCDD saha_r_Page_115.QC.jpg
f7f041ce03a22610c8ad97c6d0d5763b
13da1ddff027028ba5696455c664d0c069c0ca75
F20101118_AACCCO saha_r_Page_087thm.jpg
0544ffed646af8fddcc63045ec34b840
3601423c83145439c0e4451ae7b0eb2669897b4c
3705 F20101118_AACBXI saha_r_Page_155thm.jpg
ba68be5c53bfdf230a56a3fbd4bdcddb
d2a60c51f555c4bcaaba19b36a67b36f4057f86b
4688 F20101118_AACBWU saha_r_Page_125thm.jpg
9f0774866c3de7d81d10ca6987dca067
1852d964d42d4f20ff8564fef8eb978911419d2a
80764 F20101118_AACAUG saha_r_Page_096.jpg
abbaf50400f9296a8c2658627281c629
28643d435b850706853a8d96089f86b2598d1e3a
24344 F20101118_AACATS saha_r_Page_015.QC.jpg
5fc1d09ea3a3ed5d73c4bb4660c5d315
689bc0d9af48d78ff77347ebc951c253e65cc720
17896 F20101118_AACBAB saha_r_Page_090.QC.jpg
8679f8a0ce4e38efb7a5dc3b3a5eec57
03d988b6c20da2dd33c0f0b6093d0f93e3e7b7de
24199 F20101118_AACCDE saha_r_Page_117.QC.jpg
cea197a1bb4d9fd3dc8222d6724709b2
f92c0a9a1c9d250fec516c27188d4a544276d658
5367 F20101118_AACCCP saha_r_Page_091thm.jpg
aca4c8ef758eaec4dd77601cb9fb210d
2a7989a8391f065d017fa4f3732d9df9b2072635
20379 F20101118_AACBXJ saha_r_Page_071.QC.jpg
05e3751f771d07fe927aec7e7506e502
e258766cc15136d36fc3e1be09a9494ad1708a55
20732 F20101118_AACBWV saha_r_Page_137.QC.jpg
6f2303d743aac3d8f2c28f5fe5591f64
8282a1012ea574e45787393b2aea868256151519
1859 F20101118_AACAUH saha_r_Page_010.txt
daf6c4591cc09f95881b1231615ebf77
8b4c17ac2c1708de394debac259b61dbf73b2946
66768 F20101118_AACATT saha_r_Page_060.jp2
703dc8b842f359343abf1ba979bcc2a5
6729bc0098615c118dc99108b80a8006d194abd9
211102 F20101118_AACBAC UFE0015302_00001.mets
f8905c3aac103847c504e3f7a5b32b16
51c8e40e5e65adc1b40ed81bee3b843afb902972
6169 F20101118_AACCDF saha_r_Page_117thm.jpg
3e713eb59b2df0f87e78a1cec256d481
7fa3ebeb4aae540b481b568ae68309a9ff7d5fd3
24753 F20101118_AACCCQ saha_r_Page_096.QC.jpg
811c4f8f26b57ff0077a3ddbb054908a
9adc4c97a2b33c58b8d9cca25ec0e13f77cbae77
1316 F20101118_AACBXK saha_r_Page_003thm.jpg
a7777b18e968750caad876e9b9864ff7
87777cbf9b2e7a71626743e06dceebc164e4b707
16691 F20101118_AACBWW saha_r_Page_167.QC.jpg
f2296f59bd08f32ef2dbd1d574b00759
1b01f689333b08d3c567dcb0a73e924d39d91598
2867 F20101118_AACAUI saha_r_Page_008.txt
c8c8a4e245bda0b487600ceb97ffb268
c6a33fcc08dbb8c96a076f87541a40d3f0d6c58e
26093 F20101118_AACATU saha_r_Page_041.QC.jpg
7b7ebefe2aaa41a29719df87b038a318
bcc0d82cd7a3a8656384fb58b754f60eb7268dab
F20101118_AACCDG saha_r_Page_121thm.jpg
f74a5d63b79115102377a3504817cc4d
19f8ada94cfa9f8c09e9ea6c9808d8134dd71542
18853 F20101118_AACCCR saha_r_Page_097.QC.jpg
9d1bb5b63db3a6409e9ada5175a12308
8661ad20c617a0216fbcf69d3f53a83d77fb5cb2
23485 F20101118_AACBYA saha_r_Page_178.QC.jpg
dff5db906cfffe196dd6c343e703826b
a57ab76036528cb8df15d0ef70643cea3cdac2c6
5141 F20101118_AACBXL saha_r_Page_140thm.jpg
0ef7bf54b1858eac6abb9e4a8a936268
716d12c63fd832bf2f6e6802f0af8f3dd6bf5af0
5415 F20101118_AACBWX saha_r_Page_057thm.jpg
ab37411daee15b919cf06ff7846622bc
ede028789f13f5f65a85656b69053c69ada02300
917086 F20101118_AACAUJ saha_r_Page_122.jp2
42a282a02d91bb04a8ebc4dde718dd6b
9d18f14136bbee96d1a0e1910906c52e8f27f5d2
39356 F20101118_AACATV saha_r_Page_138.pro
8e3c953016dad228cf7cf8577a878c2d
a2d93fdc440493d40a15c0e9c9b9dd558cc4a571
21899 F20101118_AACCDH saha_r_Page_127.QC.jpg
0d3f9c9c38981c5c31756f88694fe1ed
c5b9c56103f69adfb8a3d9c34d07d8d689743306
5886 F20101118_AACCCS saha_r_Page_098thm.jpg
dfd18dd0381070fbb607437bb2e7044a
49e3f57b9bcf797e3f6f1e7419d0562dd9e8d1d7
4901 F20101118_AACBYB saha_r_Page_167thm.jpg
d6786d35c2943c17f17275fa0743ea13
08a029338825c68a1b32944fd9778307f5d1ee08
5893 F20101118_AACBXM saha_r_Page_084thm.jpg
79104a07a806b91ffde84d11f1985069
cafb798ae6118def3aa1179587ffeeab803721e4
17429 F20101118_AACBWY saha_r_Page_148.QC.jpg
fb6557a857a160fa873f5ef52c3913f6
13d2d8af7b73f52077911369b7701c1c6405e248
60924 F20101118_AACAUK saha_r_Page_138.jpg
4501a5852997e08eed69dec0d3bb2c00
ba25e35ce4131e067184dae01d836a8dc11fe32e
717489 F20101118_AACATW saha_r_Page_042.jp2
d14d9ded91a44ce74becd6287de5768d
e858bd8386d9fb3b422b0f14f0af01348a5c50f8
24204 F20101118_AACBAF saha_r_Page_001.jpg
08ae4c293295ef36d5f9e97bdfa6c70b
a42304a90712d33ab377f1a8605cf60fd2289cf1
5270 F20101118_AACCDI saha_r_Page_129thm.jpg
8dac385a7a7b3d01bfb88e8e53bcaf0c
05877de5ce8c641709af89408218589d1fd71eeb
20619 F20101118_AACCCT saha_r_Page_099.QC.jpg
4e91fbe05d06e46f75280823f7b259a4
19e04b8e7177ff3c0448c7cd4ed47905ccfb16ec
3113 F20101118_AACBYC saha_r_Page_002.QC.jpg
6c3506d7b0798b6fe1ca582e0ff7f518
7463e3835fcb1dc3ef6630cc460603184a9f653b
4383 F20101118_AACBXN saha_r_Page_136thm.jpg
45e62bdf83050fd4a327658bd920603d
33d542facd20b905ba0554af127b6dccb69eb754
25222 F20101118_AACBWZ saha_r_Page_055.QC.jpg
ccf00fb47055eca72df30b5f4f2e631e
e2d3b698bd5a21c0819b3e89d0baab627786d4e4
85151 F20101118_AACAVA saha_r_Page_179.jp2
0c8f101c5fd122f0b138e9da784cc75d
1b67884e2b79f25deb2c1a067e1edfe820e8d2b8
F20101118_AACAUL saha_r_Page_168.tif
f9b3d5ffa4c59f0aaccbafb3ab5f6f86
75e53012e6b84722e38bef772400d72b3fa55d7e
F20101118_AACATX saha_r_Page_067.tif
77b16b5039c36a12227d55f596254c9b
c808ce84b62d3531fccfbde1f14f55be3d2aa432
10099 F20101118_AACBAG saha_r_Page_002.jpg
b40ba2e2fa1e6c595bbe4f56ed17d185
71f2cba700e722c244ca04e16c1ba2e2672fda1a
4714 F20101118_AACCDJ saha_r_Page_132thm.jpg
bdb55d43f9cd1326962280a29ee20002
421a430eeec630edf77084f16ceb631a214a5729
5636 F20101118_AACCCU saha_r_Page_099thm.jpg
864ee8c82bcc1f2305e191e974496844
70163a5498bab7b7dbd37d8cb98d28731af62045
19399 F20101118_AACBYD saha_r_Page_120.QC.jpg
4301b5f8d14b4e042ce32db2eb487055
f2c9f7d6f0f5370593f27767a4ebf3d6f31a74e4
19637 F20101118_AACBXO saha_r_Page_039.QC.jpg
8b3621538b32fcf85df8ff334485ec79
b7c7fc989b1e59869691875988e0d32c8302edb8
21153 F20101118_AACAVB saha_r_Page_183.QC.jpg
d8be6dc69cb8e01d1e9d4c3e43fd94d6
40ff6e7ccfcd6c92aae51c2663f29d8cae269f00
F20101118_AACAUM saha_r_Page_146.txt
ee0cf6733e415b410183778b6fcfca25
bc03555149c8344254e7bdcea8920187c6527626
678449 F20101118_AACATY saha_r_Page_140.jp2
6866790fcc2b114f4315bb2df86ac094
ec246258e6cddc6b01d1fbf4c332bbbd6efbde6a
9238 F20101118_AACBAH saha_r_Page_003.jpg
40ac754e00af195a70d4b1c4a20ab559
55a210ac1e6009d62f1f35c6b786a7a4c8648467
5252 F20101118_AACCDK saha_r_Page_134thm.jpg
0f99debe23c6b589bc0f90561fae1760
c5ccf870d176cb6277952fac8be7220222bc5437
5431 F20101118_AACCCV saha_r_Page_100thm.jpg
ad9a19b8580b7f3eb49107f96670568a
5ecdeec76c13243b47ccd0659936a2912616d088
6163 F20101118_AACBXP saha_r_Page_094thm.jpg
4c91417f32d5d43b68d36ff7b37447e8
ab407f7698bceaad7e68ea3a9e9e13d8af248195
3604 F20101118_AACAUN saha_r_Page_185thm.jpg
06f35b4faa4052a552bf75044eb3e0a8
1e2e19f74336deb4e71ed6359c4c69e7dc9c305f
5242 F20101118_AACATZ saha_r_Page_169thm.jpg
59fe10df75e2570f1fbbafbe799414cc
a34d0731417aa3675d587b127d61b492f8d10de9
47265 F20101118_AACBAI saha_r_Page_004.jpg
2cceb849f10a5622f28e17e599001e8f
97d0dd130bc34df38ca3f1c8b1cd725adbd63a66
5480 F20101118_AACCDL saha_r_Page_137thm.jpg
2538a2e1e670e1b4411363a23902fd46
45673f3339dd4af01c5c7d8bc36aa1f11e3de889
5890 F20101118_AACCCW saha_r_Page_102thm.jpg
2bd72aedf890674f1215c8bcee85d396
ee577758e420184121c96b16453c06b03eb6562d
23464 F20101118_AACBYE saha_r_Page_094.QC.jpg
1707ed83713b059245a4340bcd694cac
32c3987ff0e0eec406a63751f11415ca1b2ee666
17701 F20101118_AACBXQ saha_r_Page_037.QC.jpg
fd378ac7358f7df4eecd7983b8e7c5f0
841c520d3fb107215e8ea7301fe2a6e2e0777bf3
F20101118_AACAVC saha_r_Page_005.tif
c8502e393cc1aeab1b7c017375e5a5d6
a18135070994e9e8efa6b830503a629a357de6bc
61296 F20101118_AACAUO saha_r_Page_159.jpg
6fb181136d1c25864fc120e15a5be42d
ddf62d2b928d15476ec1542d174f423163f3c095
75523 F20101118_AACBAJ saha_r_Page_005.jpg
ae1ca6014a68859c20557d3b3d7902f7
079513e6f3831e3a86993fcbe69809bf343d5f0d
4953 F20101118_AACCEA saha_r_Page_179thm.jpg
b27bb0a068e1f1bf20cff5d844cd8a9b
17f4a8cb50a8d806956ebffe147c83c9fa7464a9
15157 F20101118_AACCDM saha_r_Page_143.QC.jpg
62a00af54c5037d532282dc73ac3abd7
5d55ca2ce29ce97a201d1bcc049aa8743c9cb074
18352 F20101118_AACCCX saha_r_Page_103.QC.jpg
dd7da39a9f1d5a5ffa606ea5438e59d5
c49e1987e5ba73e84dd4d42f2d73cf02b7e2621a
6024 F20101118_AACBYF saha_r_Page_149thm.jpg
b6f42b13df478b20fe9b515d2b347245
ba3874b1ee9a973639866db59660f84e9603fd42
4717 F20101118_AACBXR saha_r_Page_184.QC.jpg
6b9f6aa089ff4fef4e2b843d6ebabf03
b69039484f204b76e34ab73c00e82aee09a463f2
55834 F20101118_AACAVD saha_r_Page_148.jpg
8ea2feaea5669e1d1daffd9418ef3013
0e30042595577a684b1b35d08455022fc15bcf95
F20101118_AACAUP saha_r_Page_166.tif
9621eff5ec6b69a19d114dbc6d95e423
8549e2cbc62628b355ad568b88d4a8b15ad9d343
59918 F20101118_AACBAK saha_r_Page_009.jpg
cd5ed0d96498f895763765bfc0c54e8b
48e6226c3a0d0cddb3dce6becbecf6a910655ed9
22257 F20101118_AACCEB saha_r_Page_180.QC.jpg
bfb88703ab40fcb85fdce44f30c64ed2
306a414a6dc3678651417b1e4f5846cbfa16b01e
6383 F20101118_AACCDN saha_r_Page_150thm.jpg
99b57c8fc6a2be34599fe596500e53bb
4dc9d60d452632a42d12c5043f53fe92bdf4a0f8
4973 F20101118_AACCCY saha_r_Page_103thm.jpg
caba74b53b8e37735664997726530a1d
98b46a1bff90025be8dd8b72f7ba5bb787fed89c
20694 F20101118_AACBYG saha_r_Page_040.QC.jpg
025abf206292ea5263b0ed7a65f89d20
d138b99cb23e0607d94c8b737b98c74a6557ab49
6524 F20101118_AACBXS saha_r_Page_041thm.jpg
cdf7ff073742d8d0c037788c36b20722
15519c03a7f6b4a0788c34d3c769edb3cbd57cba
1051985 F20101118_AACAVE saha_r_Page_021.jp2
390c4694614e43ffec6c95ec9631bbef
724fa9836aec289a4bb4dca4a72afafcc924094f
21282 F20101118_AACAUQ saha_r_Page_092.QC.jpg
db32fb5cdffeb226ec7e21a41c446836
f648ca35d32733f2bc7fb2dc39399fa3d51e8318
26831 F20101118_AACBAL saha_r_Page_011.jpg
fb85e5542c5135d0ce8adc7b9e3bf0b2
7dd8547da20003065fedad91566020694f453a79
20322 F20101118_AACCEC saha_r_Page_181.QC.jpg
072382062f15d403b884c9b7f4e2a15c
42dc7bb7abe980acd55526977e06f39b47bb0b0a
26358 F20101118_AACCCZ saha_r_Page_109.QC.jpg
30d181e3d586e17aaa17b4e5e59bf9ca
fd07dd5b04f00f40f6a1826120703e94738f6f2d
5639 F20101118_AACBYH saha_r_Page_119thm.jpg
eb7df456af8d5020896ab05e8aed7478
308ae329b00491f4fabe8ff2add1d285af1e6773
22527 F20101118_AACBXT saha_r_Page_113.QC.jpg
aa1664233d52894776db7525dd2134c7
3c1d7feecb86659a8529ab9b368348ce910f2951
1064 F20101118_AACAVF saha_r_Page_185.txt
f9973866837d99f994d03b59ed3587ca
4f8d024b3e182b24be8be6c1e33b19f90cb9170e
F20101118_AACAUR saha_r_Page_158.tif
81748cc612e0113a404300bde8989866
790f77a2bad81aaba58b7d76c102fd9c0d5a85e8
62925 F20101118_AACBBA saha_r_Page_035.jpg
9b28f6beefc5f988a9fe66a8c057443d
3f5b441a5fa533a1b5f13797086012bef201d044
1737 F20101118_AACCED saha_r_Page_184thm.jpg
a57400f495d4e44b3ab391dc54ba76fe
255625e582b6ab9814a1663a52606c1b9c4a6e24
4449 F20101118_AACCDO saha_r_Page_154thm.jpg
6478f364382d97be45d4930db1d01941
e7844f17c32ddbf249423a1c272ea3a425ce2de0
5670 F20101118_AACBYI saha_r_Page_182thm.jpg
47be7aa923247c5046f761cb3c56f95f
d3d769f2da61656f4a9ee62a350de4a89a18b1d5
6756 F20101118_AACBXU saha_r_Page_008thm.jpg
659a9f8efae682b4da745fcb07929c8d
776df3f2bdb98ee2689b9c422b29f6a1761c3978
71871 F20101118_AACAVG saha_r_Page_019.jpg
1c5c06fbd7814e1c50ae2ba67d6a3535
3b84571686416a85256733e81eb900a4538848d5
727442 F20101118_AACAUS saha_r_Page_167.jp2
364bb9daff915b876b49dcdd009bff9d
94446b20861c0f82cf584bb53a762aecbbd3c85e
66010 F20101118_AACBBB saha_r_Page_036.jpg
25f6589b0363ee7a44be2784ff6f59ec
7fd86ffd87e5f366983ddcd4b0f33d38a8c69820
80305 F20101118_AACBAM saha_r_Page_013.jpg
5918ca8fd7f6372211706125f5f45ac0
72dc86e4b9b4757051a2edcc791c0721f08f311c
5470 F20101118_AACCDP saha_r_Page_157thm.jpg
a9d4a01012df9f7872afa9f14e34ac97
03cee55790660e78faa2675cf0f4dcc1428aee4e
4915 F20101118_AACBYJ saha_r_Page_012thm.jpg
b44d9ae9c2554423f02ae8ce83f0e9e2
5c98cb0e9808622904a994ef6dced39b8e66f5e8
22201 F20101118_AACBXV saha_r_Page_019.QC.jpg
2d8f5bf4b9f0bd11e58be6e75addf7ce
6e29ae77b6b4c6d45d89d64fe90d7145f8595415
42452 F20101118_AACAVH saha_r_Page_085.pro
ebaa01655213bb6b7326c0d920b46236
8119f93bdb2b50e3a537dfe374df67299213ac7d
1540 F20101118_AACAUT saha_r_Page_159.txt
650fdf4aac40d8e754cd7ed08385593e
8aa5aaddcfaaec1997b0ded68bd10cd8b772385e
53798 F20101118_AACBBC saha_r_Page_037.jpg
2d85a59fa85d8f1274889297de50c1cc
86c662b0813ad4cd785c927c392703037871732f
79687 F20101118_AACBAN saha_r_Page_014.jpg
fabb45996719481f77666ea5b8deccfc
48c53f7c49bc69ff0665836248049abcff1ce19d
4641 F20101118_AACCDQ saha_r_Page_160thm.jpg
f16dbdca6558d99c0b837fe3e058e6ee
d590ff38f94d6454abd1df3f92d2862642fa1c17
5147 F20101118_AACBYK saha_r_Page_007thm.jpg
5efb25f069025d6201bde5766b8c1347
54c8c2970646766b95579f67830a0b007213a9f6
5422 F20101118_AACBXW saha_r_Page_145thm.jpg
da3153f07bb2fd4fd9138cc90a5a9d47
8f2517ae2552576b40eb3bbbf253f9d8acf76dee
16012 F20101118_AACAVI saha_r_Page_082.QC.jpg
3bb0597b926c2f39954638674c5e120d
74ea06c10526d63fe76fb4f2c236ef74614d749f
F20101118_AACAUU saha_r_Page_082.tif
21c9fb0c0cf0a35f44e0c5873eacf642
a99cf9d5b2a13390dda0f156c999dfec7e13ce09
62668 F20101118_AACBBD saha_r_Page_039.jpg
b583355b1d24606559b925079aa1ad2f
3bbcb3f771ace03a105c999a22566a2cd536f2e4
78103 F20101118_AACBAO saha_r_Page_015.jpg
0359a04f66e7632d7722905211f66590
26f90e947f2b3f639c411f4a590cf4b29324cdc4
4588 F20101118_AACCDR saha_r_Page_163thm.jpg
68d6f6317b1e5cc6b8362a8e7e4a84c4
8a7be94ecc09958fa6d7a2e79bd5e54bbccf04ee
24025 F20101118_AACBZA saha_r_Page_030.QC.jpg
96c233af62a54aec83bbab919b8e6b5e
276b2bdf7e4de5d46ecc9d078a1788dd9bfae5de
20254 F20101118_AACBYL saha_r_Page_145.QC.jpg
f81069730bad18a344f1724074fae7f0
5e8ed06cce323a59568cc4983f94fd7a048f7e05
17765 F20101118_AACBXX saha_r_Page_059.QC.jpg
37007ec858e69a48729aa47751cab20e
16e0a68596a9797a9c9d0f6cece6c3e17be192e7
35161 F20101118_AACAVJ saha_r_Page_011.jp2
b649da31d88e577e908c370c82055149
2f6acb3c12cedad89312447797bd24e7778c732e
F20101118_AACAUV saha_r_Page_109.jp2
28b89b07a23a09181372ee68b4d96241
c5cce42d1c6be7741d5c158e91ea396ea5fd4c40
66665 F20101118_AACBBE saha_r_Page_040.jpg
58c96d1953473ca2041351ada836146c
c908fa42ac0827334a8e3d96ba4f76594d748d46
58741 F20101118_AACBAP saha_r_Page_016.jpg
8112232e85c05848683b277bdaa5b3b7
19a7d05e23b2ff8dbe1932f0fa69ca3dd0303796
16673 F20101118_AACCDS saha_r_Page_166.QC.jpg
e601ca9ed8fdb70f19155bbb6c639de1
658c76ea64089b33bafb021b225c87070c460c5d
F20101118_AACBZB saha_r_Page_177thm.jpg
773c67b6515e09dabf0b55152b318d4f
81c2d2956aad4cd984170614eadf15cafbfe672d
23066 F20101118_AACBYM saha_r_Page_147.QC.jpg
012517709f3d7e3e76ad2f573dd98940
c6342e6f72833a879c9a06a91480733ed2e0bfdd
5736 F20101118_AACBXY saha_r_Page_047thm.jpg
e52cc8ebdaa9d8a22ada71817b28ff77
58ac60428cc39e351d845592e41e6b9184e2cee3
F20101118_AACAVK saha_r_Page_063.tif
995448c1e63af782684b4acd0ef8738a
697021f9480a44e07dd4e0a29bef89b9fa4ba302
23081 F20101118_AACAUW saha_r_Page_017.QC.jpg
f51c6946c4a6fad9c1cc04a21f4e9556
2b279fbd31247197f5d9e5dfe2038a8dcaebce12
60584 F20101118_AACBBF saha_r_Page_043.jpg
565cc0d85102bcb9e2a5c0c994c8a023
1cf4d0ef319c5713da42882dae57b91670bdb109
49550 F20101118_AACBAQ saha_r_Page_018.jpg
32045dd1d318db5cd9fe5d37cb320618
f9bb679a62698e7a2e8c5fced1184efe8be705d5
4957 F20101118_AACCDT saha_r_Page_166thm.jpg
43ebed40feefa311112194d06468cb6c
dac84545f46c5cb3834f2395fb52e7319d3234d7
19241 F20101118_AACBZC saha_r_Page_174.QC.jpg
5b6752f44248273d57a15553c1c596e4
f4288055e26262e462962fda39addb61c58fc0c9
18654 F20101118_AACBYN saha_r_Page_012.QC.jpg
bef647628e8c15b1d245a07869fe3547
046ee2a60b106c38bc7bf737c02e6fbc748f32e0
6202 F20101118_AACBXZ saha_r_Page_096thm.jpg
6a0f3b48434c4359137a31d14899e9c9
4c6e718f5d1336fbdd2601c25f85bb1551bb86d8
21357 F20101118_AACAVL saha_r_Page_061.QC.jpg
6e601dee19479d8618dfa97028199f16
324e19fd3304cd480d88004a26865df4174de1fe
20536 F20101118_AACAUX saha_r_Page_086.QC.jpg
d1453857b52a47d3c921aceedf63891c
15f0bd5a3ecd4cb02b72d629a7bcd9f7a20ba02f
43211 F20101118_AACBBG saha_r_Page_048.jpg
b4c4c140cdc46db284752e1d1b3db7e4
c0ecfe9822cdcdc79382edde448363a1e65f230a
5640 F20101118_AACAWA saha_r_Page_085thm.jpg
8c41b7c83cd6970f296a4466e83ed797
6c0db12367ba824fc756e8f567bc3ee033eb2b10
76358 F20101118_AACBAR saha_r_Page_020.jpg
14aef360b53ca415d339e46d9211c773
7cf9821d905d288b01093a3aa2132d23bf1ecd18
19952 F20101118_AACCDU saha_r_Page_168.QC.jpg
a27729bb6dadbcdbfe7d20290704557e
44e9e7481c660717251b1dc2162bc88ce79163d9
6365 F20101118_AACBZD saha_r_Page_107thm.jpg
c3989091d23493b1eacc0ff878411c8c
7f0b9e8a26891806566a40678d6bdca73732bf4a
5968 F20101118_AACBYO saha_r_Page_152thm.jpg
fb49d0e8a119f62e0a396c24f57217c4
6d14ca567981fec18ee1325e0252eacd274e792a
48580 F20101118_AACAUY saha_r_Page_115.pro
1e09be87f673332e1604374bae47f522
8fec8cbcb18a5cbd724106f382a386ec60c60b3e
59963 F20101118_AACBBH saha_r_Page_050.jpg
c3005bdf337d0f3ebf40412f6a722649
f38b6e482a245cc994dac7ff39590f17409c0ee4
F20101118_AACAWB saha_r_Page_126.txt
67b67984d8d8c322e709ca70995866cc
40f653f9e2b3814234047bd5ecc9743a3fa2a601
83101 F20101118_AACBAS saha_r_Page_021.jpg
ea0660de08198bf959e50eb36dbd8939
c5aa2bf150c22351ff2fbe6d01cd4c838754bc3a
F20101118_AACAVM saha_r_Page_001.tif
2a616751f91c49d9580a456ef9d400d1
6b4ceac2a892ada27fd2eb3cb7d17872c3c70072
17404 F20101118_AACCDV saha_r_Page_169.QC.jpg
deb723f6361f69970cba71422ae702cf
6ee196e0fe28709551e8de96e83ad5b9b63a9ed4
12115 F20101118_AACBZE saha_r_Page_101.QC.jpg
05e38544fc179407934f93198447215d
73634dd9d9ae7bda7686062e717f610fb4640f86
16215 F20101118_AACBYP saha_r_Page_140.QC.jpg
7fc0ca9c81d67f28317efa6f85016f6d
8c51c8fa356fe140dfd470aaac41bfdd8e69114c
1522 F20101118_AACAUZ saha_r_Page_016.txt
2e06fde4d316ef5386d58a865882a9fe
939611ef4f1757137c4207ca07db6eae32368bf9
85278 F20101118_AACBBI saha_r_Page_051.jpg
312daa87a29d7c964b07694eeea187c6
d1361bf56ca46986d7791f12887b956323053407
69672 F20101118_AACAWC saha_r_Page_047.jpg
ec4bcf5448267c86be0cd7b6c57b38ad
0f679844558b7877906efae30a4b10a3176dd9ac
82204 F20101118_AACBAT saha_r_Page_022.jpg
fe4e18859ed1eacf8f099c2a4893f16f
82f37aa9840634f133b0a68d40e53df6fcf7b1d4
48422 F20101118_AACAVN saha_r_Page_147.pro
78a725d0f8382a4b7ddef04a1662c0ab
8fd2c9809b46edbdc29ea9fae2986a3499210b3f
5416 F20101118_AACCDW saha_r_Page_174thm.jpg
2289344e226a27b6a9c74e40ed424282
8c615613e75e379b12baa76d644e6c4ef0478343
21589 F20101118_AACBYQ saha_r_Page_104.QC.jpg
7a2dd2e78c7bf7e4c0dc70da183067ca
fee65f264a6f7e0220381f73df814637615d8490
74730 F20101118_AACBBJ saha_r_Page_052.jpg
ea42bc79229706fccc3967483a57d240
4a6fff840d5bcb22eee28abe5a562fd4ba9ebd6c
68077 F20101118_AACBAU saha_r_Page_024.jpg
219e010dc66e300b7f8fdbaee75d4ac4
b9cae29d8e46bdb1ec247ab73de15e6f86a95dc8
F20101118_AACAVO saha_r_Page_002.tif
ef29ae0673422a7a1bf9aa32d90e1eef
402c40f5868387caf1532be9cda4dd0842f9af09
19180 F20101118_AACCDX saha_r_Page_175.QC.jpg
f1c8b80e5faa65115c4313576fa96438
7d73a7da1b0fcac661d264000934d291353df126
4715 F20101118_AACBZF saha_r_Page_162thm.jpg
efb77549c06fad8f713f11172ab76d3c
20441f55ef75bdbbc1f9d77b38d0412e8d76dd2f
18939 F20101118_AACBYR saha_r_Page_177.QC.jpg
ee7a1fd0e448d2d03311c3d269148e27
35c89cdb4ce4d3d48d0231c5acbfaa2bcf200490
73931 F20101118_AACBBK saha_r_Page_054.jpg
4ffd8dc78e5cfc3df0a91ccde43a156a
58886e2dc29dc4011f378a06ffff21be245df5c1
40263 F20101118_AACAWD saha_r_Page_185.jpg
b05daa49e9ec8446f19328f43370fe63
42aca54eb7943dcef8fdad35b61cc7cb2ed85267
63721 F20101118_AACBAV saha_r_Page_027.jpg
27581b587e19b2c4ba367fd6bbc47a8d
9ec56f6e93e5f4a2a97e7f1230340402b6489d0c
F20101118_AACAVP saha_r_Page_178.tif
4424c049747c7b54f8f2ec755e57d702
aebec48a06d671c98a976435ca30a259e3b8e008
9050 F20101118_AACCDY saha_r_Page_176.QC.jpg
95ded1ebfe19b113b1651a266702f332
ff5c4a7e837b7b71d2b2da6531152c34916d0470
5766 F20101118_AACBZG saha_r_Page_024thm.jpg
827ade9a4db87c1e2457af3713e27ed2
95f93ad0cb1b28a1d2902d60772add8b3c971723
20020 F20101118_AACBYS saha_r_Page_043.QC.jpg
297989d8669059c0ba4951886a5f244f
d27f7edde866ebee4c8fb15e62a73e3a153fb7b9
86200 F20101118_AACBBL saha_r_Page_056.jpg
189f7cebb995a212c953bf215874e197
16ed640d514dd4bea2417d4db90be4d6214a4e34
1924 F20101118_AACAWE saha_r_Page_066.txt
8a810d573f29c1d62db90a7286836b52
960ba5d10027fab23fcde962fdf8ba45442f6c52
77417 F20101118_AACBAW saha_r_Page_030.jpg
24777a77f19844605fec1f4f0babfce6
9659d4799edc5689c93c5717168fed45f5e22629
14612 F20101118_AACAVQ saha_r_Page_184.jpg
aca45c7ed50010c81f48ad413c14e0de
d4045b7199c7c9ad869d83647da9036de0aacf08
2628 F20101118_AACCDZ saha_r_Page_176thm.jpg
53dd1f6afd9aedfa0508be96b60b64ac
5afe8e6b6158da7376851b11ad3d0e0e0a35fa84
4621 F20101118_AACBZH saha_r_Page_148thm.jpg
5a88815ae9ac0861f9b2a0a6ffeeb7b6
2dbf965c3fdae468f65766a272bb49b2165b9128
21972 F20101118_AACBYT saha_r_Page_098.QC.jpg
f65e31a5c58059cc26e561b46d3624aa
c2e17559cd7bd302684430f7b8798803439fa983
76426 F20101118_AACBCA saha_r_Page_077.jpg
edbd7d7035ec43cc0124680714673d5c
b4ee6d32ed8328d4f13127bb37a3c33a9967ce68
65948 F20101118_AACBBM saha_r_Page_057.jpg
5e8e29f0559992b44ed71bbd9453d49e
53d45f33107ba708584cc84fc3cc0acb368f67e8
F20101118_AACAWF saha_r_Page_117.jp2
58a60be6a827aad2bb12903c1ea47eb6
0ef715633208eb9e7a1ddad0f43a956b3ce54216
72396 F20101118_AACBAX saha_r_Page_032.jpg
02c04cb5ed776411ca286878162f0753
a6c9225ddbb935fb35b8522908c88e09b24aadd8
23307 F20101118_AACAVR saha_r_Page_028.QC.jpg
2dd5f2a6d9643899eee9c7b42e4a5a2d
8cf536cf13ed2f79f5390213dd178cef45c19842
19121 F20101118_AACBZI saha_r_Page_100.QC.jpg
482b2fdc587625e08b8663e6d6159349
1e361645e7ec04bbbbacf77ec24ae8e822421c4b
4113 F20101118_AACBYU saha_r_Page_171thm.jpg
f2ca875ec521685c4abc97f0308f9806
b26accd578fabdb6feefad25bd014fc13299b867
63497 F20101118_AACBCB saha_r_Page_078.jpg
1d331b7d464c17afc5c59a5038298524
c2761a1f0c241dcec587cd45239f2a30ec58721a
30091 F20101118_AACAWG saha_r_Page_063.pro
ee64f3a04b44393fe1f7bf706b940c0d
93f05a27098857c84afabe85fdf191ff662fbaf8
58503 F20101118_AACBAY saha_r_Page_033.jpg
b96344a7dd64efaab5a9e3679c3d37d6
e0eed14b264fd13b60eba57deb36af6023a2ff5f
59932 F20101118_AACAVS saha_r_Page_058.jpg
f87ca81d709b2dacf5f2b4e84e9e562d
6eb94f44958325637bf87272e7d072497a4516e9
5549 F20101118_AACBZJ saha_r_Page_040thm.jpg
58c7430640f724c2bae9514b79bebe28
04c90abdb63833d17a4cd5d3ab170c977b00f2c3
26470 F20101118_AACBYV saha_r_Page_051.QC.jpg
389753b854932eadfd049c881d68ed54
4390810973415f2881ccf68034c3b07b38894c1d
64434 F20101118_AACBCC saha_r_Page_079.jpg
80baec5b832f10ae8ba162cf029bf237
53f7e15bb20d00b17ff60e37b9f3c85dbe360a7e
57181 F20101118_AACBBN saha_r_Page_059.jpg
e203e573af47047fcf9d500cad57b6ee
f3f6ad513d0a9d2887d01c5f5c139c181de68bb3
F20101118_AACAWH saha_r_Page_040.tif
cfee9c281ce91f0ea66112a4a728c3b9
83b6c8607be3da70fdb3409158f8a32e5d9bdf13
59641 F20101118_AACBAZ saha_r_Page_034.jpg
5e704bb9d6ebca93e00fc93ebb665718
8662436831e5af93cee2211001e31488896ca591
20461 F20101118_AACAVT saha_r_Page_079.QC.jpg
32ddf6d1fe3506826ad43e47f20d792e
5fde743edec524f3bad1b416d8a9491ade7193ac
14707 F20101118_AACBZK saha_r_Page_161.QC.jpg
c73b2a9d9d448861cd9a775a4c7c57e6
81ddee6c58d15f308b901f4ae3e82d039c49a0fe
5692 F20101118_AACBYW saha_r_Page_153thm.jpg
db6987420aa44179f6619a9fe8c0c741
70a49f4f385f62ef95e0b4a631c9ead05c30bfa0
53467 F20101118_AACBCD saha_r_Page_081.jpg
65c7e06a3249b84327d8abc95c0c3dd7
86db60afa245c00f87cf4f8d884ec846e76bcc4d
47029 F20101118_AACBBO saha_r_Page_060.jpg
ab5eeb49f1e7232dd65b4d6a765d3116
031c8e1973928a6410699d0ae1bdeb7202b3c805
F20101118_AACAWI saha_r_Page_006.tif
71c792c1e73f3d186f281721844f2e6d
633eeb7d3b8a75a70fbefc0b41787f1d28c06f58
32967 F20101118_AACAVU saha_r_Page_123.pro
f1f71e60896cce8b32b1fb6912e713a0
0e85be6d4cbf0bc4cbe5a9361697ce57e579225d
25280 F20101118_AACBZL saha_r_Page_111.QC.jpg
799ad40b29c5dba643080630c5ca240d
ca95f42596451956e758443f79304069d1f182db
6095 F20101118_AACBYX saha_r_Page_124thm.jpg
73d46993473b97d041f81d7916de02ee
e0954e3cc12eaee5f03bf52116c81364f6e1d025
65315 F20101118_AACBCE saha_r_Page_085.jpg
301af2f262403762bf02c12cdbec50a9
f7af165bf5a6ed45ac183f99a93908723e1e37ee
69445 F20101118_AACBBP saha_r_Page_061.jpg
fbc9aacc80565c5a1446cd66f664704f
d01f3a9d10256ce2a62a382f3ec1b8df95e974ba
F20101118_AACAWJ saha_r_Page_024.tif
4007c9afb82b2ebe9cad481733b3ebc8
d962301d8b5fffcc6ef42f691af15dc8be1cd693
988326 F20101118_AACAVV saha_r_Page_104.jp2
dd0f470bf47b3a3ea51ae72c5287d766
2c08368d06fcdfad305d078f0af12da02f504772
4935 F20101118_AACBZM saha_r_Page_062thm.jpg
83d827e3fee2fabe5a7dc155125429a7
2574e490556dda380f06c5245f0c73fd3b131d6a
5494 F20101118_AACBYY saha_r_Page_116thm.jpg
4804c3a5070b1908936c7b0280e8ead6
54edab1e80923492f2c17884774f50a326137f26
67376 F20101118_AACBCF saha_r_Page_086.jpg
ab2bdd3c080914adba016c70d762fff1
578e73d3e0e8b064208818911a0d2f6afe335711
54497 F20101118_AACBBQ saha_r_Page_062.jpg
169599dd053274883119be922f8f89fa
5e957f3847ef374e17477cf50d5948d6674b10f3
5179 F20101118_AACAWK saha_r_Page_159thm.jpg
778d5127d1f9770a1aa7e6fd9e5b3d88
842a2ef7f1ee1b0db4920c0582da40c540d7db03
4923 F20101118_AACAVW saha_r_Page_010thm.jpg
482dd95695c02e8e599c50f71da15def
bda4404934dc059265570a9e59794db87b955411
18176 F20101118_AACBZN saha_r_Page_009.QC.jpg
92135f9e62670ff6965725e4e89ed438
0a05516f6726c4b388ebcc713b2419f2e87c9db5
15542 F20101118_AACBYZ saha_r_Page_125.QC.jpg
e0cb2ba424890a21351a35b819e753b8
e4eb1a6efc3ba2e2ebc09d158e1501b4ccc58a29
58045 F20101118_AACBCG saha_r_Page_087.jpg
e537cca19d8ed615980cc7734fd92a54
6a7f7477b71d1ead1d6b55824da6b689142c48b7
20333 F20101118_AACAXA saha_r_Page_122.QC.jpg
6cf352724b233d419c8212273bdf98df
7570acb833a5b6b61ba0ad965de27d2fa2aeaee1
47753 F20101118_AACBBR saha_r_Page_063.jpg
22d82fd0b19d5190b9df25c97dc8ce45
75605d63a7934ab394cdeb9012b4e8be0241e06b
F20101118_AACAWL saha_r_Page_175.tif
13690d817cefeadfce85f7e8991f7a91
9524f6cac4257911edd27f172da0300097ada289
2208 F20101118_AACAVX saha_r_Page_029.txt
ed4d8259b77b45b89fadd2355c4b64f8
8db181de071578f21e79d7a770212c8ddf4c3f3b
F20101118_AACBZO saha_r_Page_061thm.jpg
59fa4c69899185739cea8e8a3fdf0bf8
0e0308b21291e4ea17e7c72acf9b60f165499640
61120 F20101118_AACBCH saha_r_Page_088.jpg
52533a4a73cfe5892b047704908e371e
0d82fa657f5fe3a686c6ab7b32275f17a6eb8e25
1043881 F20101118_AACAXB saha_r_Page_099.jp2
f580c5c01e1a2aeb38ac4c43de07e549
a725a306d060e77e6f21e614a4dd88c757436e88
60947 F20101118_AACBBS saha_r_Page_064.jpg
35be7539b3117df7a543b612c10b3aa4
9028c86e9361ecf249db2085c69133e03a53b952
81692 F20101118_AACAWM saha_r_Page_111.jpg
57117754d43685f65fd963cfcf32cb2b
fb6f8f2691c58a3394933f52d9a9893038e5d405
F20101118_AACAVY saha_r_Page_149.tif
aadabc55965f5a361f468efa8cd4dcb5
ae9fd7edaccc7b0f88c3ee64ffe2b62a42aeeb62
5606 F20101118_AACBZP saha_r_Page_146thm.jpg
72271db64a80de302afefc2f3cd651a1
46435d1869b1807f48092946af646d4efabb82ba
63872 F20101118_AACBCI saha_r_Page_089.jpg
4f13e4a34a5500b9d72a0c0c30ff1e99
52390dfe6b9e03dca387fbbb82be35b035d10c6b
71952 F20101118_AACAXC saha_r_Page_133.jpg
385dcd73eaea4472ec632acff55da112
5c8b76122c691d5cff170328e2ecfbce7ed4329c
68604 F20101118_AACBBT saha_r_Page_065.jpg
48e3771b6b3e670868660591f10a2a5b
170b82bbcf08e1687049802ee884e1f8890bf2d4
73288 F20101118_AACAWN saha_r_Page_006.jpg
67f77e62a01766050d5d2af56921fb3a
1fb0211657565eb17ee7d0062e5ed6942f47e57b
4964 F20101118_AACAVZ saha_r_Page_090thm.jpg
915b6d5f6ee1c8c92ec71ac614b8e922
66554d75268db65dac3769ef22135d03dbb324f2
4469 F20101118_AACBZQ saha_r_Page_018thm.jpg
edb638ef2e1b2abc0b5ec59020b638e7
7ca560374692fb68b9c6bd9fe5819a30eb2fc87b
60534 F20101118_AACBCJ saha_r_Page_090.jpg
06db6f1092695435dd217ad0904aa8eb
1de6ad377745b83e41cfe6c3282048d96db9de06
14920 F20101118_AACAXD saha_r_Page_154.QC.jpg
98630975cbc4021823a12224acd64213
992774142a06bafad9172cda13caaa59b07bc2f5
62783 F20101118_AACBBU saha_r_Page_066.jpg
e2f6ec26cb32bb5e615e4b32f5717bfb
3a77966addc56f11509aac670091168a1d02fb8b
F20101118_AACAWO saha_r_Page_015.tif
931e3d99e9587b8d53d8b791085afe85
47be67593f87158adf39c3a04924c4f73937a3dc
21897 F20101118_AACBZR saha_r_Page_133.QC.jpg
7c5f7f0b6faf1f8f8fc0a6dbba67867e
2be02f901a9245af92a50d7041e037fa340eb5e9
66286 F20101118_AACBBV saha_r_Page_070.jpg
2d03c66f590f231eb45383cfa2b6f8e0
b1d15f71253b038e1ffe4a9d21f0ef55ffdf71c6
789042 F20101118_AACAWP saha_r_Page_034.jp2
7e86f1c0a49e774fcc978d84da764cc6
d1573e581d88b3fec41e9e4d704d52471b768ad7
56919 F20101118_AACBCK saha_r_Page_091.jpg
5034ac130f3c982b4a88fe4547039e22
55bc7967b614519c2c70ac59c789943a0ca3fcb8
18975 F20101118_AACBZS saha_r_Page_058.QC.jpg
dbee791484427cbcd85b558ab8ebcc59
d98ee931199ea503e21965887d12b15005d41f30
76719 F20101118_AACAXE saha_r_Page_106.jpg
2fb11fde27711925b7fbb1a37586f70c
de87323fd1d625111a6d42e2a38a12ee06384fce
67490 F20101118_AACBBW saha_r_Page_071.jpg
d39408139504cb5e20bf87cb3f62da54
0625464a8ac85afaf03d47f0d42b3481376e32c2
F20101118_AACAWQ saha_r_Page_161.tif
a286e529838c58938f4836d86ff35dcd
63910f4ac7f74913a692f87c4a66dff3f6b725b0
65934 F20101118_AACBCL saha_r_Page_092.jpg
086cc76fced533f569739e1629aac83d
74595c6a1d1286fb2fcb9c8f357f920e017e389d
24694 F20101118_AACBZT saha_r_Page_044.QC.jpg
a5f7bbd05bfc5483c845cc432c02a34e
057cd265886dbc73713e004f42d1a23fb07c99e8
16807 F20101118_AACAXF saha_r_Page_173.QC.jpg
fe57579535391d3b83bc392430cba5c6
3913bf25a10c3edb0326e69eac5bc677a31e22ea
65289 F20101118_AACBBX saha_r_Page_072.jpg
f6e739329f6fdcbc5c255cc9cf7e91c9
43b6aa57221317a57cb9e1217516fa8fc63cbdd2
59246 F20101118_AACAWR saha_r_Page_101.jp2
59e87d316c14631a3555618b8a4c6cae
0dd6991ad79e07cdc1f9c91de851da096e674065
63423 F20101118_AACBDA saha_r_Page_119.jpg
92baffe407faba3e3209d5c0594c2d8d
060b1a50131d96da26373d80e889ed3a0115c3c1
79740 F20101118_AACBCM saha_r_Page_095.jpg
9a058f518db679addb5ecd37ff4d52ae
aabfd7600ab9f70dedafeee94b11fdc9849d0c0c
26368 F20101118_AACBZU saha_r_Page_151.QC.jpg
e4211744d2909f6d3286e9aee06ba68b
e7911a20f19fc91eb189404ef728126187e7baa0
F20101118_AACAXG saha_r_Page_073.tif
5ad34b88b42e37d1c11d65c96f7c6863
8e34d2e3e3909aeb1d5efb5996cb0e333db5ac7f
51592 F20101118_AACBBY saha_r_Page_073.jpg
74e02431bb01e2b9e8104ccefba62057
90f4bcd98ca769369fc441d3b5fd5abc53c14d97
15366 F20101118_AACAWS saha_r_Page_011.pro
7da87d167c2db33a342c54a0ac852aea
69bc37c4131687e2bfc63a65f2adcbc2f4e48db8
64369 F20101118_AACBDB saha_r_Page_120.jpg
bb4cd727b4aef478f46e7c7f51fe8502
c132a14153bd76f0dd68013b9afedce5962348bf
60629 F20101118_AACBCN saha_r_Page_097.jpg
fbcb754b626d9f06719d5d64a311e26f
f1f41885a2939553135c472e7f88b3bf706da0b3
5000 F20101118_AACBZV saha_r_Page_123thm.jpg
9c7236f82b9e4dd0be4c1201137566df
680ffc08a0d8b2bdb3be00db6a621077b176894b
20600 F20101118_AACAXH saha_r_Page_036.QC.jpg
08ffeeeacd2aebe511bea6765ada6b31
1078c37e11b21c17c4f1f19fb8d13442c3993fc1
54580 F20101118_AACBBZ saha_r_Page_076.jpg
0f22cc509efcc57b4c69450767db2692
5c2158301dcf1d711a0735a9ab56632ad698fb2b
1051937 F20101118_AACAWT saha_r_Page_041.jp2
bda30d8ed49ed56f62ed649970e78d1d
0b87b04af6682af82b33fc66a80ba4ff12e2f149
65487 F20101118_AACBDC saha_r_Page_122.jpg
3078cd075bd5592146b3c649247192a3
ddb56eb5b77643fd9ee6cbd4ff8ec9144eb4883a
F20101118_AACBZW saha_r_Page_053thm.jpg
9fa3471cfebec8acb76539e81ff9fa05
70248837ad11f12df756a6dc4138ae86cf1de1f5
28723 F20101118_AACAXI saha_r_Page_042.pro
458c0ea93fcd9f83e51b7f46567b3902
b0e61f72d84b65d3e21bb69f2048e4cf1208343e
F20101118_AACAWU saha_r_Page_157.tif
4a779395c89daf28432c0113f580a6ad
1cabbe55639c00f29a5d308087b3c55b66645be4
75681 F20101118_AACBDD saha_r_Page_124.jpg
a010d4d162f1c01fc43f7c79988e64ad
3d2490d485f78b6fb864ec660584de675767b84a
70779 F20101118_AACBCO saha_r_Page_098.jpg
95d79ef1f830b432d56d6578bc93adde
9edb4d673e393baabbeaaf0a4ff692ea11a5e9bc
2951 F20101118_AACBZX saha_r_Page_003.QC.jpg
49351c20d38cb2635144eb1e383fc25d
42b3dbde81077e739ae1c89713709f799816ec8e
F20101118_AACAXJ saha_r_Page_087.tif
10a88f27b5456ff5d70b42074528498e
32f09b116e590e0962fee7cb1bbfc5f25c46d94b
2002 F20101118_AACAWV saha_r_Page_069.txt
3e650446d133fbc5043b0219c3977ec5
9a4e81e8dc938c1ce4ac297fe7abbd49c5f73356
51219 F20101118_AACBDE saha_r_Page_125.jpg
cd2572ab4b39fc84bc560283c65b860f
a198c287b4772443e655cc401fc9b5183c798515
60738 F20101118_AACBCP saha_r_Page_100.jpg
56d0be39f4bc26253f3bb9ae1a34e73b
61dab15aae207f596f9b219b8b37c5f7cf15d3aa
5308 F20101118_AACBZY saha_r_Page_175thm.jpg
dd6acfd06786536bc63b9c73259f36a9
be7e805652e8bc7713c894a327e29ee3ee4e70fb
41388 F20101118_AACAXK saha_r_Page_157.pro
3ff591b5116f54071b57636457e01b05
36b4235c9287d458c7c295fb8fd868fc1bdf1d2d
6198 F20101118_AACAWW saha_r_Page_032thm.jpg
0487c98583b43ae47405eb05dce7b049
d4691cdbc411620dd6c06d6e05f52c8e879096f0
77640 F20101118_AACBDF saha_r_Page_128.jpg
73c7fe0937f62d6e97ce75d35c24ebd4
30fae5fcce1849730de3b36096fab2f133463502
38042 F20101118_AACBCQ saha_r_Page_101.jpg
d5cc4dc35f5d8e6355a347f4ec13df52
431401065adcd0bfc5929f6c59c371df0be483cd
4835 F20101118_AACBZZ saha_r_Page_081thm.jpg
16dd74e945e107f65f731635551ce7fd
408ad524635cf5eaac8c8ba342f2e97bd680ec99
17073 F20101118_AACAXL saha_r_Page_172.QC.jpg
6da366e956d5c0bf232a8a5a79a8b2bc
9841ade109a331d8c5ed244e55d3d5ed835ba7a4
53456 F20101118_AACAWX saha_r_Page_074.jpg
cc018a803a899951c2fdde0f3633425d
111cb7a93ebe64461d442c2ecf6f34ca89bd4539
58295 F20101118_AACBDG saha_r_Page_129.jpg
cbd678a1a93b2a772e437ba269cce54a
77dc4111cc30408b7efed31596e5d5c694718645
6551 F20101118_AACAYA saha_r_Page_045thm.jpg
858d87bdeb70f36383c1210c821cd3ae
bdeced4a7535c8606723cf3da64d6a0ad230acea
69685 F20101118_AACBCR saha_r_Page_104.jpg
a3bfb37425939703f9e8d506a4f462b9
bc10616d5776b831b52ebaef96cdd7e99285ccac
832383 F20101118_AACAXM saha_r_Page_174.jp2
11120714d3c572c79f72459b8c463387
4a859a53d5c06d9b1241724ef3345ed896de4806
1652 F20101118_AACAWY saha_r_Page_009.txt
9e81f9a8e72e3b61fdcabcbad0318256
c44171758ab6699f46aa02d4eac6ed77622b3047
53895 F20101118_AACBDH saha_r_Page_130.jpg
d8633fc3220e5eab845cf3e4f2b6acbd
1e9ceafa7366c93a3d4fcdfce1a303aaf7ca8b4e
18809 F20101118_AACAYB saha_r_Page_010.QC.jpg
4af8f30a2bd4adeb35c2cd7949ac9951
8b0811b231b0d6f497e85bac71039598ba900200
55369 F20101118_AACBCS saha_r_Page_105.jpg
24de740ded5f02793f787800244cb4a3
c9c60f050486b794343bc5244208b2b6438f916a
26106 F20101118_AACAXN saha_r_Page_029.QC.jpg
ab773dca2a164c073bc0e64ff1f47b95
f350d87198586e80700138c0f25329a1846b8299
665745 F20101118_AACAWZ saha_r_Page_160.jp2
6b9ea5dabb877bb2787bcf6d15a989a7
20ac15e11be6cef4418631a5825f981bf694e84a
53278 F20101118_AACBDI saha_r_Page_132.jpg
aa946dec35d2695ead045ff226c1d718
f07db77aba50dfc012d2913a50de611e2d257ff5
4722 F20101118_AACAYC saha_r_Page_074thm.jpg
4bfae60885925a371673e81aa7d3bcf5
2d4b255d40d31d5483da1b2d4d7dc8c570983b43
82249 F20101118_AACBCT saha_r_Page_107.jpg
b92c395d06defe27884b1a2660affa24
5d19cb56515dec89fa9f7606ba36df3d9bf6d409
48847 F20101118_AACAXO saha_r_Page_017.pro
bc3d18fd016018e01428ee3c94065eef
3d97d4ab08c2bab9878ea0d42ac847ebc937b37d
67537 F20101118_AACBDJ saha_r_Page_137.jpg
6a8c2df0bcf9d0917a306d6341119a3f
996d6d14c3072424b444d8017c118fcd95068d36
F20101118_AACAYD saha_r_Page_022.txt
12924f07196c9adce9c988b212ffa221
f4feb73119aae572763cce7ecf001c30c3078aac
86620 F20101118_AACBCU saha_r_Page_108.jpg
88e08e36e943fee2ef8d74a3df8bb8c2
c16dddd9591b7b49f382e6d4440b9dadb15101b2
5191 F20101118_AACAXP saha_r_Page_039thm.jpg
ca2e8a19c10b9c8f99d95eb5d46c1c4b
6b6b46cd8e2f4470485b364a455a48ecf48fad66
50241 F20101118_AACBDK saha_r_Page_139.jpg
d601a2e59465c02ccd0caca430ad6f77
405bc4999d6465b2603d896490393a256b25e85f
F20101118_AACAYE saha_r_Page_102.tif
d95bd27bc431c463235aae5bcdc78193
ecb98ead0f9a89d73b6112bdddd444f20261122c
84333 F20101118_AACBCV saha_r_Page_109.jpg
a05b881db61b05f6c310e1e496895f59
1350c77caf464f4b0c19ead8cbf66ef02deeae49
1434 F20101118_AACAXQ saha_r_Page_171.txt
6c103ed6db62100c598b9c5effabd765
f760f44df2172d302bdf8411ad2afd6c184cc462
76342 F20101118_AACBDL saha_r_Page_142.jpg
afa1dbfce91b5aff1868db645f205dca
c7177dd2cac533a84b37a97cc7e0901525001bfc
71307 F20101118_AACBCW saha_r_Page_113.jpg
8b002af1c749b475241f6761527d27cd
c72d935893e72af8b7dd0b36dda8b2ea081231aa
22454 F20101118_AACAXR saha_r_Page_084.QC.jpg
4bd724ec62b5c26dde4f0ddcbb736ac4
c53f2b4b81cd1c4f6c64a1087e16a94245f879a8
39974 F20101118_AACBEA saha_r_Page_171.jpg
0327055aec3ef38fdf04182393650d04
da5a9eb9a1aa6ac58fefa6e83df2ca7cb7f41fba
48372 F20101118_AACBDM saha_r_Page_143.jpg
60fa21a6f3e6ce0ee3431b519c2855c0
426ae8c616a74ec5118b3f5e06d882db7d3a0b43
75654 F20101118_AACAYF saha_r_Page_141.jpg
6035e5cfc219e6e118f821842e15ed19
15f4df7570732945c3379edbb2decf657b57ddf8
67594 F20101118_AACBCX saha_r_Page_114.jpg
2a12e98d24b0e767ce33472c1483dc83
c316ee2fd294043ea01e4e4e0f587eb52edac0b2
43974 F20101118_AACAXS saha_r_Page_012.pro
228864e02a9300aad7aee43ef0dd3e49
53a63561e3424d55162040c19b1c58bdd91bbd96
55052 F20101118_AACBEB saha_r_Page_172.jpg
1a1f5b3dffda9e11cdc98a6ecaa6c71a
05ee61736782fc60dd3b324bc32fc3c46211aada
62182 F20101118_AACBDN saha_r_Page_144.jpg
338b44ef58b3c0f28d6934c9120cf1f2
37740a9000a072475d8cd65c39b1deb65d591352
6475 F20101118_AACAYG saha_r_Page_030thm.jpg
b82474d70e6dca65198b0ffe733aa545
4f6b93d2eaadeee114cc73b08fc6c17b3f8c0d50
77888 F20101118_AACBCY saha_r_Page_117.jpg
effc40adad765b982a69800675afee7d
cd20bf449997d6c015e6ad0b96b61e5029c6337d
F20101118_AACAXT saha_r_Page_044.tif
a7a89d043f126a904787c4ad58251c9e
d8708a7681fd7163d4a5c1ba5e7bf4aee7372f9a
54545 F20101118_AACBEC saha_r_Page_173.jpg
10677cd0b2fc55efa34b234091114d6d
5eb2a88ae918f41b5e50b15ab8bbe83d5004133d
59121 F20101118_AACBDO saha_r_Page_146.jpg
1a33cd1b670ee9584135775c2d8ca0b6
0cbe4f703884408918207272f85abea6a07369d0
F20101118_AACAYH saha_r_Page_153.tif
adccbdf46bc00d2164c817bfbb099fa9
baec0fe5641ffc792d84a4d5997174716704f7f5
67334 F20101118_AACBCZ saha_r_Page_118.jpg
bf946642209a60a9fab5995906adbbc7
973bc45baa56ba0809aacd16b8b542e3be5092c2
2031 F20101118_AACAXU saha_r_Page_174.txt
30505ebc3ca3724e2cc10ea209481e0b
613ab5509bf846197859d82a064dd510b2c4ff80
60266 F20101118_AACBED saha_r_Page_174.jpg
5e5ad44ec8395e15d310fa2922616a27
b42b64a16d0e64dd71f247248d90dbda47dfaaaa
945637 F20101118_AACAYI saha_r_Page_131.jp2
3ed9bd755ce5d8e10b41dcc583492101
5284a620daab417cdd74980b06d46a8a1a795a23
1036125 F20101118_AACAXV saha_r_Page_098.jp2
6a7e69d209c0e80bee4636c7971be6dd
132bf4c93050904cf62465646d66dcae428c58ae
60991 F20101118_AACBEE saha_r_Page_175.jpg
c975ae7c29c14806d9d310f5d5e0dfc9
f68d616297ee2bc5bbb6e88e7eaa0e2a3070b959
76233 F20101118_AACBDP saha_r_Page_152.jpg
4a3e7a196ac5c26825b9eaee825049d4
fc2b8adea3266f6ca0beb31861c4b3fe76e95411
5767 F20101118_AACAYJ saha_r_Page_019thm.jpg
c74ee88cab2b4b468289a1c972aaead5
5c9b74ec9e99f5e4c2216e4ab2a9221ad4a7fe0d
32547 F20101118_AACAXW saha_r_Page_004.pro
937bb862593de22f182bcbcc5501f20d
a2cf7e9cf5645ad351ad73a7cecd0af4d4583778
29243 F20101118_AACBEF saha_r_Page_176.jpg
7e5c5fc1262d02fbcb6ee33211f0be55
4d02cdcff614c2b6b2eed4e0edcb1c92fe4839d9
47965 F20101118_AACBDQ saha_r_Page_154.jpg
e3f86f8bb2f2ea9671a9cce991593dc8
2f4fa1e7f3874ef0178270e9910efb09192e91ce
767445 F20101118_AACAYK saha_r_Page_049.jp2
b9fe932857884a82ad77657249367eae
73bcc47a0535b34f29af580c1b04bec977bc7a5a
66688 F20101118_AACAXX saha_r_Page_026.jpg
0c773cc5aeefdd4813da267ec1f93bd0
ddab641edd69d77b9b1c2abccbf7656fe9af8878
60589 F20101118_AACBEG saha_r_Page_177.jpg
e6da337d25cda2585c9ee2cf8f185a8a
4424462fd58a964a57694bd08695effaaa520cde
76303 F20101118_AACAZA saha_r_Page_135.jpg
07a23ecb26f0cceabcb91ae5327ce328
30e093b7e6217b0566e197c0732d77cbe9dd9006
35254 F20101118_AACBDR saha_r_Page_155.jpg
a6721fbdf7b432ece69583df9be19908
1e9255050b40dce734b3c9c00b233e1b3e2c1db3
5663 F20101118_AACAYL saha_r_Page_112thm.jpg
4de12bf10af067ec9d41fce6ee9277b1
46a72f4d2b60dfcaf917f5db1c3eba6b913ac1fc
45585 F20101118_AACAXY saha_r_Page_182.pro
8afcb7b239e9a769c386b5e1803f18cd
80a08a2c7fa04caed222c58479091b3d8330d255
61126 F20101118_AACBEH saha_r_Page_179.jpg
fc4d6e6f2c8af5d5bc1f92d08467b7d2
acbffc840b34c0bba7603d8861ddfd2804b33b50
84891 F20101118_AACAZB saha_r_Page_041.jpg
29a03ae2426840be9090a92010bf5ede
fe255b4d52252dac3889df4e62c1070ddd1954c0
61278 F20101118_AACBDS saha_r_Page_156.jpg
dce7577de4b6c4d2dd41c7e4cdd11bcd
1e7fb4e3ca616170b5325e7df808d52a4b03be6b
6318 F20101118_AACAYM saha_r_Page_044thm.jpg
872b9a0c36591b1f2438b1891a11529e
b8a2c291f907f1b0ba6e467d6f54c75634efc651
F20101118_AACAXZ saha_r_Page_086.tif
897fae3bf1be1d6f70e0466de22221be
38217a0d71701fd8141302775b218cc76b8b30af
70937 F20101118_AACBEI saha_r_Page_180.jpg
26a7a078cec2466d504de75f43a07d3d
2f25123e8304ce597cb3302aee737c60c3c95436
1732 F20101118_AACAZC saha_r_Page_166.txt
dce03e2b639907efc043a93ab767fa13
3d037fd1abfd6248c0b9344e9607d0c09d5da6e9
60890 F20101118_AACBDT saha_r_Page_157.jpg
675335099ec31888392b41da90e3d186
b8326e7c32b570df42861dd56dbc72d95a12ba6c
969000 F20101118_AACAYN saha_r_Page_112.jp2
b1008913b96dad29549fe4cd7c4dc4f1
7001684c718061048dd40d9b0a5c2956e343aea8
63990 F20101118_AACBEJ saha_r_Page_181.jpg
78b02b92107e94392f8ceda9c2fc6473
35ea674b77935ed44697c69e06c578c04e827db1
2273 F20101118_AACAZD saha_r_Page_056.txt
fc1c9af6b03401cb0307d8ddf3a7ecd2
944fb34acdabd602d43373a073cfba28fd45d1b0
47838 F20101118_AACBDU saha_r_Page_161.jpg
7984bc86cf8448f5db6ea8b617ad2c72
41919d2ed5230b403472f4c550a5644a63e137fa
973453 F20101118_AACAYO saha_r_Page_170.jp2
9e3088e3ce67903b0aef5dfabbd28163
4e0b5d1e473710bd62fd5b25bf56f3f5d1bd6863
69598 F20101118_AACBEK saha_r_Page_183.jpg
435c5a1600d86c2c7752509db22043ba
0e19457ee946685a8273d0402f2a94fc386cb71f
12922 F20101118_AACAZE saha_r_Page_185.QC.jpg
df3cdb98597ee2ad127b552684008cf5
0d81a5f98d0fce9b35915c5c97c8932ed529ce03
50048 F20101118_AACBDV saha_r_Page_162.jpg
857cc1538672032fcc2d876fb6056ed0
09ac0445b28cbdfcfd1147572cb78719295ca270
19114 F20101118_AACAYP saha_r_Page_005.QC.jpg
2b5df6672431b24978ff33a94f0a894c
12379597d5cab76730ac8718bf296fed992c470e
4036 F20101118_AACBEL saha_r_Page_003.jp2
aebd15b8f5a0d2adb6aca91ea8790a55
d4d81d6407046586342a2334aa20147a5d58d919
677 F20101118_AACAZF saha_r_Page_176.txt
e7cdfeab1df9348ccba97d0fa79a8f27
2f49872fe29e23bf3e83bbb3283e79e0d8839bc9
53730 F20101118_AACBDW saha_r_Page_163.jpg
2b30ddd920870cf9a62f70dd3806c10a
1c43d2f22b84f6e7bbe4c7887f4f8046280cce79
6087 F20101118_AACAYQ saha_r_Page_135thm.jpg
79afca6a772e335272d7e610fbad2595
8cb7e988112b1566eeee473a75558f70eb34954b
72313 F20101118_AACBEM saha_r_Page_004.jp2
30ce2ddbb2b0f9365f1e4dc83c8b12ed
6f0ec7fe17bae598c76c3c7dada8e8293f625332
53660 F20101118_AACBDX saha_r_Page_164.jpg
e2f965db03d05278a5f132199f0ec81d
620acdbba64120dd22c0e6cdd08a209382184676
21717 F20101118_AACAYR saha_r_Page_170.QC.jpg
9f7c1b9df13c16dbd66e8afdfc229b50
ae7cc1c4517bac95d0123ff871cc1880f2642012
943331 F20101118_AACBFA saha_r_Page_026.jp2
ccc58172bad891764925b701b5982cb4
89cc90462fa163395d0af71e4fd4ec421f7617e3
1051978 F20101118_AACBEN saha_r_Page_005.jp2
48f590cdf21e683c677748a14ea2fdb8
480164376a61a50a8172fba4ed2f8e7197e47b1c
23212 F20101118_AACAZG saha_r_Page_052.QC.jpg
5493d8db655da4eff5e5151ef7d623e4
eaccd55d9ed3cc24a03af5bc861232577a59641e
47520 F20101118_AACBDY saha_r_Page_165.jpg
9b47f4b01e5f8f66f68df05c0928afe2
b11cb9f2f3b0184d73f6ca7dd07433da7753e5ae
54418 F20101118_AACAYS saha_r_Page_166.jpg
42bb55ae82998a31d05dc422dd236e03
05061f7b8692ce7285ecaa34a8b3bfb0f7c2b4c7
1051966 F20101118_AACBFB saha_r_Page_028.jp2
8144649aeab472760f74e3a7197b3247
5af4fdbe6e6056d3490defdb3acc2a08804c6073
F20101118_AACBEO saha_r_Page_006.jp2
4690c926bb987668720bc5814cbc8813
976497d2fd1d7387a6e9a175590257267d77a28a
41523 F20101118_AACAZH saha_r_Page_086.pro
540a3090b0cef035a3ce97e5d4aedf6f
f52a67f86bc77c9eca314827a06f8430466e35e2
54411 F20101118_AACBDZ saha_r_Page_169.jpg
5cb9e115091181088f8de92489659304
be7dd19dd7b73d6cf9a3475a9b0b4391f8caabee
67721 F20101118_AACAYT saha_r_Page_025.jpg
ecb93c24acac1f4db55cf11f4d9b2319
867f4c2df2f70e2e8772dad0aa8713a1b1c1a33c
1051954 F20101118_AACBFC saha_r_Page_030.jp2
8fb1c604bc50e9b08a1989b034d010c9
eb42529987c275d1781b3cdaf0858a108c7dbc5f
F20101118_AACBEP saha_r_Page_007.jp2
ba3b0aa0eedf65c23a1919d103574154
3f3dbe184a3e1b28b8023434eb30310331f589a3
4068 F20101118_AACAZI saha_r_Page_101thm.jpg
af23fe764ebec16301c7a0ce18a47488
04d4239786004f8ac7904fd1330bf496ef2cd65d
15044 F20101118_AACAYU saha_r_Page_136.QC.jpg
4d5dc154720726f2fef2e0dd466b831e
3319fd8a0f15f7ef92912e4d985cb724537f8ac4
1051963 F20101118_AACBFD saha_r_Page_032.jp2
d8887147012233bf18b2a95aebdb0644
eeafcd8dba40be837a3e6285c2df778d1c19f64b
5003 F20101118_AACAZJ saha_r_Page_042thm.jpg
7607ab02e827a315f49c23b51b751355
414a1226c851a0231a87df77e12dd5da3e79bb0e
52321 F20101118_AACAYV saha_r_Page_082.jpg
cdf6d9db47c694167c693ea785ad3d6d
19069aadff5a168038f392bb9f05e08436371cd0
902700 F20101118_AACBFE saha_r_Page_033.jp2
01bce1fd89d36297144d9309db687151
822fdd35950bd996138deb0ded1844157c2679f7
1051965 F20101118_AACBEQ saha_r_Page_008.jp2
5e8ab017fb41fc3ae61ddcd745e6d21d
7cd00919c89847583070b287160a50880558239b
20282 F20101118_AACAZK saha_r_Page_158.QC.jpg
cc2c2103983828a8f97df5deeddf81f4
383793392c4cadfc65caa3472c9ced6349223d12
15136 F20101118_AACAYW saha_r_Page_160.QC.jpg
0062735f49fc54361fc66b33472a8883
3c14f374074e1b66524ca4ecc768c4eab8ee32d1
781250 F20101118_AACBFF saha_r_Page_037.jp2
98c2f359052774531cda7e62453a10c0
c588300897382ee1280d0670cdc67378c8244a62
F20101118_AACBER saha_r_Page_009.jp2
c47c55f4ab6bf304abb160ea0893256d
b502b978e72cfec07b6b410c9da48926d199d974
1530 F20101118_AACAZL saha_r_Page_059.txt
34c3dced68c3413f6fdc34d2fcac0ecc
03cf602b3715d1f97246030f94c85b8f9d1bca9e
1423 F20101118_AACAYX saha_r_Page_164.txt
e1fc42a1799d3f9ca7fca55a4bfa2bb1
bc329a3a37304f8ce8248f1e1b345b217648b53f
1000593 F20101118_AACBFG saha_r_Page_038.jp2
6aebfbef05fc8d1f110fbec161cbfc3f
f65d280134eee1d8c48c9464ca2dad2284980a67
F20101118_AACBES saha_r_Page_013.jp2
a2ccb644717e2872e4d3d9e6250e3c9c
8c867bb960cfed35d0f7d08b178a917f309a9e5c
92122 F20101118_AACAZM saha_r_Page_012.jp2
c5da9179226bde35a2156e479437932c
91a5a4d790c2c5a7f954556604e7f213c138e16d
36972 F20101118_AACAYY saha_r_Page_176.jp2
b3d82440d180f5bf499e0fa002100cf5
eae31699a3fc8ae80612e26cbb5d91cd83e0ac40
97241 F20101118_AACBFH saha_r_Page_039.jp2
d55d5e944044e3877bae6869ce17c114
f9b41c339b607ed4c319eef7f4602b288a0b98e0
816096 F20101118_AACBET saha_r_Page_016.jp2
9378ec8502e065b8d4f521bbfafa02cb
3229bbf8fdfe8e0c967207040482a093d4964ae5
45148 F20101118_AACAZN saha_r_Page_114.pro
362606fc34a837a2a7446a706e99c101
c735a49e3165633fd5567f06dcf30220c421edab
F20101118_AACAYZ saha_r_Page_057.tif
1273cac29b65ba91ac8e88db28b09295
f69d90f0104ef1583c72a44e0e5201cfbd20b6fb
101579 F20101118_AACBFI saha_r_Page_040.jp2
b94fcc1d7b0aca3fc55f78a5a515f255
8c4584065eb6e45d03371566e84548e79f8deaa3
1042043 F20101118_AACBEU saha_r_Page_017.jp2
a5cf227250251a6349a941c78d479146
e4b60cb7279e7f8e4af9611b165ae4e9922e465f
58422 F20101118_AACAZO saha_r_Page_012.jpg
af7ad1eb244b87c1babaaee87373be4d
a486fed7b453a0014e2f5afce64ebb7b91cdc39e
95754 F20101118_AACBFJ saha_r_Page_043.jp2
5fb51fdea94b77099d29c22eeb86ceb9
295d7044317c81881189e0321a8ee52adcbadbdb
643677 F20101118_AACBEV saha_r_Page_018.jp2
f50b2b51d839ce45e3253b16c0b34487
be169f70b3051a46be02696e6cd13e72c1b61c12
100166 F20101118_AACAZP saha_r_Page_183.jp2
f168bb741bda1bdabbb9edd32ff90e65
c5cc0d0a9109538151506b42cb9c7adf6a2eecc7
F20101118_AACBFK saha_r_Page_044.jp2
cc4c8c23512878a6066d3e0dac1a79e3
a5ba37197da5a249e611588fb8d7c29ca15aae31
987033 F20101118_AACBEW saha_r_Page_019.jp2
68d8c381e44ba7d3cbe7b0369b3540c1
9f7494d44ce7c0621dee20ab68db407f228d4c7a
5215 F20101118_AACAZQ saha_r_Page_172thm.jpg
f8ff344cfc246dbbdc3070f62cdad7f5
4f135e98a93d1227c1626da61a75b1859fbfe91f
1051959 F20101118_AACBFL saha_r_Page_045.jp2
5ba412505d11d8a09fac8c54f2636297
56ae14953f280651ff17823d22d8ef52800dece6
1051958 F20101118_AACBEX saha_r_Page_020.jp2
9f8bb4a0e09a28e8552bd11194e934e3
534b6b3ed902c0b4191ec2cd19f610debf5b5315
77021 F20101118_AACAZR saha_r_Page_094.jpg
ba81bb941c8b70dd9608cc769b464ede
8864a6a17830441671603f3e1ee9e09339563e59
616237 F20101118_AACBGA saha_r_Page_063.jp2
ec100c3f55e4001a53c96187552a2aec
783add59135885c6194b9b80375113798640a330
F20101118_AACBFM saha_r_Page_046.jp2
2ba65db2fc254fa872d5977fc590d4b2
33e8102d1794d6aa63e8661236c654d1dcaade50
716351 F20101118_AACBEY saha_r_Page_023.jp2
44662b1372b930957143dedd79c3645e
e8d38dfd0789eb13886ce76115cd4c73f576c58e
54405 F20101118_AACAZS saha_r_Page_021.pro
d83a2062b2f95953b5a9688ff40b1a2e
94d457c4234f474e3f95dd9cf1394e8423e669b5
881838 F20101118_AACBGB saha_r_Page_064.jp2
d3723c1212f93b0917acd72aa75a41db
04b7392621f6afee6ce1c4c36be5a016a159c5ec
108390 F20101118_AACBFN saha_r_Page_047.jp2
078b17c994276f34ddedacfb2f252495
4d0d5ab41a19512e2764b7e8dbcf4a3c6187a999
928313 F20101118_AACBEZ saha_r_Page_025.jp2
01004ba2b8ab66fb016a922c6b4f9658
2e7dd89743df9c3e49b16fa3d20b49dd0675b8c6
13368 F20101118_AACAZT saha_r_Page_093.QC.jpg
f279a92d84ee58cd570dafdaad62530a
fc5e9bd2af43dc67338f6124dfbda7b660e92411
927360 F20101118_AACBGC saha_r_Page_065.jp2
6ec2cb360e5c6504902d9e4a14f2328d
acda8353dafb30002d287eb7ba541647bb3e583e
576743 F20101118_AACBFO saha_r_Page_048.jp2
2780d557b7d48aaced333acd6ebb6e90
57f48d7380197bf01389b6f27afff4103e10dce3
F20101118_AACAZU saha_r_Page_036thm.jpg
65081cdc5111986095f98fd06300f5f3
39953cb64f8d4fa6bdc6ab04364d1fc5d915cd1c
643283 F20101118_AACBGD saha_r_Page_067.jp2
36853beed893d5a52e5ba7bad71caa56
a1aff6c6bd0a269194b89f8f5411dc0af1b29c64
91628 F20101118_AACBFP saha_r_Page_050.jp2
730d307e8f99cc3928f3cb28cad05aa8
daafed92988a414959e26a23e027d2cf3f83defb
43317 F20101118_AACAZV saha_r_Page_153.pro
e0e606a6806247067fddc0ab1c38441c
68d816ff52b00b11fe24e960b50dd23b4d017b3f
1051899 F20101118_AACBGE saha_r_Page_069.jp2
836d978887d6aa670cf840e8ff9e3cea
f1417f0d5048700d7f2fe100011f83750b519796
F20101118_AACBFQ saha_r_Page_051.jp2
e5b7d039c8c3377de28367ae63ea3c64
57f984979183ac1eb99abada12d4e8d759f7f4f4
F20101118_AACAZW saha_r_Page_094.tif
494a594c9e9bed155e8d2ec6c2c3f130
5a1b97dea8b024ed9f6a1a8e543bab4ba0c1fd4e
927228 F20101118_AACBGF saha_r_Page_070.jp2
0d2483bddf73fba4952d80155951b039
1a3608fe05d88f9a9ccca86479e3ab053404b959
920649 F20101118_AACAZX saha_r_Page_066.jp2
def714d2b78d8df125ac822cea3b240c
3c330a1b24fcd8a5ee28a59462593bb486ea9053
900094 F20101118_AACBGG saha_r_Page_071.jp2
b8e33b08f90f9c32ebbfab7029b40f94
1d890335def04798f6f33368369f8cc202c10cf4
1051948 F20101118_AACBFR saha_r_Page_052.jp2
ee09ace700f824ee0d4c62ce1ad1caf1
9f52218e3849b00d11547f628d73f98101d7b97e
F20101118_AACAZY saha_r_Page_095.txt
f29da54962141be74ed357a3c561b9a8
c211afe0646e172edc27c9daaffd553c03598dbf
704504 F20101118_AACBGH saha_r_Page_073.jp2
bfada64c1897ec06df4279cf810b395b
11db15e22da1f6908266869b6683112c5f026cce
1051925 F20101118_AACBFS saha_r_Page_054.jp2
3c35e1a869b708f98984526407016545
ad4309eb80db487ebf372d19f0985c2f1ca59625
F20101118_AACAZZ saha_r_Page_123.tif
4e6e137199cc1a4f94aa14246388016d
2dbe8d481ef340cde50c3d8d9ac75398d6db5757
1010946 F20101118_AACBGI saha_r_Page_075.jp2
d1da03b528b2b579c4475ddfb1f63f88
5e84183c78a5056bda83fcf8709fd3731818ef5a
F20101118_AACBFT saha_r_Page_055.jp2
d5f42aad150092f3b44b47c371565023
3399f57102f71ac9f56650dff534ac6c0882c128
F20101118_AACBGJ saha_r_Page_077.jp2
67388375f05f3e73a37aaae44290cb75
a05169e76e198e1d31a4507f153b2b4b4947ec73
1051932 F20101118_AACBFU saha_r_Page_056.jp2
b9108a622dd0c6e8b25286198d3eda8d
369b83d8268934c1e6ed4242d7e206fca78b50a1
916714 F20101118_AACBGK saha_r_Page_079.jp2
c5f0a8f093c5810a42d6cdebf15a22dd
86f2c2b9e23b85873d26c0ef2d1d39f59c00b487
862317 F20101118_AACBFV saha_r_Page_057.jp2
405ba0023e2e8c7a8739b3ff3a8f0d2f
6f5241879767b9aa6df6dbcdc9018cc809c1b3ff
880502 F20101118_AACBGL saha_r_Page_080.jp2
075018905097d26ee74fce01add6ba15
52c0cadcf69aa670aa64eaab56fec14868152978
811389 F20101118_AACBFW saha_r_Page_058.jp2
42d9604695e667562a21b71af04e88bb
80a7ff232a08b19e11cb6866fabdd17b516b9e3c



PAGE 4

FirstandformeostIwouldliketothankmyresearchsupervisor,ProfessorDmitriiMaslov,forhisconstantencouragementandguidancethroughouttheentirecourseofmyresearch.Hisenthusiasm,dedication,andoptimismtowardsphysicsresearchhavebeenextremelyinfectious.ThecountlesshoursIhavespentdiscussingphysicswithhimwerehighlyproductiveandintellectuallystimulating.IwouldliketothankProfessorJimDufty,ProfessorArthurHebardandProfessorPradeepKumar,whowerealwayswillingandopentodiscussanyphysicsrelatedquestions.IamhonoredandgratefultoProfessorRussellBowers,ProfessorAdrianRoitberg,ProfessorKhandkerMuttalib,ProfessorSergeiObukhovandProfessorArthurHebardforservingonmysupervisorycommittee.MythanksgotothePhysicsDepartmentsecretaries,Ms.Balkcom,Ms.Latimer,Ms.NicholaandMr.Williams;andtomyfriendsPartho,Vidya,Suhas,Aditi,AparnaandKarthikfortheirhelpandsupport.IwouldliketothankmywifeandbestfriendSreyaforbeingmysourceofstrengthandinspirationthroughalltheseyears.Iwouldliketothankmyfamilyfortheunconditionallove,supportandencouragementtheyprovidedthroughtheyears. iv

PAGE 5

page ACKNOWLEDGMENTS ............................. iv LISTOFFIGURES ................................ vii ABSTRACT .................................... x CHAPTER 1INTRODUCTION .............................. 1 1.1TransportinUltraStrongMagneticFields .............. 2 1.1.1WeakLocalizationQCC .................... 5 1.1.2InteractionCorrectiontotheConductivity-AltshulerAronovCorrections(classI) ....................... 10 1.1.3CorrectionstoWLQCCduetoElectron-ElectronInteractions:Dephasing(classII) ....................... 17 1.2Non-FermiLiquidFeaturesofFermiLiquids:1DPhysicsinHigherDimensions ............................... 19 1.3SpinSusceptibilitynearaFerromagneticQuantumCriticalPointinItinerantTwoandThreeDimensionalSystems. .......... 34 1.3.1Hertz'sLGWFunctional .................... 36 2CORRELATEDELECTRONSINULTRA-HIGHMAGNETICFIELD:TRANSPORTPROPERTIES ........................ 43 2.1LocalizationintheUltraQuantumLimit ............... 45 2.1.1DiagrammaticCalculationfortheConductivity ....... 46 2.1.2QuantumInterferenceCorrectiontotheConductivity .... 50 2.2ConductivityofInteractingElectronsintheUltra-QuantumLimit:DiagrammaticApproach ........................ 58 2.2.1Self-EnergyDiagrams ...................... 61 2.2.1.1DiagramFig. 2{10 (a) ................ 64 2.2.1.2DiagramFig. 2{11 (a). ................ 67 2.2.2VertexCorrections ....................... 69 2.2.2.1Diagram 2{10 (b) ................... 69 2.2.2.2DiagramFig. 2{11 (b) ................ 71 2.2.3Sub-LeadingDiagrams ..................... 72 2.2.4CorrectiontotheConductivity ................. 73 2.2.5EectiveImpurityPotential .................. 73 v

PAGE 6

..... 75 2.3.1Non-InteractingCase ...................... 76 2.3.2InteractingCase ......................... 78 2.4Experiments ............................... 84 2.5Conclusions ............................... 93 3SINGULARCORRECTIONSTOTHERMODYNAMICSFORAONEDIMENSIONALINTERACTINGSYSTEM:EVOLUTIONOFTHENONANALYTICCORRECTIONSTOTHEFERMILIQUIDBEHAVIOR 95 3.1One-DimensionalModel ........................ 99 3.2SpecicHeat .............................. 105 3.2.1SpecicHeatfromtheSecondOrderSelfEnergy ....... 107 3.2.2SpecicHeatfromtheThermodynamicPotentialatSecondOrder ............................... 112 3.2.3SpecicHeatfromThirdOrderSelfEnergy .......... 116 3.2.4SpecicHeatfromtheSine-GordonModel .......... 127 3.3SpinSusceptibility ........................... 130 3.4Experiments ............................... 136 3.5Conclusion ................................ 137 4SPINSUSCEPTIBILITYNEARAFERROMAGNETICQUANTUMCRITICALPOINTINITINERANTTWOANDTHREEDIMENSIONALSYSTEMS ................................... 138 4.1SpinSusceptibilitys(H),in2D .................... 141 4.2SpinSusceptibilitys(H),in3D .................... 145 4.3SpinSusceptibilityforaFermiLiquidin2D ............. 147 4.4SpinSusceptibilityneartheQuantumCriticalPoint ......... 156 4.4.12D ................................ 158 4.4.23D ................................ 162 4.5Conclusions ............................... 165 5CONCLUSIONS ............................... 166 REFERENCES ................................... 168 BIOGRAPHICALSKETCH ............................ 174 vi

PAGE 7

Figure page 1{1Weaklocalizationcorrections. ........................ 5 1{2LadderdiagramforM(diuson)andC(Cooperon). ............ 7 1{3Quantumcorrectionstoconductivityfornoninteractingelectrons. 7 1{4ScatteringbyFriedeloscillations. ...................... 12 1{5Self-energyatrstorderininteractionwithabosoniceld ........ 23 1{6Kinematicsofscattering.(a)\Any-angle"scatteringleadingtoregularFLtermsinself-energy;(b)Dynamicalforwardscattering;(c)Dynamicalbackscattering.Processes(b)and(c)areresponsiblefornonanalytictermsintheself-energy ............................... 25 1{7Nontrivialsecondorderdiagramsfortheself-energy ........... 26 1{8Scatteringprocessesresponsiblefordivergentand/ornonanalyticcorrectionstotheself-energyin2D.(a)\Forwardscattering"-ananalogoftheg4processin1D(b)\Forwardscattering"withanti-parallelmomenta-ananalogoftheg2processin1D(c)\backscattering"withantiparallelmomenta-ananalogoftheg1processin1D ...................... 29 1{9Typicaltrajectoriesoftwointeractingfermions .............. 31 2{1Diagram(a)istheleadingcontributiontotheselfenergyatfourthorder 48 2{2Dyson'sseries ................................. 49 2{3Drudeconductivity .............................. 49 2{4Thirdandsecondorderfandiagram. .................... 50 2{5Cooperonsequencefor3DelectronsintheUQL.Unlikein1D,eachtermintheseriescomeswithadierentcoecientcn. ............. 54 2{6Firstandsecondorderdiuson ....................... 55 2{7Interferencecorrectiontoconductivity ................... 56 vii

PAGE 8

58 2{9Firstorderinteractioncorrectionstotheconductivitywhereeectsofimpuritiesappearonlyinthedisorder-averagedGreen'sfunctions. ... 61 2{10Exchangediagramsthatarerstorderintheinteractionandwithasingleextraimpurityline.TheGreen'sfunctionsaredisorder-averaged.Diagrams(a)and(b)givelnTcorrectiontotheconductivityandexchangediagrams(c),(d)and(e)givesub-leadingcorrectionstotheconductivity. ..... 62 2{11Hartreediagramsthatarerstorderintheinteractionandwithasingleextraimpurityline.TheGreen'sfunctionsaredisorder-averaged.BothdiagramsgivelnTcorrectiontotheconductivity. ............. 63 2{12Theself-energycorrectioncontainedindiagram 2{10 (a),denotedinthetextas( 212 )+. ................................ 64 2{13Theself-energycorrectioncontainedindiagram 2{11 (a),denotedinthetextas( 213 )+ 67 2{14Diagram 2{10 (b)vsdiagram 2{11 (b). .................... 71 2{15Eectiveimpuritypotential ......................... 74 2{16Thehandlediagramcorrespondstodiagrams 2{10 (a)and 2{11 (a)andthecrossingdiagramcorrespondsto 2{10 (b)and 2{11 (b). ........ 75 2{17ProleoftheFriedeloscillationsaroundapointimpurityina3DmetalintheUQL.Theoscillationsdecayas1=zalongthemagneticelddirectionandhaveaGaussianenvelopeinthetransversedirection. ......... 79 2{18Renormalizedconductivitiesparallel(zz)andperpendicular(xx)tothedirectionoftheappliedmagneticeld.Power-lawbehaviorisexpectedinthetemperatureregion1=TW. ................. 82 2{19Temperaturedependenceoftheab-planeresistivityxxforagraphitecrystalatthec-axismagneticeldsindicatedinthelegend ........ 86 2{20Temperaturedependenceofthec-axisconductivityzzforagraphitecrystalinamagneticeldparalleltothecaxis.Themagneticeldvaluesareindicatedontheplot,withtheeldincreasingdownwards,thelowestplotcorrespondstothehighesteld ..................... 87 2{21Temperaturedependence(log-logscale)oftheab-planeresistivityscaledwiththeeldxx=B2foragraphitecrystalatthec-axismagneticeldsindicatedinthelegend ............................ 88 viii

PAGE 9

................................ 89 2{23PhasebreakingratevsTduetoelectron-phononscattering ....... 92 3{1Interactionvertices .............................. 101 3{2Non-trivialsecondorderselfenergydiagramsforrightmovingfermions 107 3{3Secondorderdiagramsforthethermodynamicpotentialwithmaximumnumberofexplicitparticle-holebubbles ................... 112 3{4Thedierentchoicesforthe3rdorderdiagram. .............. 117 3{5All3rdordersediagramsforrightmoverswhichhavetwo2kF. ..... 122 3{6AllthirdorderselfenergydiagramscontainingtwoCooperbubbles ... 123 3{7Eectivethirdorderself-energydiagrams(thedoublelineisavertex). 124 3{8Allg2andg1verticesat2ndorder. ..................... 125 3{9Allthirdorderself-energydiagramswithtwoCooperbubblesortwo2kFbubbles. .................................... 126 3{10Secondorderdiagramsforthethermodynamicpotential. ......... 132 4{1Particle-holetypesecondorderdiagramforthethermodynamicpotential. 143 4{2Particle-holetypethirdorderdiagramforthethermodynamicpotential. 144 4{3Theskeletondiagramforthethermodynamicpotential. .......... 148 4{4Fermionself-energy(a)andBosonicself-energy(b). ........... 158 ix

PAGE 10

Inthisworkwehavestudiedthefundamentalaspectsoftransportandthermodynamicpropertiesofaone-dimensional(1D)electronsystem,andhaveshownthatthese1Dcorrelationsplayanimportantroleinunderstandingthephysicsofhigher-dimensionalsystems.Therstsystemwestudiedisathree-dimensional(3D)metalsubjectedtoastrongmagneticeldthatconnestheelectronstothelowestLandaulevel.Weinvestigatedtheeectofdiluteimpuritiesinthetransportpropertiesofthissystem.Weshowedthatthenatureofelectrontransportisonedimensionalduetothereducedeectivedimensionalityinducedbythemagneticeld.Thelocalizationbehaviorinthissystemwasshowntobeintermediate,betweena1Danda3Dsystem.Theinteractioncorrectionstotheconductivityexhibitpowerlawscaling,/Twithaelddependentexponent. Nextwestudiedthethermodynamicpropertiesofaone-dimensionalinteractingsystem,whereweshowedthatthenext-to-leadingtermsinthespecicheatandspinsusceptibilityarenonanalytic,inthesamewayastheyareforhigher-dimensional(D=2;3)systems.Weobtainedthenonanalytic,TlnTterminthespecicheatin1Dandshowedthatalthoughthenonanalyticcorrections x

PAGE 11

Inthenalpartofthisworkweanalyzedthenonanalyticcorrectionstothespinsusceptibility(s(H))inhigherdimensionalsystems.Weshowedthat,althoughtherewerecontributionsfromnon-1Dscatteringinthesenonanalyticterms,thedominantcontributioncamefrom1Dscattering.Wealsoshowedthatthesecondorderferromagneticquantumphasetransitionisunstablebothin2Dand3D,withatendencytowardsarstordertransition. xi

PAGE 12

One-dimensionalinteractingsystems(Luttinger-liquids)exhibitmanyfeatureswhichappeardistinctfromtheirhigher-dimensionalcounterparts(Fermi-liquids).Ourgoalinthisthesisistohighlightthesimilaritiesbetweenhigher-Dand1Dsystems.Theprogressinunderstandingof1Dsystemshasbeengreatlyfacilitatedbytheavailabilityofexactorasymptoticallyexactmethods(BetheAnsatz,bosonization,conformaleldtheory),whichtypicallydonotworkverywellabove1D.Thedownsideofthisprogressisthat1Deects,beingstudiedbyspecically1Dmethods,looksomewhatspecialandnotreallyrelatedtohigherdimensions.Wearegoingtoarguethatthisisnottrue.Manyeectswhichareviewedasthehallmarksof1Dphysics,e.g.,thesuppressionoftunnelingconductancebytheelectron-electroninteraction,dohavehigherdimensionalcounterpartsandstemfromessentiallythesamephysics.Inparticular,scatteringatFriedeloscillationscausedbytunnelingbarriersandimpuritiesisresponsibleforzero-biastunnelinganomaliesinalldimensions.Thedierenceliesinthemagnitudeoftheeect,notinitsqualitativenature.Weillustratethissimilaritybyshowingthat1Dcorrelationsplayanimportantroleinunderstandingthephysicsofhigherdimensionalsystems.Westudiedthreeseeminglydierentproblems,butaswewillshow,allthreeofthemareconnectedbythecommonfeatureof1Dcorrelations.Ourgoalintheintroductionistoprovideabackgroundforthephysicsdiscussedinthethreechaptersofthisdissertation.WehavesethandkBequaltounityeverywhere. 1

PAGE 13

2; where!c=eH=mcisthecyclotronfrequencyandnistheLandaulevel.Thusthesystemexhibitseectscharacteristicofone-dimensional(1D)metals,whilebeingintrinsicallya3Dsystem.Thisreductionofeectivedimensionalityofchargecarriersfrom3Dto1Dismostpronouncedintheultra-quantumlimit(n=0,whenonlythelowestLandaulevelremainspopulated)andisexpectedtoresultinanumberofunusualphases.ItiswellknownthatthegroundstateofrepulsivelyinteractingelectronsintheUQLisunstablewithrespecttotheformationofachargedensitywave[ 1 { 3 ],whichhasbeenobservedexperimentallyinmagneto-resistancemeasurementsongraphiteinhighmagneticelds[ 4 ].ThemostcompleteanalysisoftheCDWinstabilityforthecaseofshortrangeinteractionswasperformedinRef.[ 3 ],bysolvingtherenormalization-group(RG)equationsfortheinteractionvertex.Ontheotherhand,ithasrecentlybeenshownthatforthecaseoflong-range(Coulomb)interactionsbetweenelectrons,a3DmetalinUQLexhibitsLuttinger-liquidlike(1D)behavioratenergieshigherthantheCDWgap[ 5 6 ].Biaginietal.[ 5 ]andTsaietal.[ 6 ]showedthatintheUQL,thetunnelingconductancehasapowerlawanomaly(nonlinearitiesinI-Vcharacteristicsatsmallbiases),whichistypicalforaonedimensionalinteractingsystem(Luttingerliquid).Themagnetic-eld-inducedLuttingerliquidphasecanbeanticipatedfromthefollowingsimpliedpicture.Inastrongmagneticeld,electrontrajectoriesarehelicesspiralingaroundtheeldlines.Abundleof

PAGE 14

suchtrajectorieswithacommoncenteroforbitcanbeviewedasa1Dconductor(\wire").Inthepresenceofelectron-electroninteractions,each\wire,"consideredseparately,isintheLLstate.Interactionswithsmallmomentumtransfersamongelectronsondierent\wires"donotchangetheLLnatureofasinglewire[ 7 ].Inchapter2ofthisdissertationwestudythetransportpropertiesofadisordered3DmetalintheUQL,bothwithandwithoutelectron-electroninteractions.Boththelocalizationandinteractioncorrectionstotheconductivityshowsignaturestypicalforone-dimensionalsystems.Beforewegetintothedetailsofourstudy,wewillbrieyreviewthephysicsoftheinterplaybetweentheinteractioneectsanddisorderinducedlocalizationindiusivesystemsoflowdimensionality. Atlowtemperatures,theconductivityofdisorderedconductors(normalmetalsandsemiconductors)isdeterminedbyscatteringofelectronsoquencheddisorder(e.g.,impuritiesanddefects).TheresidualconductivityisgivenbytheDrudeformula, m; wherenistheelectronconcentration,eistheelectroncharge,isthetransportmeanfreetime,andmistheeectivemass.TheDrudeformulaneglectsinterferencebetweenelectronwavesscatteredbydierentimpurities,whichoccurascorrectionstoEq. 1{2 ,intheparameter(kF`)11(wherekFistheFermimomentumand`ismeanfreepath).Inlowdimensions(d2),these(interference)quantumcorrectionstotheconductivity(QCC)divergewhenthetemperatureTdecreasesandeventually,drivethesystemtotheinsulatingregime.Thequantumcorrectionstotheconductivityareofsubstantialimportanceevenforconductorsthatarefarfromthestronglocalizationregime:inawiderangeofparametersQCC,thoughsmallerthantheconductivity,determineallthetemperatureandelddependenceoftheconductivity.Thesystematicstudyof

PAGE 15

QCCstartedalmostthreedecadesago.Acomprehensivereviewofthestatusoftheproblemfromboththeoreticalandexperimentalviewpointscanbefoundinseveralpapers[ 8 { 11 ]. Accordingtotheirphysicalorigin,QCCcanbedividedintotwodistinctgroups.Thecorrectionofthersttype,knownastheweaklocalization(WL)correction,iscausedbythequantuminterferenceeectonthediusivemotionofasingleelectron.Forlow-dimensional(d=1;2)innitesystemstheWLQCCdivergeatT!0;thisdivergenceisregularizedeitherbyamagneticeldorbysomeotherdephasing(inelasticscattering)mechanism.WewillelaborateonthistypeofQCCinsection 1.1.1 belowandalsoseehowitchangesfora3DmetalinUQLinchapter2. ThesecondtypeofQCC,usuallyreferredtoastheinteractioneects,isabsentintheone-particleapproximation;theyareentirelyduetointeractionbetweenelectrons.Thesecorrectionscanbeinterpretedastheelasticscatteringofanelectronotheinhomogeneousdistributionofthedensityoftherestoftheelectrons.OnecanattributethisinhomogeneousdistributiontotheFriedeloscillationsproducedbyeachimpurity.Theroleoftheelectron-electroninteractionsinthistypeofQCCistoproduceastaticself-consistent(andtemperaturedependent)potentialwhichrenormalizesthesingleparticledensityofstatesandtheconductivity.Suchapotentialdoesnotleadtoanyrealtransitionsbetweensingle-electronquantumstates(thoserequirerealinelasticscattering).Therefore,itdoesnotbreakthetimereversalinvarianceofthesystemandneitherdoesitaectnorregularizetheWLcorrections.WewillelaborateonthistypeofQCCinsection1.1.2,andalsostudyitforourcaseof3DmetalinUQLinchapter2. HowevertheinteractionbetweenelectronsisbynomeansirrelevanttotheWLQCC.Indeed,theseinteractionscausephaserelaxationofthesingleelectron

PAGE 16

states,andthusresultinthecut-oofthedivergencesintheWLcorrections.Thisdephasing(describedbybythephasebreakingtime(T))requiresrealinelasticcollisionsbetweentheelectronsandcanbeobtainedexperimentallyfromthetemperaturedependenceofmagneto-resistancemeasurements.Wewilldiscussthephasebreakingtimeduetoelectron-electroninteractionsinsection1.1.3,ofthisintroduction.Thereforetherearetwoclassesofinteractioncontributiontotheconductivity:thegenuineinteractioncorrections(elasticscatteringofFriedeloscillation:Altshuler-Aronovcorrections)-ClassI,andcorrectionstoWLQCCduetointeractions(inelasticscattering-dephasing)-ClassII. Figure1{1. Weaklocalizationcorrections. alongdierenttrajectories(Fig. 1{1 ).ThetotalprobabilityWforatransferfrompointAtopointBis TherstterminEq. 1{3 describesthesumoftheprobabilitiesforeachpathandthesecondtermcorrespondstointerferenceofvariousamplitudes.Theinterference

PAGE 17

termdropsoutwhenaveragingovermanypathsbecauseofitsoscillatorynature.However,thereexistsspecialtypeoftrajectories,i.e.,theself-intersectingones,forwhichinterferencecannotbeneglected(seeFig. 1{1 ).IfA1istheamplitudefortheclockwisemotionaroundtheloopandA2istheamplitudefortheanticlockwisemotion,thentheprobabilitytoreachpointOis i.e.,twicethevaluewewouldhaveobtainedbyneglectinginterference.Enhancedprobabilitytondtheparticleatapointoforiginmeansreducedprobabilitytonditatnalpoint(B).Thereforethiseectleadstoadecreaseintheconductivity(increaseinresistivity)inducedbyinterference. TherelativemagnitudeofweaklocalizationQCC,=,isproportionaltotheprobabilitytoformalooptrajectory ZdPZdtv2 leadingto(2d)=2(lnford=2),whichdivergesasTisloweredford2,leadingtostrongAndersonlocalization.Herevistheelectronvelocity,Disthediusioncoecient,istheelectronwavelengthand(T)isthephasebreakingtime.Phasecoherenceisdestroyedbyinelasticscattering(electron-electron,electron-phonon)orbymagneticanda.celectricelds.ThetemperaturedependenceoftheWLcorrectionisdeterminedby(T).Typically,Tp,wheretheexponentpdependsontheinelasticscatteringmechanism(electron-electron,electron-phonon)anddimensionality.Interferenceeectsoccurfor(T)i.e.,atlowtemperatures. InthelanguageofFeynmandiagrams,theWLQCC[ 12 ]isobtainedbyincludingthemaximallycrossedladderdiagram,theCooperon(seeFig. 1{2 ),intheconductivitydiagram.Theothertypeofladder(vertex)diagram,theDiuson

PAGE 18

(Fig. 1{2 ),whenincludedintheconductivitydiagramchangestheelasticscatteringtimetothetransporttime.Intheeld-theoreticlanguage,theWeakLocalization Figure1{2. LadderdiagramforM(diuson)andC(Cooperon). correction,whicharisesduetointerferenceoftimereversedpathsisdeterminedbythe\Cooperon"modeC(Q;!),i.e.,theparticle-particlediusionpropagator, 21 torstorderin(kf`)1.Calculationofthesingularcontributionstoconductivity(interferenceeect)atsmall!;QshouldincludediagramscontainingasaninternalblockthegraphswhichyieldaftersummationC(Q;!)( 1{3 ).TheWLQCCis Figure1{3.

PAGE 19

whichgives !+1 indierentdimensions.Perturbationtheorybreaksdowninone-andtwo-dimensions(for!1 13 ]hadrstshownthatatsucientlyhighimpurityconcentration,electronicstatesbecomelocalizedandthesystembecomesaninsulator.MottandTwose[ 14 ]hadpredictedthattheconductivityforaonedimensionalsystemshouldvanishinthelimitoflowfrequencies(Mott'slaw)whichwaslaterrigourouslyprovedbyBerezinskii[ 15 ],whoshowedthatelectronstatesin1Darestronglylocalizedandthereisnodiusiveregimein1D.Thelocalizationlengthin1Disoftheorderofthemeanfreepath(`),thereforein1Dforlengthscalesshorterthan`,theelectronmotionisballisticandforlengthslongerthan`thenelectronmotionislocalized.2Dsystemsarealsostronglylocalizedbutthelocalizationlengthisverylarge(Lloc`ekf`)ascomparedto1D(Lloc`).Thusin2D,theballisticregime(L`)crossesovertothediusiveregime(for`
PAGE 20

Inthemetallicregimeg1,theconductanceshowsohmicbehaviorforwhich(g)=d2.Correctionsto(g)inthemetallicregimeareobtainedbyperturbationtheoryin1 1{1 )acquireadditionalphasefactors,A1!A1expie 0;A2!A2expie 0; 17 ].ThecharacteristictimescaleforphasebreakingistHlH2=DwherelH=p Aweakmagneticelddestroysphasecoherenceandincreasestheconductivity.Iftheeldisincreasedfurther,wereachtheclassicalmagneto-resistanceregime,

PAGE 21

wheretheconductivitydecreaseswiththeeld.WhathappensatevenhighereldswhenLandauquantizationbecomesimportant?Weaddressthisissueinchapter2ofthisdissertation.Weshowthatathree-dimensionaldisorderedconductorintheUltraquantumlimit,whereonlythelowestlandaulevelispopulated,exhibitsanewphenomenon:intermediatelocalization.ThequantuminterferencecorrectionisoftheorderoftheDrudeconductivityD(asin1D)whichindicatesabreakdownofperturbationtheory.However,theconductivityremainsniteatT!0(asin3D).Itisdemonstratedthattheparticle-particlecorrelator(Cooperon)ismassive.Itsmass(inunitsofthescatteringrate)isoftheorderoftheimpurityscatteringrate. 18 ](thewavefunctionrenormalizationZ,eectivemassm?,etc.).Firstwenotethatwithinthetransportequation,electron-electroncollisionscaninnowayaecttheconductivityinthecaseofasimplebandstructureandintheabsenceofUmklappprocesses,sinceelectron-electroncollisionsconservethetotalmomentumoftheelectronsystem.InclusionoftheFermiliquidcorrectionsrenormalizestheresidualresistivitywhilenotresultinginanydependenceoftheconductivityonthetemperatureandfrequency.HoweveronefrequentlyencountersthesituationthattheresistivityscalesasT2.Thisdependenceisofteninterpretedasthe\Fermi-liquid"eect,arisingfromelectron-electronscatteringwithcharacteristictimeee/T2.Infact,theresistivityisduetoUmklappscattering.Ingoodmetals,normalprocesses(whichconservethetotalelectronmomenta)andUmklappprocesses(whichconservethemomentauptoareciprocal

PAGE 22

latticevector)areequallyprobableandtheUmklappscatteringrateenteringtheresistivityalsoscalesasT2.Notethatatlowtemperaturesthisresistivityduetoelectron-electronscattering(Umklapp)givesthedominantcontributionbecausetheelectron-phononcontributiontotheresistivityscalesasT5(Bloch'slaw,eph1=T5). Aswasmentionedpreviously,takingintoaccounttheinterferenceofelasticscatteringbyimpuritieswiththeelectron-electroninteractionproducesnontrivialtemperatureandfrequencydependencesoftheconductivity.ThiscorrectionarisesfromcoherentscatteringofanelectronfromanimpurityandtheFriedeloscillationitcreates[ 19 ].Wewillrststudythiscorrectiontotheconductivityintheballisticlimit,(T1,whereistheelasticscatteringlifetime)andtheninthediusiveregime(T1).Inthediusivelimitanelectronundergoesmultiplecollisionswithimpuritiesbeforeitscattersfromanotherelectron,whereasintheballisticlimittheelectron-electroncollisionrateisfasterthanelectron-impurityrate,thussingleimpurityeectsareimportantintheballisticlimit.Inchapter2ofthisdissertationwewillevaluatethisinteractionQCCintheballisticlimitina3DmetalintheUQL.TherehasbeenarecentrenewalofinterestintheinteractionQCC,(classI)duetothemetaltoinsulatortransitionobservedintwo-dimensional(highmobility)Si-MOSFETsamples[ 20 ].ThequalitativefeaturesofthistransitionwasunderstoodbyZala,NarozhnyandAleiner[ 19 ]whoshowedthattheinsulating(logarithmicupturnintheresistivity)behaviorinthediusiveregimeandmetallic(linearriseintemperature)behavioroftheresistivityintheballisticlimit(2D),areduetocoherentscatteringatFriedeloscillations.Below,werstoutlinetheirsimplequantummechanicalscatteringtheoryapproachtoshowhowtemperaturedependentcorrectionstoconductivityariseforscatteringatFriedeloscillations,andthenextendtheiranalysistoobtaintheinteractionQCCin

PAGE 23

3Dballisticlimit.Inchapter2weevaluatethiscorrectionfora3DsystemintheUQL. Figure1{4. ScatteringbyFriedeloscillations. 1{4 ).Consideranimpurityattheorigin;itspotentialUimp(~r)inducesamodulationofelectrondensityaroundtheimpurity.IntheBornapproximationonecanndtheoscillatingcorrection,n(r)=n(r)n0totheelectrondensityn(~r)=Pkjk(~r)j2: Hereristhedistancefromtheimpurity,kFistheFermimomentum,g=RUimp(~r)d~risthematrixelementforimpurityscatteringandn0istheelectrondensityintheabsenceofimpuritiesanddisdimensionality.Takingintoaccountelectron-electroninteractionsV0(~r1~r2)onendsadditionalscatteringpotentialduetotheFriedeloscillationsEq. 1{11 .Thispotentialcanbepresentedasasumofthedirect(Hartree)andexchange(Fock)terms[ 21 ]

PAGE 24

2V0(~r1~r2)n(~r1;~r2);(1{14) whereby(~r)wedenotediagonalelementsoftheoneelectrondensitymatrix, Asafunctionofthedistancefromtheimpurity,theHartree-FockenergyVoscillatessimilarlytoEq. 1{11 .Theleadingcorrectiontoconductivityisaresultofinterferencebetweentwosemi-classicalpathsshowning.4.Ifanelectronfollowspath\A,"itscattersotheFriedeloscillationcreatedbytheimpurityandpath\B"correspondstoscatteringbytheimpurityitself.Interferenceismostimportantforscatteringanglescloseto(orforbackscattering),sincetheextraphasefactoraccumulatedbytheelectrononpath\A"(ei2kR)relativetopath\B"iscanceledbythephaseoftheFriedeloscillationei2kFR,sothattheamplitudecorrespondingtothetwopathsarecoherent.Asaresult,theprobabilityofbackscatteringisgreaterthantheclassicalexpectation(takenintoaccounttheDrudeconductivity).Therefore,accountingforinterferenceeectsleadtoacorrectiontotheconductivity.Wenotethattheinterferencepersiststolargedistances,limitedbytemperatureRjkkFj1vF=T.Thusthereisapossibilityforthecorrectiontohavenontrivialtemperaturedependence.Thesignofthecorrectiondependsonthesignoftheeectivecouplingconstantthatdescribeselectron-electroninteraction.First,wewillstudythecontributionarisingfromtheHartreepotential.ConsiderascatteringprobleminthepotentialgiveninEq. 1{13 .Theparticle'swavefunctionisasumoftheincomingplanewaveandtheoutgoingsphericalwave(3D),=ei~k:~r+f()eikr

PAGE 25

wheref()isthescatteringamplitude,whichwewilldetermineintheBornapproximation.Fortheimpuritypotentialitselftheamplitudef()weaklydependsontheangle.AtzerotemperatureitdeterminestheDrudeconductivityD,whiletheleadingtemperaturecorrectionisT2(whenthescatteringtimeenergydependent),asisusualforFermisystems.WenowshowthatthisisnotthecaseforthepotentialinEq. 1{12 .Infact,takingintoaccountEq. 1{12 leadstoenhancedbackscatteringandthustotheconductivitycorrectionwhichdependsontemperatureas/T2lnT(in3D),T(in2D)and,aswewillseelaterT2(in3DUQL,istheinteractionparameter)allfortheballisticlimit. Farfromthescattererthewavefunctionofaparticlecanbefoundintherstorderofperturbationtheoryas=ei~k:~r+(~r),wherethecorrectionisgivenby[ 22 ] SubstitutingtheformoftheHartreepotentialfromEq: 1{13 ,andintroducingtheFouriertransformoftheelectron-electroninteractionV0(q),weobtainforthescatteringamplitude(atlargedistancesfromtheimpurity) 2Zd~rn(r)ei~q:~r;(1{17) where~q=~kk~r=randj~qj=2ksin(=2).Weseethatthescatteringamplitudedependsonthescatteringangle(),aswellastheelectron'senergy(=k2=2m).Thedensityoscillationin3D,withhardwallboundaryconditionattheorigin(impenetrableimpurity),isn(r)=Zd~kf(k)[jk(~r)j2j0k(~r)j2];=2kF 2kF(ra)sin(2kFr) 2kFr;

PAGE 26

whereaisthesizeoftheimpurityandf(k)istheFermidistributionfunction.Wemaketheswavescatteringapproximation(slowparticles,kFa1)toobtain r2cos(2kFr) 2kFrsin(2kFr) (2kFr)2:(1{18) SubstitutingthedensityfromEq. 1{18 inEq. 1{17 ,weobtainforthescatteringamplitude 2sin( 4lnj1sin( 1+sin( Inthelimit+xwherex1,thescatteringamplitudebehavesasf(x)V0(2kF)[xlnx].Thetransportscatteringcrosssectionisnow wheref0istheamplitudeforscatteringattheimpurityitself(whichdoesnotdependonintheBornlimitandgivesaconstant(Tindependent)valuefortheDrudeconductivity).Theleadingenergydependencecomesfromtheinterference(crossterm),whichisproportionaltof().Themaincontributiontotheintegralcomesfrom(backscattering).Expandingnear,i.e.,=+1where1issmall[ 19 ],1p 1 ThenoneobtainstheinteractionQCCfromtheHartreechannel[ 23 ]in3D(using=D==D) DV0(2kF)T EF2ln(EF Oneobtainsasimilarcontributionfromtheexchange(Fock)potential,exceptnowthecouplingconstantinfrontoftheT2lnTtermisV0(0).TheHartreeandexchangecontributioncomewithoppositesigns.In2DtheinteractionQCCis

PAGE 27

linearintemperature[ 19 ] D[2V0(2kF)V0(0)]T EF:(1{23) In1DYue,GlazmanandMatveev[ 21 ]usedthesameapproachandcalculatedthecorrectiontothetransmissioncoecientduetoscatteringattheFriedeloscillationandobtainedalogarithmictemperaturecorrectionatthelowestorder where[V02V2kF]=vF.Usingapoormanrenormalizationgroupprocedure,theyshowedthattherstorderlogarithmiccorrectionisinfactaweakcouplingexpansionofthemoregeneralpowerlawscalingformofthetransmissioncoecient,t=t0T W; W2:(1{25) Thisresultwasalsoobtainedindependently(viabosnization)byKaneandFisher[ 24 ].Eq. 1{22 1{23 ,andEq. 1{25 givetheinteractionQCCintheballisticlimitin3D,2Dand1Dsystemsrespectively.Inchapter2ofthisdissertationweshowthatin3DUQL,thisinteractioncorrectiontotheconductivitybehavessimilartothatofatrue1Dsystem. TheinteractioncorrectiontoQCCinthediusivelimitalsoarisesfromthesamephysics(namelyscatteringatfriedeloscillations)butnowonehasaverageovermanyimpurities(diusivemotion).ThiscorrectiontotheconductivitywasevaluatedbyAltshulerandAronovin3D[ 8 ]andbyAltshuler,AronovandLeein

PAGE 28

2D[ 25 ]. whereFisthedependsonthestrengthoftheinteraction,and 33F D;(3D):(1{27) ScatteringattheFriedeloscillationsalsoresultsinasingularenergy(temperature)dependenceofthelocaldensityofstateswhichcanbeobservedasazerobiasanomalyintunneling.ThelocalDOScanbeobtainedfromtheelectronsGreen'sfunctionusing()=ImRd~pGR(~(p);).ThecorrectiontotheGreen'sfunctioncanbeevaluatedthesamewayasweevaluatedthecorrectiontothewavefunctionduetoFriedeloscillationoritcanalsobeevaluateddiagrammaticallybycalculatingtheelectron'sselfenergyinthepresenceofdisorderandinteraction[ 8 ].

PAGE 29

excitationdecayshouldlikewisebeproportionalto2.Itturnsout,however,thatexcitationindisorderedsystemsdecaysfaster,whichraisesthequestionofvalidityofquasiparticledescriptionofdisorderedconductorsinlowdimensionalsystems.Apartfrombeingimportantinthedevelopmentofthetheory,thedecaytimeforoneelectronexcitations(thephaserelaxationtime),governsthetemperaturedependenceoftheWLQCC. ItwasshownbyAltshulerandAronovthatfor3Ddisorderedsystems,thephaserelaxationtimeisgovernedbylargeenergytransferprocessesandinthisregimeee(whereeeistheoutrelaxationtime).TheoutrelaxationtimecanbecalculatedfromtheBolztmannequation(withdiusivedynamicsfortheelectrons).Thisgives1 D)3=2in3D,[ 8 ].Howeverinlowerdimensions(d=1;2)electron-electroncollisionswithsmallenergytransfersisthedominantmechanismfordephasing.TheBolzmannapproach(whichisgoodforlargeenergytransfers)failsin2Dand1Dcase.Technically,therewouldbedivergencesforsmallenergytransfers[ 8 ]bothin2D(logarithmic)andquasi-1D(powerlaw)intheBolztmann-equationresultfortheoutrelaxationrate.Thesedivergencesmustberegularizedinaself-consistentmanner.Thephasebreakingtimeinlowerdimensionscanalsobeobtainedbysolvingtheequationofmotionfortheparticle-particle(Cooperon)propagatorinthepresenceofspaceandtimedependentuctuatingelectromagneticeldswhichmodelthesmallenergytransferprocesses[ 26 ].Thisgives()1T(in2D)and()1T2=3(inquasi-1D). Intrueone-dimensionalsystems,thissubjectiscontroversialastrue1Dsystemsdonothaveadiusiveregime(theballisticlimitcrossesovertothelocalizedregime)andthequasiparticledescriptionbreaksdownforaninteracting1DsystemwhichisintheLuttingerliquidstate.Asaresultonecannotdeneee.Inarecentworkonthissubject[ 27 ],itwasshownthatevenfora1DdisorderedLuttingerliquid,thereexistsaweaklocalizationcorrectiontotheconductivity

PAGE 30

whosetemperaturedependenceisgovernedbythephaserelaxationrate,()1/p );(1{28) whereD=e2vF2istheDrudeconductivityin1D,whichdependsonTthrougharenormalizationofstaticdisorder,0==(EF=T)2.Here0isnon-interactingscatteringtimeandistherenormalized(byFriedeloscillation)scatteringtimeandcharacterizestheinteraction. Atpresenttherearenotheoreticalpredictionsforin3DUQL.TheFermiliquidapproachesforcalculatingthephasebreakingtimearenotexpectedtoworkherebecausetheCooperonisnotasingulardiagram(itacquiresamassin3DUQLasshowninchapter2)and,onceagain,therearenosingleparticlelikeexcitationsasthegroundstateisacharge-density-waveandexcitationsabovethegroundstateareLuttingerliquidlike.Howeverinchapter2wewillshowthatsomerecentmagneto-resistancemeasurementsongraphiteinUQLqualitativelyagreewithpredictionsofduetoelectron-phononinteractionsin1D. 28 ].AsearchforstabilityconditionsofaFermiliquidanddeviationsfromaFermiliquidbehavior,[ 29 { 32 ]particularlynearquantumcriticalpoints,intensiedinrecentyearsmostlyduetothenon-Fermi-liquidfeaturesofthenormalstateofhighTcsuperconductors[ 33 ]andheavyfermionmaterials[ 34 ]. ThesimilaritybetweentheFermi-liquidandaFermigasholdsonlyfortheleadingtermsintheexpansionofthethermodynamicquantities(specicheat

PAGE 31

Inthisintroduction,wewilldiscussthedierencebetweenthe\regular"processeswhichleadtotheleadingFermi-liquidformsofthermodynamicquantitiesand\rare"1Dprocesseswhichareresponsibleforthenonanalytic(non-Fermiliquid)behavior.Wewillseethattheroleoftheserareprocessesincreasesasthedimensionalityisreducedand,eventually,therareprocessesbecomenormalin1D,wheretheFermi-liquiddescriptionbreaksdown. InaFermigas,thermodynamicquantitiesformregular,analyticseriesasafunctionofeithertemperatureT,ortheinversespatialscaleqofaninhomogeneousmagneticeld.ForTEFandqkF, where=2F=3,s0=gB2FandFmkFD2isthedensityofstates(DOS)ontheFermisurface,gistheLandefactorandBistheBohrmagnetonanda:::faresomeconstants.EvenpowersofToccurbecauseoftheapproximateparticle-holesymmetryoftheFermifunctionaroundtheFermienergy.Theaboveexpressionsarevalidinalldimensions,exceptD=2.ThisisbecausetheDOSisconstantin2D,theleadingcorrectiontotheTterminC(T)isexponentialinEF=Tandsdoesnotdependonqforq2kF.Howeverthisanomalyisremovedassoonaswetakeintoaccountanitebandwidthoftheelectronspectrum,uponwhichtheuniversal(T2nandq2n)behaviorisrestored.

PAGE 32

AninteractingFermisystemisdescribedbyLandau'sFermi-liquidtheory,accordingtowhichtheleadingtermsinC(T)andsaresameasthatoftheFermigaswithrenormalizedparameters(replacebaremassbyeectivemassm?,baregfactorbyeectiveg-factorg?intheaboveFermigasresults), whereFc;FsarechargeandspinharmonicsoftheLandauinteractionfunction:^F(~p;~p0)=Fc()^I+Fs()~:~0,where~,arethePaulimatrices.TheFermi-liquidtheoryisanasymptoticallylow-energytheorybyconstruction,anditisreallysuitableonlyforextractingtheleadingterms,correspondingtothersttermsintheFermigasexpression.Indeed,thefreeenergyoftheFermi-liquidofanensembleofquasiparticlesinteractinginapairwisemannercanbewrittenas[ 35 ]FF0=Xk(k)nk+1 2Xk;k0fk;k0nknk0+O(n3k); Strictlyspeaking,anonanalyticdependenceoffk;k0onthedeviationsfromtheFermisurfacekkF,accountsforthenon-analyticTdependenceofC(T)[ 36 ].HigherordertermsinTorqcanbeobtainedwithinmicroscopicmodelswhichspecifyparticularinteractionandemployperturbationtheory.SuchanapproachiscomplimentarytotheFL:theformerworksforweakinteractionsbutatarbitrarytemperatureswhereasFLworksbothforweakandstronginteractions,

PAGE 33

butonlyinthelimitoflowesttemperatures.Microscopicmodels(Fermigaswithweakrepulsion,electron-phononinteraction,paramagnonmodel,etc.)showthatthehigherordertermsinthespecicheatandspinsusceptibilityarenonanalyticfunctionsofTandq[ 37 { 48 ].Forexample, whereallcoecientsarepositiveforthecaseofelectron-electroninteraction.Asseenfromtheaboveequationsthenonanalyticitybecomesstrongerasthedimensionalityisreduced.Thestrongestnonanalyticityoccursis1D,where-atleastaslongassingleparticlepropertiesareconcerned-theFLbreaksdown[ 49 50 ]: Thesenonanalyticcorrectionstothespecicheatandspinsusceptibilityin1Dareobtainedinchapter3.Itturnsoutthattheevolutionofthenon-analyticbehaviorwiththedimensionalityreectsanincreasingroleofspecial,almost1Dscatteringprocessesinhigherdimensions.Thusnon-analyticitiesinhigherdimensionscanbeviewedasprecursorsof1DphysicsforD>1. WewillrststudythenecessaryconditiontoobtainaFLdescriptionandthenseehowrelaxingtheseconditionsleadtothenonanalyticformfortheself-energyandthermodynamicproperties.WithintheFermiliquid

PAGE 34

Landau'sargumentforthe"2(orT2)behaviorofImRrequirestwoconditions:(1)quasiparticlesmustobeyFermistatistics,i.e.,thetemperatureissmallerthanthedegeneracytemperatureTF=kFvF?,wherevF?istherenormalizedFermivelocity,(2)inter-particlescatteringisdominatedbyprocesseswithlarge(generally,oforderkF)momentumtransfers.Oncethesetwoconditionsweresatised,theself-energyassumesauniversalform,Eq. 1{40 andEq. 1{41 ,regardlessofaspecictypeofinteraction(electron-electron,electron-phonon)anddimensionality.Considertheself-energyofanelectron(1storder)asitinteractswithsomeboson(seeFig. 1{5 ).Thewavylinecanbe,e.g.,adynamicCoulombinteraction,phononpropagator,etc.Onthemassshell("=k;wherek=k2=2mkF2=2m)atT=0andfor">0 Figure1{5. Self-energyatrstorderininteractionwithabosoniceld ImR(")Z"0d!ZdD~qImGR("!;~k~q)ImVR(!;~q) (1{42) Theconstraintonenergytransfers(0
PAGE 35

Asafunctionofq,Fhasatleasttwocharacteristicscales.Oneisprovidedbytheinternalstructureoftheinteraction(screeningwavevectorfortheCoulombpotential)orbykFwhicheverissmaller.Thisscale,Q,doesnotdependon!andprovidestheultra-violetcutointhetheory.Inadditionthereisasecondscalej!j=vF,and,since!isboundedfromaboveby"andforlowenergies("!0),onecanassumeQj!j=vF.Thusinadimensionlessform ImVR(!;q)=! QDUq Q;j!j TheangularintegrationoverImGRyieldsonthemassshell vFq); wherethesubscriptDstandsforthedimensionality,andA3(x)=(1jxj);A2(x)=(1jxj) 1{45 andEq. 1{44 intoEq. 1{42 ,oneobtains ImR(")Z"0d!!ZQqj!j=vFdqqD2Uq Q;j!j NowifthemomentumintegralisdominatedbylargemomentaoftheorderofQ,thenthefunctionUtoleadingordercanbeconsideredtobeindependentoffrequency(sinceQj!j=vF),andonecanset!=0inU,andalsoreplacethelowerlimitoftheqintegralbyzero.Themomentumandfrequencyintegralsthendecouple,(themomentumintegralgivesapre-factorandthefrequencyintegralgives"2),andoneobtainsananalytic"2dependenceforIm.Thenthelinearin"terminRecanbeobtainedbyusingtheKramers-Kronigrelation.Thus

PAGE 36

weseethatlargemomentum(andenergyindependent)transfersanddecouplingofthemomentumandfrequencyintegralareessentialtoobtainaFLbehavior.The"2resultseemstobequitegeneralundertheassumptionsmade.Whenandwhyaretheseassumptionsviolated?Long-rangeinteraction,associatedwith Figure1{6. Kinematicsofscattering.(a)\Any-angle"scatteringleadingtoregularFLtermsinself-energy;(b)Dynamicalforwardscattering;(c)Dynamicalbackscattering.Processes(b)and(c)areresponsiblefornonanalytictermsintheself-energy small-anglescattering,isknowntodestroytheFL.Forexample,transverselongrange(current-current[ 51 ]orgauge[ 52 ])interactionswhich,unliketheCoulombinteractionarenotscreened,leadtothebreakdownoftheFermi-liquid.Buttheseinteractionsoccurunderspecialcircumstances(e.g.,nearhalf-llingforgaugeinteractions).Foramoregenericcase,itturnsoutthatevenifthebareinteractionisofthemostbenignform,e.g.,adelta-functioninrealspace,therearedeviationsfromaFLbehavior.Thesedeviationsgetampliedasthedimensionalityisreduced,and,eventually,leadtoacompletebreakdownoftheFLin1D.Alreadyforthesimplestcaseofapoint-likeinteraction,thesecondorderself-energyshows

PAGE 37

Figure1{7. Nontrivialsecondorderdiagramsfortheself-energy anontrivialfrequencydependence.Foracontactinteractionthetwoself-energydiagramsofFig.canbelumpedtogether(theseconddiagramis1=2therstone).Twogivenfermionsinteractviapolarizingthemediumconsistingofotherfermions.Hencetheeectiveinteractionatthesecondorderisproportionaltothepolarizationbubble,whichjustshowshowpolarizablethemediumis, ImVR(!;q)=U2ImR(!;q): Forsmallanglescatteringq2kF;!EF,theparticle-holepolarizationbubblehasthesamescalingforminallthreedimensions[ 53 ], vFqBD! vFq; whereD=aDmkFD2isthedensityofstatesinDdimensions(a3=2;a2=1;a1=2=)andBDisadimensionlessfunctionwhosemainroleistoimposeaconstraint!vFqin2Dand3D,and!=vFqin1D.TheaboveformofthepolarizationoperatorindicatesLandaudamping:Collectiveexcitations(spinandchargedensitywaves)decayintoparticle-holepairs,thisdecayoccursonlywithintheparticle-holecontinuumwhoseboundaryforD>1isat!=vFqforsmall!;q,therefore,decayoccursfor!
PAGE 38

getsin3D, ImR(")U2Z"0d!ZQkFj!j=vFdqq21 vFqU2Z"0d!![kF|{z}FL! vF|{z}beyondFL];a"2bj"j3; wherethersttermoriginatesfromthelargemomentumtransferregimeandistheFermi-liquidresultwhereasthesub-leadingsecondtermoriginatesfromthesmall-momentum-transferregimeandisnonanalytic.Thefractionofphasespaceforsmallanglescatteringissmall:mostoftheself-energycomesfromlarge-anglescatteringevents(qQ),butwealreadystarttoseetheimportanceforsmallangleprocesses.ApplyingKramers-Kronigtransformationtothenon-analyticpart(j"j3)inImR,wegetacorrespondingnon-analyticcontributiontotherealpartas(ReR)non-an/"3lnj"jand,nally,usingthespecicheatformula(seeEq. 3{14 inchapter3)wegetanonanalyticT3lnTcontributionwhichhasbeenobservedexperimentallybothinmetals[ 54 ](mostlyheavyfermionmaterials)andHe3[ 55 ].Similarlyin2D ImR(")U2Z"0d!ZkFj!j=vFdqq1 vFqU2"2lnEF andReR(")/"j"jandthisresultsintheT2non-analyticityforthespecicheatwhichhasbeenobservedinrecentexperimentsonmonolayersofHe3adsorbedonsolidsubstrate[ 56 ]. In1D,asweshowinchapter3,thesituationisslightlydierent.EventhoughthesamepowercountingargumentsleadtoImR/j"jandReR/"lnj"jforthesecondorderself-energy,C(T)islinear(analytic)inTatsecondorderandthenonanalyticTlnTshowsuponlyatthirdorderininteractionandonlyforfermionswithspin.Thisdierenceisduetothefactthatin1D,smallmomentumtransfers(hereparticle-holecontinuumshrinkstoasingleline!=vFq,sodecayof

PAGE 39

collectiveexcitationsispossibleonlyonthisline)donotleadtothespecicheatnonanalyticitywhichoccurssolelyfromthenonanalyticityofthebackscattering(atq2kF)particle-holebubbleortheKohnanomaly.Thus,wehavethesamesingularbehaviorofthebubble(responsefunctions)andtheresultsfortheself-energydiersbecausethephasevolumeqDismoreeectiveinsuppressingthesingularityinhigherdimensionsthaninlowerones. Inadditiontotheforwardscatteringnonanalyticity,thereisalsoanonanalyticcontributiontotheself-energyandthermodynamicsarisingfromq2kF,partoftheresponsefunction,i.e.,theKohnanomaly.Usually,theKohnanomalyisassociatedwiththe2kFnonanalyticityofthestaticparticle-holebubbleanditsmostfamiliarmanifestationistheFriedeloscillationinelectrondensityproducedbyastaticimpurity(seesection1.1.2,ofthisthesis).HerethestaticKohnanomalyisofnointerestaswearedealingwithdynamicalprocesses.However,thedynamicalbubbleisalsosingularnear2kF,e.g.,in2D ImR(q2kF;!)/! Duetotheone-sidedsingularityinImRasafunctionofq,the2kFeectiveinteractionoscillatesandfallsoasapowerlawinrealspace.Bypowercounting,sincethestaticFriedeloscillationfallsoassin(2kFr) ~U/!sin(2kFr) DynamicalKohnanomalyresultsinthesamekindofnon-analyticityintheself-energy(andthermodynamics)astheforwardscattering.Thesingularitynowcomesfromjq2kFj!=vF,i.e.,dynamicbackscattering.Thereforethenonanalytictermintheself-energyissensitiveonlytostrictlyforwardor

PAGE 40

backscatteringevents,whereasprocesseswithintermediatemomentumtransferscontributetotheanalyticpartoftheself-energy. Figure1{8. Scatteringprocessesresponsiblefordivergentand/ornonanalyticcorrectionstotheself-energyin2D.(a)\Forwardscattering"-ananalogoftheg4processin1D(b)\Forwardscattering"withanti-parallelmomenta-ananalogoftheg2processin1D(c)\backscattering"withantiparallelmomenta-ananalogoftheg1processin1D Wewillnowperformakinematicanalysisandshowthatthenonanalytictermsintheself-energyandspecicheatin2Dcomesfromonly1Dscatteringprocesses.Considertheself-energydiagramofFig.1-7.(a).Thenonanalytic"2ln"termintheself-energycamefromtwoq1singularities:onefromtheangularaverageofImGRandtheotheronefromthedynamic,!=vFqpartoftheparticle-holebubble.Thisformofthebubblearisesonlyinthelimit!vFq, ImR(!;q)ImZZdD~pd"G("!;~p~q)G(";~p)! vFqZd(cos! vFq);! vFqq vFq(for!vFq): Fromthedeltafunction,cos=!=vFq1,whichmeansthattheanglebetween~pand~qis=2or~pand~qareperpendiculartoeachother.SimilarlytheangularaveragingofImGR(~k~q;";!)alsopinstheanglebetween~kandqto90degrees.hImGR(~k~q;";!)i1Zd1("!qvFcos1)=)cos1"! vFq! vFq1=)1=2 Thus~pand~k(thetwoincomingmomentaofthefermions)arealmostperpendiculartothesamevector~q.In2D,thismeansthattheyareeitheralmostparalleltoeachotheroranti-paralleltoeachother,andsincethemomentumtransferiseither

PAGE 41

small,~q0ornear2kF,i.e.,jq2kFj0,weessentiallyhavethree1Dscatteringprocesses(seeFig. 1{8 )whichareresponsibleforthenonanalyticcorrectionstotheself-energy.Thesethreeprocessesare(a)twofermionswithalmostparallelmomenta(~k1~k2)collideandtransferasmallmomentum(~q0)andleavewithoutgoingmomentumwhicharealmostparalleltoeachother(~k10~k20)andparalleltotheirincomingmomenta(~k10~k1;~k20~k2):analogoustothe\g4"scatteringmechanismin1D(seeFig. 1{8 (a)andchapter3)(b)twofermionswithalmostanti-parallelmomentacollide(~k1~k2)andtransferasmallmomentum(~q0)andleavewithoutgoingmomentumwhicharealmostanti-paralleltoeachother(~k10~k20)butparalleltotheirincomingmomenta(~k10~k1;~k20~k2):analogoustothe\g2"scatteringmechanismin1D(seeFig. 1{8 (b)andchapter3),(c)twofermionswithanti-almostparallelmomentacollide(~k1~k2)andtransferalargemomentum~q2kFandleavewithoutgoingmomentumwhicharealmostanti-paralleltoeachother(~k10~k20)andalsoanti-paralleltotheirincomingmomenta(~k10~k1;~k20~k2):analogoustothe\g1"scatteringmechanismin1D(seeFig. 1{8 (c)andchapter3).Thereforethenonanalytic"2ln"termintheself-energyin2Dcomesfrom1Dscatteringevents,theonlydierenceisthat2Dtrajectoriesdohavesomeangularspread,whichisoftheorderofj!j=EF.Itturnsout(Ref.[ 44 ]),thatoutofthethree1Dprocesses,theg2processandg1processaredirectlyresponsiblefornonanalyticcorrections(NAC)toC(T)in2Dandonlytheg1processleadstoNACtoC(T)in1D.Theg4processalthoughleadstoamass-shellsingularityintheself-energyinboth2Dand1D,butdoesnotgiveanyNACtothermodynamics. In3Dthesituationisslightlydierent,~p?~qand~k?~qmeanthatboth~pand~klieinthesameplane.However,itisstillpossibletoshowthatforthethermodynamicpotential,~pand~kareeitherparalleloranti-paralleltoeachother.Hence,thenonanalyticterminC(T)alsocomesfromthe1Dprocesses.In

PAGE 42

addition,thedynamicforwardscatteringevents(markedwithastarinFig.1-9.)which,althoughnotbeing1Dinnature,doesleadtoanonanalyticityin3D.ThustheT3lnTanomalyinC(T)comesfromboth1Dandnon-1Dprocesses[ 47 ].Thedierenceisthattheformerstartalreadyatthesecondorderininteractionwhereasthelatteroccuronlyatthirdorder.In2D,theentireT2nonanalyticityinC(T)comesfrom1Dprocesses.Thenonanalyticcorrectiontothespinsusceptibilitywillbethesubjectofdiscussioninchapter4ofthisthesis,wherewewillshowthatthenonanalyticityins,bothin2Dand3Dcomesfromboth1Dandnon1Dscatteringprocesses. Typicaltrajectoriesoftwointeractingfermions Ourkinematicargumentscanbesummarizedinthefollowingpictorialway.Supposewefollowthetrajectoriesoftwofermions,asshowninFig. 1{9 .There

PAGE 43

areseveraltypesofscatteringprocesses.First,thereisa\any-angle"scatteringwhich,inourparticularexample,occursatathirdfermionwhosetrajectoryisnotshown.Thisscatteringcontributesaanalytic,FLtermsbothtotheself-energyandthermodynamics.Second,therearedynamicforwardscatteringevents,whenqj!j=vF.Thesearenon-1Dprocesses,asthefermionsentertheinteractionregionatanarbitraryangletoeachother.In3D,athirdorderinsuchaprocessleadstoaT3lnTterminC(T).In2Ddynamicforwardscatteringdoesnotleadtoanonanalyticity.Finallythereare1DscatteringprocessesmarkedwithaSiriusstarwherefermionsconspiretoaligntheirmomentaeitherparalleloranti-paralleltoeachother.TheseprocessesdeterminethenonanalyticpartofandC(T)in2Dand1D. Thereforethenonanalytictermsinthetwo-dimensionalself-energyandthermodynamicsarecompletelydeterminedby1Dprocesses,2Dscatteringdoesnotplayanyroleinthenonanalyticterms.Asaresult,ifthebareinteractionhassomeqdependence,onlytwoFouriercomponentsmatter:U(0)andU(2kF)e.g.,in2D ImR(")/[U2(0)+U2(2kF)U(0)U(2kF)]"2lnj"j; ReR(")/[U2(0)+U2(2kF)U(0)U(2kF)]"j"j; whereaandbaresomecoecients.TheseperturbativeresultscanbegeneralizedfortheFermi-liquidcase,wheninteractionisnotweak.ThentheverticesU(0)andU(2kF),occurringintheperturbativeexpressionsarereplacedbyscatteringamplitude()atangle=, ^(~p;~p0)=c()^I+s()~:~0;

PAGE 44

wherecandsrefertothechargeandspinsectorsrespectively.Thusin2D[ 45 ], Theadditional(lnT)2factorinthedenominatorcomesfromtheCooperchannelrenormalizationofthebackscatteringamplitude[ 47 48 ].In3D,theT3lnTnonanalyticityinthespecicheatarisesfromboth1D(excitationofasingleparticle-holepair)andnon-1D(excitationofthreeparticle-holepairs)scatteringprocesses[ 47 ]. {z }1D,onep-hpair+a;12a;0+3a;1+:::| {z }non1D,threep-hpairs wheresubscripta=c;sand0;1;2:::indicatetheharmonicsoftheexpansion.Again,theadditional(1+glnT)2factorinthedenominatorcomesfromtheCooperchannelrenormalizationofthebackscatteringamplitude[ 47 48 ]. WesawthatthenonanalyticcorrectionstothespecicheatinD=2;3,arisefromonedimensionalscatteringprocesses,(andtheyshowupatsecondorderinperturbationtheory),andthedegreeofnonanalyticityincreaseswithdecreaseindimensionality.Thispredictsthatthestrongestnonanalyticityinthespecicheatshouldoccurin1D.However,itwasshowninRef.[ 57 ],thatthespecicheatin1DislinearinT,atleastinsecondorderinperturbationtheory.Inaddition,thebosonizationsolutionofaone-dimensionalinteractingsystem,predictsthattheC(T)islinearinT.Weresolvethisparadoxbyshowing(inchapter3)thatthegeneralargumentfornonanalyticityinD>1atthesecondorderininteractionbreaksdownin1D,duetoasubtlecancelationandthenonanalyticTlnTterminthespecicheatoccursatthirdorderandonlyforelectronswithspin.ThisisveriedbyconsideringtheRGowofthemarginallyirrelevantoperatorintheSine-Gordontheory(whichappearsinthebosonizationschemeforfermionswithspin).Forspinlesselectronsweshowthatthenonanalyticitiesinparticle-particle

PAGE 45

andparticle-holechannelscompletelycanceloutandtheresultingspecicheatislinearinT(thebosonizedtheoryisgaussian).Thesingularityintheparticle-holechannelresultsinanonanalyticbehaviorforthespin-susceptibilitys/lnmax[jQj;jHj;T],presentatthesecondorder. Thespinsusceptibilitybothin2Dand3Dgetsnonanalyticcontributionsfromboth1Dandnon-1Dprocesses.ThesecorrectionswillbedescribedindetailinChapter4ofthisthesiswherewealsostudythenonanalyticcorrectionsnearaferromagneticquantumcriticalpoint. 58 ]derivedaLandau-Ginzburg-Wilson(LGW)functionalforthistransitionbyconsideringasimplemodelofitinerantelectronsthatinteractonlyviatheexchangeinteraction.HertzanalyzedthisLGWfunctionalbymeansoftherenormalizationgroup(RG)methodsthatgeneralizetheWilson'streatmentofclassicalphasetransitions.Heconcludedthattheferromagneticorderinanitinerantsystemsetsinviaacontinuous(or2ndorder)quantumphasetransitionandtheresultingstateisspatiallyuniform.Furthermore,heshowedthatthecriticalbehaviorinthephysicaldimensionsd=3andd=2ismean-eld-like,

PAGE 46

sincethedynamicalcriticalexponentz=3,(whicharisesduetothecouplingbetweenstaticsanddynamicsinaquantumproblem),decreasestheuppercriticaldimensionfromd+c=4fortheclassicalcasetod+c=1inthequantumcase.Hertz'stheorywhichwaslaterextendedbyMillis[ 59 ]andMoriya[ 60 ],(itiscommonlyreferredastheHertzMillisMoriya(HMM)theory),isbelievedtoexplainthequantumcriticalbehaviorinanumberofmaterials[ 61 ];however,thereareothersystemswhichdonotagreewiththeHMMpredictionsandshowarstordertransition,(e.g.,UGe2),totheorderedstate.Thiscontradictionmotivatedthetheoriststore-examinetheassumptionsmadeintheHMMtheory. TheHMMscenarioofaferromagneticquantumphasetransitionisbasedontheassumptionthatfermionscanbeintegratedoutsothattheeectiveactioninvolvesonlyuctuationsoftheorderparameter.Thisassumptionhasrecentlybeenquestioned,asmicroscopiccalculationsrevealnon-analyticdependencesofthespinsusceptibilityonthemomentum(q),magneticeld(H),andforD6=3,temperature(T)[ 42 44 ]bothawayandnearthequantumcriticalpoint(seethediscussioninsection1.2).Forexample,in2D s(H;Q;T)=const.+max(jHj;jQj;T); andin3D s(H;Q)=const.+(q2;H2)ln[max(jHj;jQj)]; whereH,qandTaremeasuredinappropriateunits.ThedependenceonTisnonanalyticinthesensethattheSommerfeldexpansionfortheFermigascanonlygenerateevenpowersofT.Ofparticularimportanceisthesignofthenonanalyticdependence:s(H;Q)increasesbothasafunctionofHandq(at2ndorderinperturbationtheory)forsmallH,q.Ass(H;Q)mustdenitelydecreaseforHandqexceedingtheatomicscale,thenaturalconclusionisthenithasamaximum

PAGE 47

atniteHandq.Thismeansthatthesystemshowsatendencyeithertoarstordertransitiontoauniformferromagneticstate(themetamagnetictransitionasafunctionoftheeld),ororderingatniteq,(toaspiralstate).Thechoiceoftheparticularscenarioisdeterminedbyaninterplayofthemicroscopicparameters.InChapter4ofthisthesis,wewillobtainthenonanalyticcorrectionstos(H)insecondandthirdorderinperturbationtheoryandshowthatthesecorrectionsoscillatebetweenpositiveat2ndorder,(whichpointstowardsametamagnetictransition),andnegativeat3rdorder(whichpointstowardsacontinuoussecondorderphasetransition)values.Thusitisimpossibletopredictthenatureofthephasetransitionbyinvestigatingthenonanalytictermsatthelowestorderinperturbationtheory.Furthermore,inrealsystemsinteractionsarenotweakandonecannotterminatetheperturbationtheorytoafewloworders.Tocircumventthisinherentproblemwithperturbativecalculationsandtomakepredictionsforrealisticsystems(e.g.,He3),weobtainthenonanalyticelddependenceforagenericFermiliquidbyexpressingourresultintermsofthelowestharmonicsoftheLandauinteractionparameters.Wealsodescribethenonanalyticelddependencenearthequantumcriticalpointusingtheself-consistentspin-fermionmodel,andshowthatthesignofthecorrectionsismetamagnetic.Here,intheintroduction,webrieyreviewHertz'stheoryofthesecondorderphasetransition. ThepartitionfunctionisobtainedbyperformingaHubbard-Stratonovichtransformationtodecouplethefour-fermioninteractioninthechargeandspinchannel.Thechargechannelisassumedtobenon-criticalandisthusdiscarded,

PAGE 48

whereasthepartitionfunctionforthespinchanneltakesthefollowingform; where Theeldlistheconjugateeldtothenl"nl#,whichcanbeconsideredasthemagneticeldactingonthefermions.Performingthefunctionalintegrationoverthefermionoperators(C;Cy)hearrivedatthepartitionfunctionZ=ZDeSeff(); TheMean-eld-theorywouldcorrespondtothesaddlepointapproximationtothefunctionalintegrationwithrespectto.Todeduceaneective(LGW)functional,oneexpandstheinteractionterm(Trlnterm)inl.Thematrix(M)intheTrlnterminEq. 1{66 intheFourierspacebecomes (M)(k;i!n;;k0;i!m;0)=0[(i!n+k)!n;!m~k;~k0+U

PAGE 49

ThersttermontherighthandsideoftheaboveequationistheinverseGreen'sfunctionforfreefermions(G10),thesecondtermisthe\interaction"(V).ThenTrln[M]=Trln[G10(1G0V)]=Trln[G10]+Trln[1G0V]=Trln[G10]1Xn=11 2X~q;i!lv2(~q;i!l)(~q;i!l)(~q;i!l)+1 4VX~qi;i!iv4(~qi;i!i)(~q1;i!1)(~q2;i!2)(~q3;i!3)(~q4;i!4)(4Xi=1~qi)(4Xi=1!i): ThecoecientsvminEq. 1{68 aretheirreduciblebarem-pointverticesinthediagrammaticperturbationtheorylanguage.Thequadraticcoecientisv2(~q;i!l)=1U0(~q;i!l),where0(~q;i!l)isthefreeelectronsusceptibilitygivenbytheLindhardfunction(Polarizationbubble),whichatsmallqandsmall!=qvFbehavesas 3q Hertzassumedtheallthehigherordercoecientsvmstartingwithv4canbeapproximatedasconstantsastheyvaryonthescaleofq2kFand!EF.InappropriateunitsHertz'sformoftheeectiveLGWfunctionalis 2X~q;i!m(r0+q2+j!mj (1{70)

PAGE 50

wherer0=1UF2(isthecorrelationlengthwhichdivergesatthephasetransition),isthedistancefromthecriticalpointandu0=U400F=12isaconstant.Thus,Hertz'seectiveactionisalmostofthesameformastheclassicalLGWfunctional(forthe4theory),exceptforthepresenceofthefrequencydependentterminv2whichcontainstheessentialinformationaboutthedynamics.Theactionthereforedescribesasetofinteracting,weaklyLandau-damped(duetothej!mj=qvFterm)excitations:paramagnons. HertzthenappliedWilson'smomentumshellrenormalizationgrouptransformationtotheabovequantumfunctional.Here,qand!havetobere-scaleddierently.Thisisduetothefactthatintheparamagnonpropagator(v21),qandj!mjappearinanon-symmetricway.Therefore,thesystemisanisotropicinthetimeandspacedirections.Asaresultitbecomesnecessarytointroduceanewparameter,thedynamicalcriticalexponentzforscaling Forthequantumferromagnetictransitionwhichwestudyhere,z=3.IntheRGprocedureconsistsofthefollowingsteps(a)highenergystates(withqand!)inthe"outershell"(>q>=b;>!>=b;b>1;isacut-o)areintegratedout;(b)variablesqand!,arere-scaledasq0=qeland!0=!ezl,withlbeinginnitesimal.(c)eldsarealsore-scaledsothatintermsoftheneweldsandre-scaledqand!,theq2andj!j=qtermsinthequadraticpartoftheactionlookslikethoseintheoriginalfunctional.Performingallthesesteps,HertzobtainedthefollowingRGequations dl=2r+12uf2; dl=u18u2f4;

PAGE 51

where=4(d+z)andtheexpressionsforf2andf4canbefoundinRef.[ 58 ].ThesecondRGequationshowsthattheGaussianxedpoint,withu=0,isstableifisnegative,thatis,ifd>4z.Forz=3,weshouldthereforeexpectastableGaussianxedpointandLandauexponentsind=2;3. ThetwomainassumptionsthatHertzmadeinarrivingathisLGWfunctional(Eq. 1{68 and 1{70 )were:(1)thecoecientsvm;m4arenonsingularandcanbeapproximatedbyconstantsand(2)thestaticspinsusceptibilityhasregularq2momentumdependence.Forthe2Dferromagnetictransition,nonanalytictermsinvmwerefoundbyChubukovetal.,[ 62 ],however,theauthorsclaimedthatthesenonanalyticitiesdonotgiverisetoananomalousexponentinthespinsusceptibilityandthereforewerenotdangerous.Inchapter4ofthisdissertationweexaminethesecondassumption(2)morecarefully.ThereasoningbehindHertz'ssecondassumptionwasthebeliefthatinitinerantferromagnetstheqdependenceofthe2termcomessolelyfromfermionswithhighenergies,oftheorderofEForbandwidth,inwhichcasetheexpansioninpowersof(q=pF)2shouldgenerallyholdforqpF.ThisreasoningwasdisputedinRefs.[ 42 44 ].Theseauthorsconsideredastaticspinsusceptibilitys(q)inaweaklyinteractingFermiliquid,i.e.,farawayfromaquantumferromagnetictransition,andarguedthatforD3andarbitrarysmallinteraction,thesmallqexpansionofs(q)beginswithanonanalyticjqjd1term,withanextralogarithminD=3.Thisnonanalyticityoriginatesfromthe2pFsingularityintheparticle-holepolarizationbubble[ 42 { 44 ]andcomesfromlowenergyfermions(inthevicinityoftheFermisurface),withenergiesoftheorderofvFqEF.ThesenonanalytictermsarisewhenoneconsidersthereferenceactionS0astheonewhichcontainstheparticle-holespinsingletchannelinteraction(chargechannel)andtheCooperchannelinteraction,whichwereneglectedintheHertzmodel(Hertz'sreferenceactionwasjustthenoninteractingone).Furthermore,thepre-factorofthis

PAGE 52

termturnsouttobenegative,whichindicatesthebreakdownofthecontinuoustransitiontoferromagnetism.ThusaccordingtoRef.[ 42 63 ]themodiedeectiveactionnearthecriticalpointshouldbe 2X~q;i!m(r0jqjD1+q2+j!mj withanextralogarithminD=3.TheweakpointofthisargumentisthatwithintheRPA,oneassumesthatfermionicexcitationsremaincoherentatthequantum-criticalpoint(QCP).Meanwhile,itisknown[ 64 ]thatuponapproachingtheQCP,thefermioniceectivemassm?divergesaslninD=3and3Dinsmallerdimensions.Itcanbeshownthatm=m?appearsasaprefactorofthejqjD1term;whichwouldmeanthatthenonanalytictermvanishesattheQCP.ThisstilldoesnotimplythatEq. 1{70 isvalidatthetransitionbecause,asweshowinchapter4,thedivergenceinm?doesnotcompletelyeliminatethenonanalyticterm,itjustmakesitweakerthanawayfromtheQCP. Ourapproachwillbetousethelow-energyeectivespin-Fermionhamiltonian,whichisobtainedbyintegratingthefermionswithenergiesbetweenthefermionicbandwidthWandalowercut-o(withW),outofthepartitionfunction[ 64 65 ]: HereSqdescribethecollectivebosonicdegreesoffreedominthespinchannel,andgisresidualspin-fermioncoupling.InHertz'sapproach,allfermionswereintegratedout,whereasintheSpin-Fermionmodelonlythehigh-energyfermionsareintegratedoutwhilekeepingthelow-energyones.Thiswillturnouttobeimportantbecausethespinuctuationpropagatorisrenormalizedbythefermions,andthefermionselfenergyisrenormalizedbyinteractionwithbosons.Thismodelhastobesolvedself-consistentlyasittakesintoaccountthelow-energy(mass)

PAGE 53

renormalizationofthespinuctuationpropagator.Inchapter4ofthisdissertationweusethismodeltoobtainthemagneticelddependenceofthespinsusceptibilitynearthequantumcriticalpoint,andanalyzethestabilityofthesecondorderquantumphasetransition.

PAGE 54

One-dimensionalsystemsexhibituniquephysicalpropertieswhichreecttheinuenceofstrongcorrelations.Theeectivedimensionalityofchargecarriersinabulkmetalmaybereducedfrom3Dto1Dbyapplyingastrongmagneticeld.Ithasrecentlybeenshownthatthisreductionleadstoformationofastronglycorrelatedstate,whichbelongstotheuniversalityclassofaLuttingerliquid[ 5 ].Thetunnelingdensityofstatesexhibitsacharacteristicscalingbehaviorforthecaseoflong-rangerepulsiveinteraction[ 5 6 ].Thiseectismostpronouncedintheultra-quantumlimit(UQL),whenonlythelowestLandaulevelremainsoccupied.Here,inthischapterweinvestigatetheeectofdiluteimpuritiesonthetransportpropertiesofthesystem.Forgoodmetals,thequantizingeldistoohigh(oftheorderof104Tesla),butsemi-metalsanddopedsemiconductorshavealowcarrierdensityandquantizingeldsoftheorderof110Teslaandallowforaexperimentaltestofthetheoreticalpredictionsmadehere. Insection2.1wediscusslocalizationeectsfornon-interactingelectronsintheUQL.Wendthatthelocalizationbehaviorisintermediatebetween1D(D:stronglocalization)and3D(D:weaklocalization).Weshowthattheparticle-particlecorrelator(Cooperon)ismassiveinthestrongmagneticeldlimit.It's\mass"(inunitsofthescatteringrate)isoftheorderoftheimpurityscatteringrate.Therefore,localizationinthestrong-eldlimitproceedsasifastrongphase-breakingprocessisoperatingasfrequentlyasimpurityscattering.EvenatT=0,thisphase-breakingexistsasitisprovidedbythemagneticeldandasaresultcompletelocalizationneveroccursin3DUQL.Ontheother 43

PAGE 55

hand,theparticle-holecorrelator(thediuson)remainsmassless,whichmeansthatnormalquasi-classicaldiusiontakesplace.Ourndingsareinagreementwithpreviousworkonthissubject[ 66 67 ],wherethelocalizationproblemwasanalyzedforthecaseoflongrangeddisorder,whereasinourstudywehaveanalyzedthecaseofshortrangeddisorder.OurresultforconductivityintheUQLiscoop+Drude=Drude=2. Insection2.2wecalculatethecorrectionstotheconductivityduetoelectron-electroninteractionsusingnite-temperaturediagrammatictechniquewheredisorderistreatedintheballisticlimit.Duetothisreducedeectivedimensionality,torstorderininteraction,theleadingcorrectionsarelogarithmicintemperature.Anotherwayofobtainingtheconductivityistocalculatetheinteractioncorrectiontothescatteringcross-sectionthroughanimpurity(inaHartree-Fockapproximation)anduseaDruderelationbetweenthecross-sectionandtheconductivity.Weshowinsection2.3that,torstorderintheinteraction,thetwoapproachesareequivalent.Thisisimportantsince,whileahigherordercalculationusingthediagrammatictechniquewouldbeextremelylengthy,theinteractioncorrectiontothecross-sectionisobtainedtoallordersviaanexactmappingontoa1Dproblemoftunnelingconductanceofinteractingelectronsthroughabarrier[ 21 ].WendthattheDrudeconductivitiesparallel(=+1)andperpendicular(=1)tothemagneticeldexhibitthescalinglaws/T2,whereisafunctionofthemagneticeld.Thephysicalreasonforsuchabehavioroftheconductivityisanearly1DformoftheFriedeloscillationaroundanimpurityinthestrongmagneticeld. ThegroundstateofrepulsivelyinteractingelectronsintheUQLisknowntobeunstabletotheformationofacharge-densitywave(CDW)[ 1 { 3 ].Thishasbeenconrmed,forexample,byexperimentsongraphiteinhighmagneticelds[ 4 ].BoththeHartree-Fockandthediagrammaticcalculationspresentedherearedone

PAGE 56

withouttakingintoaccountrenormalizationcorrectionsfortheinteractionverticesthemselves.ThisisjustiedatenergiesmuchlargerthantheCDWgapbutbreaksdownatlowenoughenergies.Inorderforourresultstohold,thereshouldexistanintermediateenergyintervalinwhichtherenormalizationoftheinteractionverticesduetoCDWuctuationsisnotyetimportantbutthepower-lawrenormalizationofthescatteringcross-sectionisalreadysignicant.Thatsuchanintervalexistsforthecaseoflong-rangeelectron-electroninteractionwasshownbysolvingthefullRGequationsfortheverticesandforthecross-section.Wehavenotincludedthisdiscussionhereforbrevity.Wediscusspossibleexperimentalvericationofourresultsinsection2.4andconcludeinSection2.5. 66 68 ],atleastforshort-rangeimpurities.Inthiscase,whilescatteringatanimpurity,anelectronmovestransversetotheeldbyadistanceoftheorderofthemagneticlengthlH=1=p

PAGE 57

followingsubsectionweillustratethedierentscenariowhicharisesfor3DelectronsintheUQL. 22 ]pz;m(;;z)=eipzzeim where(r?=;).ThereasonforthisseparabilityisthedegeneracyoftheLandaulevel;theenergydoesnotdependonthetransversequantumnumbers.The1Dpart,G1D,isinthemomentumspaceandthetransversepart,G?,hasbeenkeptinrealspace.ThedisorderaveragedGreen'sfunctionisobtainedbydoing

PAGE 58

perturbationtheoryintheimpuritypotentialU(forweakdisorderkF`1,sothatthesmallparameteris1=kF`)andemployingstandardcrosstechnique[ 18 ]fordisorderaveraging.Theperturbative(inU)solutionoftheSchrodingerequationfortheGreen'sfunctionisG(r;r0;i")=G0(r;r0;i")+Zd3r1G0(r;r1;i")U(r1)G0(r1;r0;i")+ZZd3r1d3r2G0(r;r1;i")U(r1)G0(r1;r2;i")U(r2)G0(r2;r0;i")+::: Theleadingcontributiontotheself-energycomesfromthesecondorderdiagram.TherstandthirdordercorrectionsarezeroashU(r)i=0.WeworkintheBornlimit,neglectingprocesseswhereanelectronscattersfrommorethantwo(same)impurities.ThesecondordercorrectionishG2(r;r0;i")i=ZZd3r1d3r2G0(r;r1;i")G0(r1;r2;i")G0(r2;r0;i")hU(r1)U(r2)i; (i"p)2#"1Xm=0eim(0) (i"p)3#"1Xm=0eim(0)

PAGE 59

wherethetransversepartineachoftheseexpressionsissimplyG?(r?;r0?).Atfourthorder,therearethreediagrams,therainbowdiagram(Fig.2-1,(b))andtheintersectingdiagram(Fig.2-1,(c))aresmallbyafactor1=kF`comparedtotheleadingone(hG4i)(Fig.2-1,(a))forshort-rangeweakdisorder.IntheBorn Figure2{1. Diagram(a)istheleadingcontributiontotheselfenergyatfourthorder approximation,thescatteringrateinamagneticeld,is1==2niu02H,whereH=1=(22vFlH2)isthe3Ddensityofstatesinthepresenceofamagneticeld,andtheself-energyis=isgn(")=2.ThefullDyson'sseries(Fig. 2{2 )canbesummedtogive: 2!G?(r?;r0?) (2{3) Therefore,theeectofimpurityscatteringentersonlyinthe1DpartoftheGreen'sfunction.UsingtheaboveformoftheGreen'sfunctionandtheKuboformulawenowevaluatetheDrudeconductivity.TheKuboformulaforthelongitudinald.c.conductivity(EkHkz)inthekineticequationapproximationiszz=lim!!01 2{3 ).Duetothe

PAGE 60

Figure2{2. Dyson'sseries Figure2{3. Drudeconductivity factorizabilityoftheGreen'sfunction,theconductivityisalsoseparableintoa1Dandatransversepart:zz=1d?.The1DpartisinthestandardformandgivesthefamousEinstein'srelationforthed.cconductivity1d=e21D=e2`=,whereD=v2Fisthediusioncoecientand1=1=(vF)isthedensityofstatesin1D.Usingtheorthonormality:Rd[Rm()]2=1andcompleteness:Pn[Rn()]2=1=(lH2)propertiesofthewavefunction,thetransversepartcanbeshowntobeequalto1withthedegeneracyfactorofthelowestlandaulevel.?=1Xm;n=0Zd00Zd0eim(0) 2l2H1:

PAGE 61

magneticeldandevaluatetheCooperoncorrectiontotheconductivity.TheDiusonremainsmasslesswhichmeansnormalquasiclassicaldiusionoccursintheparticle-holechannel.Implicationsoftheseresultsonelectronlocalizationwillbediscussed. Figure2{4. Thirdandsecondorderfandiagram. WecalculatethesecoecientsforthelowestorderCooperondiagrams(2ndand3rdorderfandiagramshowninFig. 2{4 )explicitlyandthenstatethegeneralargumentbywhichthesenumberscanbeobtainedforallhigherorderdiagrams.

PAGE 62

Forthesecondorderfandiagram, Theone-dimensionalpartofEq.( 2{4 )isgivenby whereR(q)istheonedimensionalrungintheparticle-particlechannelandX(q;!)isthepartcontainingthevertices: Weusethelinearspectrumapproximation,Rdp1 Forthevertex,linearizingandusingqpF,(wecannotsetq=0inthevertexaprioribecauseourcooperonwillacquireamass)followedbythepoleintegrationin,andthe"-integration,weobtainfor!>0,

PAGE 63

andnallyforthe1Dpartoftheconductivity Notethat1dD,soperturbationtheorybreaksdownintheUQL.Thetransversepartoftheconductivityis, wherer?=(;)andG?isdenedinEq.( 2{2 ).Afterperformingtheazimuthalintegrations,weobtain (2)3Zd00R2l(0)Zd11Rm(1)Rn(1)Rl(1)Rn0(1)21Xl;m=0l+mXn=0R2m() (2)3[Almnn0]2: wheren0=l+mn.Noticethatthesecondorderfandiagramhastworadialintegrations,likewisethirdorderfandiagramswillhavethreeradialintegrations,andsoon.UsingtheintegralrepresentationoftheGammafunction,Almnn0is 2l2H1 2m+l1 andthetransversepartoftheconductivitybecomes: 42lH231Xl;m=0l+mXn=0 lH2me2 2lH2

PAGE 64

Thesumovernisdoneusingthebinomialpropertyandthesumoverlisatabulatedsum[ 69 ], 42lH231Xm=02 2lH2 =1 22lH23: Thecoecientofthetransversepartofthesecondorderfandiagramfortheconductivityisc2=1=2.CombiningEq.( 2{10 )and( 2{16 ),theQCCfromthefandiagramatsecondorderiszz=e2DH=8. Similarly,thehigherorderdiagramscanbeevaluated.ThethirdorderfandiagramhasthesamevertexasthesecondorderonebuthasoneextrafactoroftherungR(q).Thisgivesfortheonedimensionalpart:1D=(3e2`=16)(2l2H)3.Thetransversepartnowhasthreeradialintegrations(3factorsofA's)isgivenas: (2)4[Amnqs0][Anss0n0][Asqmn0]: wheres0=m+qnands=m+qn0.TheradialintegrationscanbeperformedasbeforetoobtaintheA's.Performingthesums,wend (2)42lH231Xm;q=0R2m()(m+q)! 42lH24: Thusforthethirdorderfandiagram,c3=1=4,andtheQCCiszz=3e2DH=64.Thenth-orderfandiagramhasnradialintegrations,eachofwhichgivesafactorof1=2sothatonehasacoecient(1=2)n.Thesummationoverangularmomentumindicesgivesafactor2regardlessofthediagram'sorder.So,theoverallcoecientinthenth-orderfandiagramduetothetransverseintegrationiscn=1=2n1.

PAGE 65

Figure2{5. Cooperonsequencefor3DelectronsintheUQL.Unlikein1D,eachtermintheseriescomeswithadierentcoecientcn. WeconstructtheCooperonsequenceinUQLasshowninFig.( 2{5 ),withtheprefactorsindicatingcn,thenumbersobtainedaftertransverseintegrationateachorder.ThesenumbersareresponsibleforthemassoftheCooperon.TheDOSfactoratnthorderis1=2lH2n+1.Thedashedlineintheguredenotesg(g1),thecorrelatorin3D(1D),whereg=1=2H=niu02=2lH2=(21)=g12lH2.Ristheonedimensionalrungintheparticle-particlechannel(smalltotalmomentum)evaluatedinthediusivelimit. For3DelectronsintheUQL,thecooperonsequencegives: andusingEq.( 2{19 ),thisbecomes, Inthelimitq;!!0,Cbecomesaconstant.Therearenoinfrareddivergence,becausewehaveamassiveCooperon.Themassinunitsofthescatteringrateisapurenumber(1/2).Itcanbeinterpretedas=H,sothatHisoftheorderoftheimpurityscatteringtime.Thisindicatesthatlocalizationinastrongeldproceedsasifastrongphase-breakingprocessisoperatingsimultaneouslywithimpurityscattering.ThisisdephasingbytheeldanditpersistsevenatT!0.

PAGE 66

WenowcontrastthissituationwhicharisesintheUQLwiththatofanyotherdimensions(1D,2D,3D)withoutthemagneticeld.Intheabsenceoftheeldthecn'sareallone(foralldimensions)andthecooperonsequenceissingular(thereisadiusionlikepoleforrealfrequencies,forq;!0). Dq2+j!j: Thisgivestheweak-localizationcorrectiontotheconductivity(WLQCCdiscussedinchapter1)in2Dand3D[ 12 ].In1Dalthoughthecooperondiagramhasapole,allnon-cooperondiagramsarealsoofthesameorder,andoneneedstosumoverallthediagramstogetstronglocalization[ 15 ]. Intheultraquantumlimitthetransversenumbersfortheparticle-holediusionpropagator(thediuson)areallequaltounity(cn=1).Thereforediusonremainsmasslessinastrongeldandnormalquasi-classicaldiusionoccursintheparticle-holechannel.Wewillnowevaluatethetransversepart Figure2{6. Firstandsecondorderdiuson oftherstandsecondorderdiusoncorrectionofFig.( 2{6 ),assumingalongrangedimpuritypotentialsuchthat1d6=0.Wedonotattempttocalculatethelongitudinalpartoftheconductivity,(1d)asthiswillbemorecomplicatedduetothelongrangedisorderpotential.Wejustassumethatthelongitudinalpartisnite.Intheshortrangedimpuritycasethediusoncorrectiontoconductivityiszero(becausethe1d=0).ThetransversepartfortherstorderDiuson

PAGE 67

correctionis, whereG?isdenedinEq.( 2{2 ).Performingtheazimuthalintegrations, (2)21Xm;n=0[Rm()]2Zd11[Rm(1)]2[Rn(1)]2Zd00[Rn(0)]2: Usingorthonormalityandcompleteness,weget?=1=2lH221,sothatc1=1.Forthesecondorderdiusonweperformtheazimuthalintegrationsandobtain, (2)31Xl;k;n=0[Rl()]2Zd11[Rk(1)]2[Rl(1)]2Zd22[Rk(2)]2[Rn(2)]2Zd00[Rn(0)]2; andusingorthonormalityandcompleteness,weobtain?=1=2lH231,andc2=1.Anynth-orderdiagramcanbecalculatedinthesameway,givingcn=1.Thereforethelongitudinaldiusionisfreeandthediusonremainsmassless. Figure2{7. Interferencecorrectiontoconductivity Nextwecalculatethequantuminterferencecorrectiontotheconductivityintheultraquantumlimit(seeFig.( 2{7 )):

PAGE 68

wherethetransverseintegrationshavebeenperformed.ThevertexpartofthisdiagramhasalreadybeenevaluatedinEq.( 2{9 ).UsingEq.( 2{21 ),Zdq 2lH2=HD TheaboveresultindicatesthatperturbationtheoryfailsintheUQL(coop=HD=1=2)inthesamemannerasitdoesinaonedimensionalsystem.However,contrarytowhathappensin1D,thereisnostronglocalizationintheUQL.Thecrosseddiusondiagram(nextorderin1=kF`,seeFig. 2{8 ),isalsononsingularandmassive.Thetransversecoecientforthelowestordercrosseddiusondiagramisc=1=3p 2<1:

PAGE 69

Thelowerboundforisbasedonthefactthatcoop+HD=1 2HD.Thenon-cooperontypediagrams,atleastinthelowestorder,havetheoppositesignasthatofthecooperon.Theyarealsoofthesameorderasthecooperon,soitisnotclearwhattheywilladdupto.Itmayhappenthatallthenon-Cooperondiagramsmodifyourpredictionforandmaymakeanywherefrom0!1.ToobtainabetterestimateforoneneedstogeneralizeBerezinskii's[ 15 ]diagramtechnique(developedforthe1Dlocalizationproblem)to3DUQL.OurresultsalsoagreewiththoseobtainedbytheauthorsofRef.[ 66 67 ].Theseauthorsconsideredlongrangedisorder,`?lH,andobtainedk(`? Inthenextsectionweusethenitetemperaturediagrammatictechniquetocalculatethecorrectionstotheconductivityduetoelectron-electroninteractions(interactionQCC).Wewillshowthatthesecorrectionsarelogarithmicintemperatureandthustheyconrmthatthesystembehaviorisone-dimensional. Figure2{8. Crosseddiusondiagrams.Left,adouble-diusondiagram,whichalsoacquiresamass.Right,athird-ordernon-cooperondiagramwhich,uptoanumber,givesthesamecontributionasthethirdorderfandiagram. (2{29)

PAGE 70

forthesingle-electronwavefunction,where (2nn!p HerelH=1=p (2{31) wherethesumisoverallLandaulevels,n(pz)=(p2zp2F)=2m+n!c,and!c=eH=mistheelectroncyclotronfrequency.WewillneedonlyexcitationsneartheFermilevelforourcalculation,sointheUQL(EF0termsinthesuminEq.( 2{31 )arenegligibleduetothelargemassterm(oforder!c)inthedenominator.Neglectingtheseterms,thetotalGreen'sfunctioniswrittenastheproductofalongitudinalandaperpendicularpart withG?(px;yy0)=0(y+pxl2H)0(y0+pxl2H).Asshownintheprevioussectionthedisorder-averagedlongitudinalGreen'sfunctioncorrespondstoG1D(";pz)=1=(i"pz+isgn(")=2)where1==2Im.CalculatingtheconductivityusingthisGreen'sfunctiongivestheDrudeformula,withdensityofstatesH=1D=2l2H. The(dynamically)screenedCoulombpotentialintheultra-quantumlimitisgivenby[ 70 ] wherethescreeningwavevectorisrelatedtothedensityofstatesviatheusualrelation2=4e2HandR(!;qz)isthepolarizationbubbleof1Delectrons.Inwhatfollows,wewillneedonlysomelimitingformsofthepotential.For

PAGE 71

1=!EFand1=`qkF; independentofthetemperature(upto(T=EF)2-terms).Inthestaticlimit,whenthetransversemomentaaresmall(q2?l2H1),thepotentialreducestoanisotropicform whichdiersfromacorrespondingquantityinthezeromagneticeldonlyinthatscaleswithHasp 5 6 ])comeswithprocesseswithq?l1H:Therefore,theGaussianfactorinthedenominatorofEq.( 2{33 )canbereplacedbyunityforallcasesofinterest. Thepolarizationbubbleexhibitsa1DKohnanomalyforqznear2kF.SuchlargemomentumtransfersareimportantonlyinHartreediagrams,wherethepotentialistobetakenat!=0intheballisticlimit.NeartheKohnanomaly,thestaticpolarizationbubblecanbewrittenas 2kF(0;qz)=1 21DlnEF tologarithmicaccuracy. Finally,thepoleofthepotentialinEq.( 2{33 )correspondstoacollectivemode{magnetoplasmon.For!;qvFEFandq?lH1;thedispersionrelationofthemagnetoplasmonmodeisgivenby where!p0=p

PAGE 72

transverseones,jqzjq?;sothatonecanwrite Figure2{9. Firstorderinteractioncorrectionstotheconductivitywhereeectsofimpuritiesappearonlyinthedisorder-averagedGreen'sfunctions. Wenowproceedtocomputetherstorderinteractioncorrectiontotheconductivityintheballisticlimit(T1).ThisincludescontributionsfromdiagramsshowninFig. 2{9 ,whereeectsofimpuritiesappearonlyinthedisorder-averagedGreen'sfunctions.Italsoincludesdiagramswithoneinteractionlineandoneextraimpurityline.Thesecanbeseparatedfurtherintoexchange(Fig. 2{10 )andHartree(Fig. 2{11 )diagrams.Inthissectionweshowthatdiagrams 2{10 (a), 2{10 (b), 2{11 (a)and 2{11 (b)givealeading{jln(T=EF)j{correctiontotheconductivity,whereasallotherdiagramsgivesub-leadingcontributions. 2{9 (a), 2{10 (a), 2{10 (c),and 2{11 (a)involvecorrectionstotheself-energyduetoelectron-electroninteraction.Diagram 2{9 (a)describesinelasticscatteringofanelectrononacollectivemode(plasmon),whichwouldhaveexistedevenforasystemwithoutdisorder.Astheelectron-electroninteractioncannotleadtoaniteconductivityinthetranslationallyinvariantcase,thisdiagramiscanceledbythecounter-correctionofthevertextype[Fig. 2{9 (b)].DiagramsFig. 2{10 (a), 2{10 (c),and 2{11 (a)describecorrectiontotheself-energydueto

PAGE 73

Figure2{10. Exchangediagramsthatarerstorderintheinteractionandwithasingleextraimpurityline.TheGreen'sfunctionsaredisorder-averaged.Diagrams(a)and(b)givelnTcorrectiontotheconductivityandexchangediagrams(c),(d)and(e)givesub-leadingcorrectionstotheconductivity. interferencebetweenelectron-electronandelectron-impurityscattering.Ageneralformofthecorrectiontotheconductivityforalldiagramsoftheself-energytypecanbewrittenas m"Zdpz where1D("n;pz)isthecorrectiontothe(Matsubara)self-energyoftheeective1Dproblem,towhichtheoriginalproblemisreduceduponintegratingouttransversecoordinates.ThisispossibleduetothefactthattheGreen'sfunctionsarefactorizedintoa1Dandatransversepart,asshowninEq.( 2{32 ),andtheintegrationsovertransversevariablescanbecarriedoutandsimplygivethe

PAGE 74

Figure2{11. Hartreediagramsthatarerstorderintheinteractionandwithasingleextraimpurityline.TheGreen'sfunctionsaredisorder-averaged.BothdiagramsgivelnTcorrectiontotheconductivity. degeneracyfactor1=2l2H.Inthiseective1Dproblem,electronsinteractviaaneectivepotential whereaseachimpuritylinecarriesafactorniu20=2l2H=vF=2,whereniistheconcentrationofimpuritiesandu0istheimpuritypotential.Theoverallfactorof2inEq.( 2{39 )isthecombinatorialcoecientassociatedwitheachdiagramoftheself-energytype. Substituting( 2{33 )into( 2{40 )andusingtheconditionlH1;weobtain PerformingtheanalyticcontinuationinEq.( 2{39 ),weobtain whereGR1D=1=("p+i=2)andR1Distheinteractioncorrectiontotheretardedself-energyofthenon-interactingelectronswhichis0=i=2:(Forbrevity,wesuppressedtheargumentsofGR1DandR1D;whichare";p):

PAGE 75

2{10 (a) 2{10 (a)whichcorrespondstoacorrectionintheself-energyasshowninFigure 2{12 .As!andqzareexpectedtobelargecomparedto1=and1=`,respectively,itsucestoreplacetheGreen'sfunctionsintheself-energybythoseintheabsenceofdisorder.Intherestofthediagramfortheconductivity,theGreen'sfunctionsaretakeninthepresenceofdisorder.In1D,itisconvenienttoseparatetheelectronsintoleft-andright-moversdescribedbytheGreen'sfunctionsG("n;p)=1=(i"nvFp+isgn"n=2),wherep=pzpF:Accordingly,therearealsotwoself-energies;forleft-andright-movingelectrons.Thecontributionfor+isshowninFig. 2{12 .TheGreen'sfunctionsofright/leftelectronsarelabeledbyinthediagram.Processesinwhichanelectronisforward-scatteredtwiceatthesameimpuritydonotcontributetotheconductivityandarethereforenotconsideredinthiscalculation.ThediagramwithbackscatteringbothatanimpurityandotherelectronsinvolvesstatesfarawayfromtheFermisurfaceandisthusneglected.TheonlyimportantdiagramistheoneshowninFig. 2{12 wheretheelectronisbackscatteredbyanimpurityandforwardscatteredbyotherelectrons. Figure2{12. Theself-energycorrectioncontainedindiagram 2{10 (a),denotedinthetextas( 212 )+. Atrst,weneglectthefrequencydependenceofthepotential.Themomentumcarriedbytheinteractionlineissmall,qz'"=vF'T=vF,andatlowtemperatures,suchthatT=vF,onecanneglectqzcomparedtoinV1DandreplaceV1Dbya

PAGE 76

constant,V1D!2g0vF,where isadimensionlesscouplingconstant.Theperturbationtheoryisvalidforg01: 212 )+("n;p)=2ig0vF [i!+vFqz][i("n!m)vF(pqz)]; ( 212 )+("n;p)=2g0 Nowweseethattologarithmicaccuracyitissafetocutthesumat!MvFqmaxEFandomitthefactorinthecurlybracketsinEq.( 2{44 ): ( 212 )+("n;p)=2g0 =g0 2+"nivFp ReR( 212 )+(";p)=g0 (2{46) ImR( 212 )+(";p)=g0 2i"+vFp

PAGE 77

ToobtaintherealpartinaformgiveninEq.( 2{46 )weusedanidentity1 2ix2==coshx;whereasthelastlineinEq.( 2{47 )isvalidtologarithmicaccuracy.Theself-energyofleft-movingelectronsisobtainedfromEqs.( 2{46 ),( 2{47 )bymakingareplacement"+vFp!"vFp: 2{46 )and( 2{47 ).Eq.( 2{46 )showsthatthecorrectiontotheeectivemassisT-dependent:forj"+vFpjT;m/T1:Inprinciple,suchacorrectionmightresultinanadditionalT-dependenceoftheconductivity.However,thisT-dependenceoccursonlyinthenext-to-leadingorderintheparameter(T)11oftheballisticapproximation.Theleadingcorrectiontotheconductivitycomesfromtheimaginarypartoftheself-energy,Eq.( 2{47 ).Thiscorrectionexhibitsacharacteristic1Dlogarithmicsingularity,whichsignalsthebreakdownoftheFermiliquid(tothelowestorderintheinteraction). Themaincontributiontotheconductivitycomesfromthecorrectiontotheimaginarypartoftheself-energy[Eq.( 2{47 )].SubstitutingEq.( 2{47 )intoEq.( 2{42 )andaddingasimilarcontributionfromtheleft-movingelectrons,weobtain 210 Wenotethattheaboveresultwasobtainedusingthestaticformoftheinteractionpotential.WenowreturntothefulldynamicpotentialandshowthatthefrequencydependenceofthepotentialdoesnotchangetheresultsgivenbyEqs.( 2{46 )and( 2{47 ),tologarithmicaccuracy.Foradynamicpotentialitismoreconvenienttoperformtheintegrationoverq?attheveryendsothatthepotentialenteringthecalculationisofthe3Dform

PAGE 78

whereweusedthatqzq?andintroduced2(q?)=q2?=(q2?+2):Theintegraloverp0givesthesameresultasforthestaticpotential.Integratingoverqz;weobtainfortheeective1Dself-energyinsteadof( 2{45 ),~( 212 )+("n;p)=e2 2+"nivFp 212 2{46 )and( 2{47 ).ComingbacktoEqs.( 2{49 )and( 2{50 ),wecaninterpretthisresultinthefollowingway.Thedierencebetweenthedynamicpotentialandthestaticoneisinthepresenceofthedynamicpolarizationbubblemultiplying2inthedenominatorofEq.( 2{49 ).Ifthepotentialistakeninthestaticform,typicalfrequenciesarerelatedtotypicalmomentaas!vFqz;whichmeansthatthisfactorisoforderofunityandmustbereplacedby:Butbecausethenalresultfordependsononlyviaa(large)logarithmicterm,log(jln`Bj);sucharenormalizationofisbeyondthelogarithmicaccuracyofthecalculation. 2{11 (a). Theself-energycorrectioncontainedindiagram 2{11 (a),denotedinthetextas( 213 )+

PAGE 79

DiagramFig. 2{11 (a)isaHartreecounter-partoftheexchangediagramofFig. 2{10 (a).Separatingthecontributionsofleft-andrightmovers,thediagramcorrespondingtobackscatteringatthestaticimpuritypotentialisshowninFig. 2{13 .Again,diagramscorrespondingtoforwardscatteringattheimpuritypotentialdonotcontributetotheconductivityanddonotneedtobeconsideredhere.ThediagramofFig. 2{13 alsoincludesbackscatteringataFriedeloscillation.Althoughthisdiagramcontainsaparticle-holebubble,itismoreconvenienttolabelthemomentaasshowninFig. 2{13 ,integrateoverp0rst,andthenoverk.Forbackscattering,the1DpotentialofEq.( 2{41 )becomes wherethelastlineisvalidtologarithmicaccuracy.Asarstapproximation,weneglecttheq-dependenceoftheinteractionpotential,replacingV2kF1DinEq.( 2{51 )byaconstantV2kF1D!2g2kFvF.Thisresultsin R( 213 )+("n;p)=2g2kF =g2kF 2+"nivFp which,uptoasignandoverallfactorofthecouplingconstant,isthesameastheexchangecontributionR( 213 )+inEq.( 2{45 ). WhenthedependenceofV2kF1Donq0isrestored,theresultinEq.( 2{53 )changesonlyinthatthecouplingconstantacquiresaweakTdependence

PAGE 80

CalculatingthecontributionofEq.( 2{53 )withEq.( 2{54 )totheconductivity,wendthecorrectiontotheconductivityfromdiagramFig. 2{11 (a)tobe: 211 2lnEF NoticethatinthelimitofverylowTand/orverystrongelds,thescreeningwavevectordropsoutoftheresultandthenetT-dependenceoftheconductivitybecomeslnxln(lnx);wherexEF=T: 2{10 (b)andFig. 2{11 (b).Thesearethevertexcorrectionscounterpartsoftheself-energydiagramsinFig. 2{10 (a)andFig. 2{11 (b),correspondingly. 2{10 (b) 2{10 (b)canbeshowntogivethesamecontributionas 2{10 (a).InthisSectionweshowthisbyreducingdiagram 2{10 (b)to 2{10 (a)withoutdoingexplicitintegrationsoverqzandMatsubarasummations. Decomposingdiagram 2{10 (b)intocontributionsfromleftandrightfermions,weobtain 210 2l2Hlim!0e2v2F whereMarethevectorverticesM=Zdp (2{57)

PAGE 81

wehave (2{59) respectively.Forallothercases,theresultscanbeshowneithertovanishbecauseofthelocationsofthepolesortocanceleachother.Intheballisticlimit,theproductM+Mcanbesimpliedinbothcasesto (m+1=)2[!2l+(vFqz)2]:(2{61) Thesubsequentintegrationofthisexpressiongivesaj!lj1-singularityanditisthissingularitywhichgivesthelnT-dependenceofthecorrectiontotheconductivity. Nowwegobacktodiagram 2{10 (a).InSec. 2.2.1.1 ,wefoundthecontributionofthisdiagrambyevaluatingtheself-energyrstandthensubstitutingtheresultintotheKuboformula.Toprovetheequivalenceofdiagrams 2{10 (a)and(b)itisconvenienttoconsiderthefulldiagram 2{10 (a)withoutsinglingouttheself-energypart.Summingupthecontributionofleftandrightfermions,weobtain 210 2l2Hlim!0e2v2F whereP=Zdp 2{62 )isobtainedonlyforthecasegiveninEq.( 2{57 ),when

PAGE 82

withQ=i v2F1 (i!lvFqz+i=)2(i(!lm)vFqz)(i!+vFqz+i=)(qz!qz): 2{63 ),weseethatitcoincideswithEq.( 2{61 ).TheMatsubarasummationgoesoveratwicesmallerintervaloffrequenciescomparedtothatinEq.( 2{56 ).WeseethatEqs.( 2{62 )and( 2{56 )givethesameresultandthus 210 210 2{11 (b) Diagram 2{10 (b)vsdiagram 2{11 (b). ThediagraminFig. 2{11 (b)isavertexcorrectioncounterpartoftheHartreeself-energydiagramFig. 2{11 (a),anditgivesthesamecontributionasFig. 2{11 (a).Toseethis,wecomparethediagramsinFigs. 2{10 (b)and 2{11 (b)

PAGE 83

labelingthemasshowninFig. 2{14 .Foraq-and!-independentinteraction,diagramFig. 2{14 (b)isofthesamemagnitudebutoppositesignasdiagramFig. 2{14 (a).Foraq-dependentinteraction,theT-dependentpartsofthesediagramsdieralsointheoverallfactorofthecouplingconstant:diagram 2{14 (a)containsg0whereasdiagram 2{14 (b)containsg2kF:Electron-electronbackscatteringindiagram(a)andelectron-electronforwardscatteringindiagram(b)giveeithersub-leadingorT-independentcontributions.Thus 211 210 210 211 2{10 (c)givesaself-energytypecontributiontotheconductivitysoweuseEq.( 2{42 ).Iftheinteractionpotentialistakentobestatic,thecontributionfromthisdiagramiszero.Usingthedynamicalpotential,theleadingcontributionfromthisdiagramisalnT-correctiontotheconductivity 210 2e2 ThiscontributionissmallerthanthatfromdiagramsFig. 2{10 (a)[Eq.( 2{48 )]andFig. 2{11 (a)[Eq.( 2{55 )](anddiagramsFig. 2{10 (b)and 2{11 (b))byaT-independentlog-factor. Diagrams 2{10 (d)and 2{10 (e)givemutuallycancelingcontributionsoftheform: 210 (2{67) 210 Eachofthesecontributionsissmallsinceweareintheballisticlimit(T1).

PAGE 84

Allthecalculationsshownherearedoneconsideringthedynamicinteractionpotentialatsmallfrequencies.Athighfrequencies,i.e.,atfrequenciesclosetothemagnetoplasmonfrequency,thecontributionsfromalltheconductivitydiagramscancelout.ThatthishastobethecasewaspointedoutrecentlyinRef.[ 19 ].Thisisaveryusefulresultbecauseeachindividualdiagram,takenseparately,maygivesingularcorrections.Inourcasewehavealsoexplicitlycheckedthatthiscancellationindeedoccurs.Contributionsfromdiagrams( 2{9 a),( 2{10 a)and( 2{10 c)canceleachother.Contributionfrom( 2{9 b)cancelsthatof( 2{10 b),andnally( 2{10 d)and( 2{10 e)canceleachother. 2{48 ),( 2{55 ),( 2{64 ),and( 2{65 ),wendtheleadingcorrectiontotheconductivity =( 210 211 210 211 2lnEF Eq.( 269 )isthemainresultofthisSection. 2{10 (a,b)andFigs. 2{11 (a,b),determinetheleadingcorrectiontotheconductivitysuggeststhattheremustbesomesimplereasonforthesediagramstobethedominantones.Indeed,onlythesediagramsariseifoneconsidersscatteringofelectronsby\eective"impuritiesthatconsistofacombinationofbareimpuritiesandtheCoulombeldsofelectronssurroundingthebareimpurities.Forweakdelta-functionbareimpurities,theeectiveimpuritypotentialcorrespondsto\dressing"theimpuritywiththemeaneldofHartreeandexchangeinteractions(seeFig. 3{1 ).~V0(";p;p0)=V0+VH(pp0)+Vx(";p;p0):

PAGE 85

Therstterminthisequationisthestrengthofabareimpurity,thesecondoneistheCoulombpotentialofelectronswhosedensityismodulatedduetothepresenceofthebareimpurity,andthethirdoneisanexchangepotentialforelectronsinteractingandscatteringthroughaweakimpurity. Figure2{15. Eectiveimpuritypotential Duetotheexchangecontribution,theeectiveimpuritypotentialisnon-local,anditmaydependontheenergy,iftheinteractionisdynamical.Performingtheimpurityaveraging,weobtainthecorrelationfunctionoftheeectiveimpuritypotential whereg=e2=vFistheinteractionstrength.Diagrammatically,Ccorrespondstoadashedlineofthecross-technique[ 18 ].Therstterm(bareimpurities)istakenintoaccountintheleadingorderin1=EF1bysumminginniteseriesforthesingle-particleGreen'sfunctionandthenusingtheKuboformulafortheconductivity.Becausethebareimpuritiesareshort-range,thereisonlyonediagramfortheconductivity{theusual\handle"diagram;thevertexcorrectiontothisdiagramvanishes.CorrectionstotheconductivityresultfromtheHartreeandexchangetermsinEq.( 2{70 ).Torstordertherearetwodiagrams,showninFig. 2{16 .Althoughthebareimpurityispoint-like,theHartreeandexchangepotentialsitgenerateshaveslowlydecayingtailsandalsooscillateinspace.Thusthevertexcorrection,Fig. 2{16 (b),isnotzero.Theself-energydiagram,Fig.

PAGE 86

2{16 (a),correspondstotwodiagrams:Fig. 2{10 (a)andFig. 2{11 (a).DiagramFig. 2{16 (b)correspondstothediagramsinFig. 2{10 (b)andFig. 2{11 (b).Foranarbitraryimpuritypotential,itcanbeshownthatcontributionsof 2{16 (a)and 2{16 (b)comingfromforwardscatteringcancelseachother.Forbackscattering,thecontributionfrom 2{16 (a)and 2{16 (b)arethesame. Figure2{16. Thehandlediagramcorrespondstodiagrams 2{10 (a)and 2{11 (a)andthecrossingdiagramcorrespondsto 2{10 (b)and 2{11 (b). 2.2.5 wedemonstratedthat,torstorderintheinteraction,theonlydiagramswhichareimportantforcorrespondtoscatteringataneectiveimpuritypotential.Thissuggeststhattheresultforcanbeobtainedbycalculatingtheinteractioncorrectiontotheimpurityscatteringcross-sectionandthensubstitutingthecorrectedcross-sectionintotheDrudeformula.InthissectionweshowthattorstorderthisproceduregivesaresultidenticaltothatofthediagrammaticapproachofSec. 2.2 .Unlikethediagrammaticseriesintheinteractionfortheconductivity,theperturbationtheoryforthescatteringcross-sectioncanbesummeduptoallordersviaarenormalizationgroupprocedure.ThiswillleadtoaLuttinger-liquid-likepower-lawscalingoftheconductivity,discussedattheendofthissection.

PAGE 87

withmz=0;1;2:::. ElectronsarerestrictedtothelowestLandaubandandthereforethereareonlytwotypesofscatteringevents:forwardandbackward.Onlybackscatteringeventscontributetothescatteringcross-section,whichcanbewrittenasA_N J,where_NisthenumberofelectronsbackscatteredperunittimeandJisthetotaluxofincomingelectrons.UsingaLandauer-typescheme,thescatteringcross-sectionineachchannelofconservedmzcanberelatedtoareectioncoecientinthischannelviaAmz=2l2Hjrmzj2:Thetotalcross-sectionisobtainedfromthesumofthecross-sectionsineachchannel[ 71 ]:

PAGE 88

Thecoecientsrmzarethereectionamplitudesof1Dscatteringproblems,givenbyasetof1DSchrodingerequations1 2m@2 witheective1DimpuritypotentialsVmz(z)=hmzjVimp(r)jmziobtainedbyprojectingtheimpuritypotentialontheangularmomentumchannelmz.Thekineticenergyoftheelectronisdenotedby"=p2z=2m.Thecross-sectionAisrelatedtothebackscatteringtimeviatheusualrelation,1=H=nivFA,whereniisthedensityofimpurityscatteringcenters.WhentheelectriceldisalongthemagneticeldandforT=0,thecorrespondingcomponentoftheconductivityisrelatedtoHvia AnimpurityofradiusalHcanbemodeledbyadelta-function:Vimp(r)=V0(r).Foradelta-functionpotential,onlythemz=0componentofVmz(z)isnon-zero,Vmz(z)=(V0=2l2H)mz;0(z).Inthiscase,thescatteringcross-sectionfornon-interactingelectronsissimply wherevz=pz=m:Consequently,atT=0theconductivityisgivenby nie2 2l2H"1+2l2HvF IntheBornlimit(whenV02l2HvF)werecovertheresultfortheconductivityasfoundbyusingtheKuboformulaforweak,delta-correlateddisorder[Eq.( 2{73 )].Intheopposite(unitary)limitA=2l2Hand

PAGE 89

wherefarawayfromtheimpuritysite,theasymptoticformofthez-componentoftheunperturbedwave-functionis: (2{77) Bycalculatingtheelectron-electroninteractioncorrectiontothewavefunction,oneobtainsthecorrectiontoamplitudest0andr0;andthereforetothescatteringcross-sectionviaEq.( 2{72 ).Sincenowtheproblemhasbeenmappedontoa1Dscatteringproblem[ 21 24 ],onecananticipatethatthisinteractioncorrectionhasaninfraredlogarithmicsingularity,asitdoesinthepure1Dcase. The1DnatureofthesystemintheUQLisalsoclearlymanifestedbythebehavioroftheFriedeloscillationsaroundtheimpurity.Theproleoftheelectrondensityaroundtheimpuritysiteisgivenbyn(r)=Rdr0(r;r0)Vimp(r0),where(r;r0)isthepolarizationoperator.Foraweakdeltaimpuritypotential,weobtain 2l2Hsin(2pFz) whichshowsonlyaslow,1=zdecay(seeFig. 2{17 ),characteristicofone-dimensionalsystems(incontrasttothe1=r3decayin3Dsystems).Correspondingly,theHartreeVH(r)andexchangeVex(r;r0)potentials,thatanincomingelectronsfeels

PAGE 90

whenbeingscatteredfromanimpurity,alsoexhibit2pF-oscillationsanddecayas1=zawayfromtheimpurityandalongthemagneticelddirection.Inthetransverseplane,thedensity,andthusthepotentials,haveGaussianenvelopeswhichfalloonthescaleofthemagneticlength(seeFig. 2{17 ). Figure2{17. ProleoftheFriedeloscillationsaroundapointimpurityina3DmetalintheUQL.Theoscillationsdecayas1=zalongthemagneticelddirectionandhaveaGaussianenvelopeinthetransversedirection. TheinteractioncorrectiontothewavefunctionduetotheHartreeandexchangepotentialsis Asdiscussedintheprevioussection,fortheUQLtheGreen'sfunctionistheproductofalongitudinal(1D)andatransversepart,G(r;r0;E)=G1D(z;z0;pz)G?(r?;r0?),wheretheasymptoticformofthelongitudinalpartasz!1is ipz8><>:t0eipz(zz0);z0<0eipz(zz0)+r0eipz(z+z0);z0>0(2{80)

PAGE 91

and,inthesymmetricgauge,thetransversepartis 2l2Hexp(jj2+j0j220) 4: Forz>0,Eq.( 2{79 )directlygivesthecorrectionforthetransmissionamplitudet.Werstconsidertheexchangepotential, whichcanbefactoredas (2{83) where Fromtheformoff(z;z0)onecanseethattheexchangepotentialalsohastermswith1=(zz0)and1=(z+z0)decay.Forexample,forz;z0>0, 2i(z+z0)r0(eipF(zz0)1) 2i(z+z0) (2{85) The1=(z+z0)decayleadstoalog-divergentcorrectiontojtj.DecomposingthescreenedCoulombpotentialV(rr0)intoFouriercomponents,allthedependenceofEq.( 2{79 )onthetransversecoordinatesr?canbecollectedintothefactor PerformingtheintegralswhichappearinEq.( 2{86 )fortheexchangecontribution,wendthatthepartcontainingperpendicularcoordinatessimplyenterstheinteractioncorrectionasaformfactor:

PAGE 92

Therefore,thetransversepartofthefreewavefunctionmz=0(r?)simplyremainsunchangedintherhsofEq.( 2{79 ).Theremainingexponentialtermappearsinthedenitionoftheeective1Dpotential,asinEq.( 2{40 ) ThesameresultisobtainedforthetransversepartoftheHartreecontributioninEq.( 2{79 ).Oncethetransversepartissolvedandtheeective1Dpotentialisdened,therestofthecalculationisexactlyequivalenttothecalculationofYueetal.[ 21 ]fortunnelingofweakly-interacting1Delectronsthroughasinglebarrier.Theinteractioncorrectiontothetransmissionamplitudetisdirectlyobtainedfromthecorrectiontothewavefunction,Eq.( 2{79 ).Justasin1D,alogarithmicallydivergentcorrectionfortisobtainedfromthelongitudinalpartofthisequation,afterintegratingoverzandz0. Itisstraightforwardtoseewhythereisalog-divergentterm.TheHartreetermofEq.( 2{79 ),afterintegrationofthetransversecoordinates,is (2{89) where (2{90) Let'sconsiderforsimplicityjr0j1andjt0j1,inwhichcaseEq.( 2{80 )givesG1D(z;z0)=(2m=pz)exp(ipzz)sinpzz0.TheHartreepotentialbehavesasVH(z)'V1D(2pF)sin(2pFz)=zsothatEq.( 2{89 )gives The1=ztermgivesalogarithmicsingularityonlyinthelimitpz!pF,sothatt=t0/V1D(2pF)ln[1=(pzpF)].TheHartreecontributioncorresponds

PAGE 93

toenhancementoft0.TheexchangecontributionhasoppositesignandisproportionaltoV1D(0).Thegeneralanswer,canbewrittenas[ 72 ] t0=jr0j2ln1 where=(g0g2kF)=2,andg0andg2kFaredenedinEqs.( 2{43 )and( 2{54 ),respectively. Figure2{18. Renormalizedconductivitiesparallel(zz)andperpendicular(xx)tothedirectionoftheappliedmagneticeld.Power-lawbehaviorisexpectedinthetemperatureregion1=TW. Thesecond-ordercontributiontothetransmissionamplitudewascalculatedexplicitlyinRef.[ 21 ].Thehigher-ordercontributionscanbesummedupbyusingarenormalizationgroup(RG)procedure.WithoutrepeatingallthestepsofRef.[ 21 ],wesimplystateherethatinourcasethetransmissionamplitudesatisesthesameRGequation,asinthepurely1Dcase.i.e., d=t(1jtj2);(2{93)

PAGE 94

whereln(1=jppFjlH)andt(=0)=t0:ThesolutionofEq.( 2{93 )ist0=t(0)0j(pzpF)lHj 2{72 ),butnowwrittenintermsoftherenormalizedreectioncoecientjrj2=1jtj2.ThenalresultfortheconductivitycanbecastinaconvenientformbyexpressingthebarereectionandtransmissioncoecientsviabareconductivitiesintheBornandunitarylimits,0zzand0zz;U,givenbyEqs.( 2{75 )and( 2{76 ),respectively: W2;(2{94) whereWisanultravioletcut-ooftheproblemand=(g0g2kF)=2,andg0andg2kFaredenedinEqs.( 2{43 )and( 2{54 ),respectivelywhichshowsthatscaleswiththemagneticeldasHlnH.Weareinterestedintemperaturedependenceoftheconductivitiesduetoelectron-electroninteractioncorrectionsandweassumeherethatthebareconductivities0zzand0zz;UhaveonlyweakT-dependencewhichcanbeneglected. Eq.( 2{94 )isthemainresultofthisSectionandisshowninFig. 2{18 .Ithasasimplephysicalmeaning:AtT=W;theconductivityisequaltoitsvaluefornon-interactingelectrons.AttemperaturesTW;theconductivityapproachesitsunitary-valuelimit,whichmeansanyweakimpurityiseventuallyrenormalizedbytheinteractiontothestrong-couplingregime.However,iftheimpurityisalreadyattheunitarylimitatT=W;itisnotrenormalizedfurtherbytheinteractions.WeemphasizethatEq.( 2{94 )isapplicableonlyforhigh-enoughtemperatures,i.e.,Tmax[1=;]:Therstconditionsisnecessarytoremainintheballistic(single-impurity)regime,thesecondoneallowsonetoconsideronlytherenormalizationoftheimpurity'sscatteringcross-sectionsbytheinteractionwithoutrenormalizingtheinteractionvertex.Thelatterprocessleads

PAGE 95

eventuallyforacharge-density-waveinstabilityatatemperatureT,whereisthecharge-density-wavegap[ 1 { 3 ].Forthepower-lawbehavior[Eq.( 2{94 )]tohavearegionofvalidity,thereshouldbeanintervalofintermediateenergiesinwhichtherenormalizationoftheinteractioncouplingconstantsduetoCDWuctuationsisnotyetimportantbutthecorrectionstothecross-sectionleadingtotheformationofpower-lawisalreadysignicant.Suchanintervalexistsforalong-rangeCoulombinteraction(jlHj1)bothfortheconductivityandtherenormalizationofthetunnelingdensityofstates[ 6 ]. Thedissipativeconductivityinageometrywhenthecurrentisparalleltotheelectriceldbutbothareperpendiculartothemagneticeld,xx;occursviajumpsbetweenadjacentcyclotrontrajectories.Intheabsenceofimpurities,electronsarelocalizedbythemagneticeldandxx=0:Inthepresenceofimpurities,xxisdirectly,ratherthaninversely,proportionaltothescatteringrate.Inparticular,forshort-rangeimpurities,xx/1=/1zzandthetemperaturedependenceofxxisoppositetothatofzz.Inthescalingregime,zz/T2andxx/T2:ThissituationisillustratedinFig. 2{18 ,where=(g0g2kF)=2,andg0andg2kFaredenedinEqs.( 2{43 )and( 2{54 ),respectivelywhichshowsthat(H)HlnH.Inthenextsectionwediscusspossibleexperimentalstudiesforobservingthelocalizationandcorrelationeectsmentionedintherstthreesections.

PAGE 96

likebehavioristhatthesystemsberelativelyclean,sothatthereisasizablerangeoftemperaturesinwhichthesystemisintheballisticregime(1=TEF).Thisrulesoutdopedsemiconductors[ 73 { 75 ]sincethechargecarrierscomefromdopantswhichactasimpuritycentersinthesystem.Anadditionalconditionforoccurrenceofthepower-lawscalingbehaviorandformationofcharge-density-waveorWignercrystal,isthattheelectron-electroninteractionisstrongenough.Bismuthcrystalscanbemadeextremelypure;however,thechargecarriersinbismuthareextremelyweaklyinteractingduetoalargedielectricconstant(100)oftheionicbackground.Therefore,thelog-correctionscalculatedherecanbeestimatedtobeverysmallandwouldbediculttobeobservedexperimentally.Charge-densitywaveinstabilityhavebeenobservedingraphite[ 4 ]suggestingthatinteractionofchargecarriersinthissystemisimportantinstrongmagneticeldsandatverylowtemperatures.Thusgraphitewouldbeanidealmaterialtoobservethecorrelationandlocalizationeectsmentionedhere.Belowwepresentsomerecentexperimentalresultsoftransportmeasurementsingraphiterstinweakmagneticelds[ 76 ]andtheninultraquantumregimeandtrytointerprettheminviewofourndings. Graphitehasalowcarrierdensity,highpurity,relativelylowFermi-energy(220K),smalleectivemass(alongthec-axis)andanequalnumberofelectronsandholes(compensatedsemi-metal).ThemetallicTdependenceofthein-planeresistivityinzeroeldturnsintoaninsulatinglikeonewhenamagneticeldoftheorderof10mTisappliednormaltothebasal(ab)plane.UsingmagnetotransportandHallmeasurements,thedetailsofthisunusualbehaviorwereshown[ 76 ],tobecapturedwithinaconventionalmultibandmodel.Theunusualtemperaturedependencedisplayedin(Fig. 2{19 )canbeunderstoodforasimpletwo-bandcase

PAGE 97

Figure2{19. Temperaturedependenceoftheab-planeresistivityxxforagraphitecrystalatthec-axismagneticeldsindicatedinthelegend wherexxisgivenby[ 77 ], withi,andRi=1=qini(i=1;2)beingtheresistivityandHallcoecientofthetwomajorityelectronandholebands,respectively.Atnottoolowtemperatures(wherethemeasurementswereperformed)electron-phononscatteringisthemainmechanismfortheresisitivityintheband.Assumingthat1;2/Tawitha>0,wendthatforperfectcompensation,(R1=R2=jRj),Eq. 2{95 canbedecomposedintotwocontributions:aeld-independentterm/Taandaeld-dependentterm/R2(T)H2=Ta.AthighT,thersttermdominatesandmetallicbehaviorensues.AtlowT,R(T)/1=n(T)saturatesandthesecondtermdominates,givinginsulatingbehavior/Ta.Althoughthisinterpretationexplainsthequalitativefeaturesoftheeldinducedmetal-insulatorbehaviorshowninFig. 2{19 ,theactualsituationissomewhatmorecomplicatedduetothepresence

PAGE 98

ofathird(minority)band,Tdependenceofthecarrierconcentrationandimperfectcompensationbetweenthemajoritybands.FormoredetailsseeRef.[ 76 ]. LetusnowdirectourattentionontransportmeasurementsintheultraquantumregimeinwhichweexpecttoseethepowerlawconductivitybehaviorsimilartowhatisshowninFig. 2{18 .Belowwepresentsomerecentdataonthesamegraphitesamplesonwhichtheweak-eldmeasurementswereperformed. Figure2{20. Temperaturedependenceofthec-axisconductivityzzforagraphitecrystalinamagneticeldparalleltothecaxis.Themagneticeldvaluesareindicatedontheplot,withtheeldincreasingdownwards,thelowestplotcorrespondstothehighesteld Withintheexperimentallystudiedtemperaturerange(5K
PAGE 99

Figure2{21. Temperaturedependence(log-logscale)oftheab-planeresistivityscaledwiththeeldxx=B2foragraphitecrystalatthec-axismagneticeldsindicatedinthelegend eld-inducesLuttingerliquidmodeltheexponentofthepowerlawshoulddependontheeld(seeEq.2.94,HlnH).Also,theexponentsoftheTscalinginzz(whichis1),andxx(whichis1=3)aredierent(asseeninexperiments)whereastheywerepredictedtobethesameintheLuttingerliquidmodel. Wearegoingtoarguethattheunusualtemperaturebehaviorofzzandxx,canbeunderstoodwithinamodelwhichincludesphonon-induceddephasingofone-dimensionalelectrons(intheUQL)andthecorrelatedmotioninthetransversedirectionduetothememoryeectofscatteringatlongrangeddisorder.Beforewegetintothedetailsofthemodel,letuskeepinmindafewnumbersforthesystemweareabouttodescribe.ForourgraphitesamplestheFermienergyisEF=220K,theBloch-Gruniesentemperature(whichseparatestheregionofTandT5contributiontotheresistivity)is!0=2kFs10KandtheDingletemperature(whichgivestheimpurityscatteringrate)is3K.Alsothetransportrelaxationtimeismuchlonger(byafactorof30),thanthetotalscatteringtime(orlifetime)tr,indicatingthelongrangenatureoftheimpurities.

PAGE 100

Figure2{22. Temperaturedependence(onalog-logscale)oftheab-planeresistivityxx=B2atthehighestattainedc-axismagneticeldof17:5Tforthesamegraphitecrystal WerstoutlineanargumentbyMurzin[ 73 ]whichshowsthatthetransversemotionoftheelectroniscorrelatedduetodriftmotioninacrossedmagneticandelectriceld.Thedisordermodelisassumedtobeionizedimpuritytypeandisthereforelongranged.Thetransversedisplacement(afterasinglescatteringact)isassumedtosatisfyr?lHrD(rDbeingthescreeningradius).Electronsareassumedtodiuseinthezdirection.Anelectronre-enterstheregionwheretheimpurity'selectriceldisstrong,(r?
PAGE 101

intheeldofagivenimpurityisWDrD HWD=cerD H2NDzz1=2t3=2: H4=3N2=3rD2=3Dzz1=3;xxce3F

PAGE 102

resistivityxx1 wherec=4=3 Nowwewillshowthatthec-axistransportbehavior(onlythetemperaturedependence)canbeexplainedwithinthecontextofapower-lawhoppingmechanism,inwhichphononscauselocalizedelectronstohopoverdistancesoftheorderofthelocalizationlengthwithafrequencyof1 2{96 ,oneobtainstheT1=3powerlawtemperaturedependenceforxx.Inastronglylocalizedsystem(happensonlyin1D)oneshouldexpecttheabsenceofstaticconductivity.Wheninelasticscatteringisallowedthesituationchangesconsiderablyasjumpsbetweenindividuallocalizedstatesbecomepossibleandareaccompaniedbyphonon(orelectron-holepair)emissionorabsorption.ViolationoflocalizationleadstoaniteconductivityofadiusiontypewhichisthecalledthesuperdiusionregimeorPowerlawhoppingregime[ 78 ].Inthisregimeelectron'smovebydiusionbutaretrappedinsidealocalizationlength`.Inelasticscatteringallowstheelectrontojump/hopoverdistancesoftheorderofthelocalizationlengthwithafrequencyof((T))1,thephasebreakingrate.ThediusionconstantisthenD=`2=((T))andtheconductivity(T)=e2D.Thisoccursfori(T),whereiistheelasticscatteringtime.AsthetemperatureisfurtherloweredsuperdiusiongiveswaytoMott'svariablerangehoppingtransportregimewhere(T)/Deq

PAGE 103

Figure2{23. PhasebreakingratevsTduetoelectron-phononscattering Thebehaviorofthephasebreakingtimeduetoelectron-phononscatteringisillustratedschematicallyinFig. 2{23 .InthelowtemperatureregimeT2kFs10K,electron-phononscatteringisinelasticandthephasebreakingtimeisoftheorderoftheelectron-phononscatteringtime1=1=/T3.ButtheexperimentalplotswerenotinthisTrange.Inthehightemperatureregime,T>2kFs,theelectron-phononscatteringtimeis1 76 ],duetoitslowdensityandsmalleectivemass.ThehighTregimecanbefurthersubdividedintotheballisticregime(epht1 2kFs=)2kFsT2kFs A)andthediusiveregime(epht1 2kFs=)2kFs ATEF),wheret,isthedurationofasinglecollisionact,whereasistherelaxationtime.Noticethattheballisticregimeexistsonlyfor(semi-metals)materialswithA1.FormetalsA1andtheballisticregimedoesnotexist.Inthehightemperaturediusivelimitthephase

PAGE 104

breakingrateisgivenby[ 26 ] 1 whichcorrespondstotheT1=3temperaturedependenceinFig. 2{23 andoccursatveryhightemperatures(T>200K)insemi-metalsduetothesmallvalueofA.Intheballisticregimeepht(!D)1,where!DistheDebyefrequency.Thismeansthationsperformmanyoscillationsduringasingleactofelectron-phononscattering.Thusscatteringatmovingionsworksasdephasingduetodynamicpotential[ 79 ]Thephase-breakingtimeisthenoftheorderoftherelaxationtime,i.e.,eph.Therefore,thegraphitesamplesintheballisticregime(5K
PAGE 105

thebareDrudeconductvityitself.Thereforeperturbationtheorybreaksdownjustasitdoesin1D.However,unlikeinthe1Dcase,theconductivityremainsniteatzerotemperature.Therefore,wecallthisregimeintermediatelocaliza-tion.Thesecondimportantmanifestationofelectroniccorrelationsandlowerdimensionalityisthattherstorderinteractioncorrectiontotheconductivityislogarithmicallydivergentintemperature,justasfor1Dsystems.Wethencalculatetheinteractioncorrectiontothecross-sectionofelectronsscatteringoasingleimpurity.UsingaDruderelationtoobtaintheconductivityfromthecross-section,wendthatthisresultisequaltotheresultobtainedfromthefulldiagrammaticcalculation.Thissuggeststhatthedominantdiagramsfortheconductivitycanbedescribedintermsofscatteringoaneective\dressed"impuritypotential.Arenormalization-groupcalculationforthecross-sectionallowsforthesummationofaseriesofmostdivergentlog-correctionsatallordersintheinteraction.Justasin1Dthissummationinourcaseleadstopower-lawscaling.Howeverinthesystemof3DelectronintheUQLthisisahigh-energybehaviorwhichexistsprovidedthattheelectron-electroninteractionissucientlylong-ranged.Somerecenttransportmeasurementsingraphitewerecomparedwiththeabovetheoreticalndingsandshowntodisagree.Toresolvethedisagreement,weinvokedamodelwithlongranged-disorderandphononinduceddephasingtoexplaintheexperimentalobservations.

PAGE 106

TherehasbeensubstantialrecentinterestinthethermodynamicsofaFermiliquid[ 41 { 47 80 ].Therevivalofinterestintheproblemistwofold.Ontheexperimentalside,technicaladvancesnowallowonetomeasurethetemperaturedependenceofthethermodynamicparameterssuchasspecicheatandspinsusceptibilityofatwo-dimensional(2D)Fermiliquidwithshortrangeinteraction,suchasmonolayersofHe3[ 56 ],aswellas,two-dimensionalsemiconductorstructureswithlongrangeinteractionandrelativelylowFermitemperatures(1K).Onthetheoryside,itturnsoutthattheleadinginteractioncorrectionsarenonanalyticfunctionsoftemperatureormagneticeld,makingthesubjectparticularlyinteresting.Thefateofthesenonanalyticcorrectionsinthespinsusceptibility,nearaquantumcriticalpointisimportantforourunderstandingofthenatureofparamagnetictoferromagneticquantumphasetransitionandwediscussthisindetailinchapter4. Asithasbeenmentionedintheintroduction,naivepowercountingargumentssuggestthatthetemperaturedependenceofanythermodynamicquantity,includingthespinsusceptibilityandthespecicheatcoecientC(T)=T=,shouldstartwithtermsquadraticintemperature.ThisconjectureisbasedontheobservationthatathermodynamicquantityatnitetemperaturecanbewrittenasRa()nF()d,wherenF()istheFermidistributionfunctionanda()issomefunction.Ifthelatterissmooth,thetemperaturedependencestartswithatermoforderT2[ 28 ].Suchacorrectioniscalled"analytic".Thisisalsoconsistent 95

PAGE 107

withtheintuitiveexpectationoftheone-to-onecorrespondencebetweenthenoninteractingFermigasandtheinteractingFermiliquidsinceintheFermigas,theSommerfeldexpansionleadstoasimplequadratictemperaturecorrections. However,wealsosawthattheassumptionabouttheanalyticityofthefunctionsinvolvedinthecalculationofvariousthermodynamicpropertiesoftheFermiliquidisquitegenerallynotjustied,becauseinanyFermiliquid,thedynamicinteractionbetweenparticlesgivesrisetoanonanalyticenergydependenceofa().ThisleadstotemperaturecorrectionsthatdonotscaleasT2andarethereforecalled"nonanalytic".Collectingthesenonanalyticcorrectionsisasubtletheoreticalproblemandinthischapterwewillevaluatethesecorrectionsforaone-dimensionalinteractingsystem. InagenericFermiliquid,thefermionicself-energybehavesasReR(";k)=A"+Bk+:::andImR(";k)=C("2+2T2)+:::.Thisformoftheself-energyimpliesthatquasiparticlesarewelldened,andthedominanteectoftheinteractionsatlowenergiesistherenormalizationofthequasi-particlemassandtheresidueofthequasiparticleGreen'sfunction.Thisbehaviorhasaprofoundeectonobservablequantitiessuchasthespecicheatandstaticspinsusceptibility,whichhavethesamefunctionaldependenceasforfreefermions,e.g.,thespecicheatC(T)islinearinT,whilethesusceptibilitys(Q;T)approachaconstantvalueatQ=0andT=0. Ithasbeenknownforsometimethatthesub-leadingtermsinthe"andTexpansionsofthefermionself-energydonotformregular,analyticseriesin"2orT2[ 81 ].Inparticular,powercounting(dimensionalanalysis)showsthattherstsubleadingtermintheretardedon-shell("=k)self-energyatT=0,isR(")/"3ln(i")in3DandR(")/"2ln(i")in2D.Thesesingularitiesintheself-energygivecorrectionstotheFermi-liquidformsofthespecicheatandstaticspinsusceptibility(CFL/T;FLs=const.),whicharenonanalyticinD3

PAGE 108

andscaleasC(T)/TDands(Q)/QD1,withextralogarithmsinD=3and1.Itwasshownrecently[ 44 ]thatthenon-analyticcorrectionstothespecicheatandspinsusceptibilityin2Doriginatefromessentiallyone-dimensionalscatteringprocesses,embeddedinahigherdimensionalphasespace.Inparticular,these1Dscatteringevents(strictlyforwardorbackscattering)givenon-analyticsub-leadingtermsintheelectronself-energy,withthedegreeofnon-analyticityincreasingasthedimensionalityislowered,simplybecausethephasevolume(qD)ismoreeectiveinsuppressingthesingularityinhigherdimensionsthaninlowerones.Thusnon-analyticitiesinhigherdimensionscanbeviewedasprecusorsof1DphysicsforD>1andthestrongestnonanalyticityshouldoccurin1D. Thepurposeofthisworkistoobtainthenonanalytic,TlnT,correctiontothespecicheatin1D,andtoelucidatethesimilaritiesanddierencesbetweenhigherdimensionalandonedimensionalnon-analyticities.Thenonanalyticterminthespecicheatin1D,wasmissedinanearlierwork[ 57 ],astheauthorsanalyzedtheself-energyuptosecondorderinperturbationtheory.Weshowthatthespecicheatandspinsusceptibilityin1Dacquirenonanalyticcorrectionsfromthesingularitiesintheonedimensionalbosonicresponsefunctions,justastheydidforhigherdimensions.Themajordierencebetweenthenon-analyticitiesinC(T)inD>1andD=1isthattheformeroccursatthesecondorderininteraction,whereasthelatterstartsonlyatthirdorder(contrarytotheexpectationthatthedegreeofnonanalyticityincreaseswithreductionofdimensionality),andthenonanalyticityin1Doccursonlyforfermionswithspin.Inhigherdimensionsthespecicheatisnonanalyticevenforspin-less(i.e.,fullyspinpolarized)fermions.Naivepowercountingbreaksdownin1DbecausethecoecientinfrontoftheTlnTterminC(T)vanishesinsecondorder,andonehastogotothirdorder.AlthoughbosonizationpredictsthatC(T)/Tin1D,thisistrueonlyforspin-lessfermions,inwhichcasethethirdorderdiagramsforthenon-analytic

PAGE 109

temperaturedependenceexactlycancelout.Forfermionswithspin,thebosonizedtheoryisofthesine-Gordontypewiththenon-gaussian(cos)termcomingfrombackscatteringoffermionsofoppositespins.Eventhoughthistermismarginallyirrelevantandowsdowntozeroatthelowestenergies,atintermediateenergiesitleadstoamultiplicativelnTfactorinC(T)andalnmax[Q;T;H]correctiontos.Thespinsusceptibilityisnonanalyticatsecondorderininteractionasinhigherdimensions.Theadvantageofusingthefermionicdiagrammaticapproachin1D,isthatinadditiontocorrectlypredictingthenon-analyticitiesitalsoelucidatestheunderlyingphysics:thenon-analyticitiesarisefromuniversalsingularitiesinthebosonicresponsefunctions,thusestablishingtheconnectionwithhigherdimensions. Thischapterisorganizedasfollows.Insection 3.1 ,wediscussthemainphysicsofaonedimensionalinteractingsystemandstatethe1DmodelweusedinourcalculationswhichistheTomonaga-Luttingermodelwithbackscattering.Insection 3.2 ,weevaluatethespecicheatforaone-dimensionalinteractingsystem.Insubsection 3.2.1 ,weexplicitlyobtainthenonanalyticformsofthe2ndorderself-energyin1D,butweshowthatthesenon-analyticitiesofthefermionself-energydoesnotleadtoanonanalyticcorrection(henceforthreferredtoasNAC)tothespecicheatatsecondorderininteraction.ThisisanimportantdierencebetweenhigherdimensionalandonedimensionalNACtothespecicheat,andweshowthisintwodierentways,rstinsubsection 3.2.1 ,wherewecalculatethespecicheatfromthefermionself-energyandthenagainfromthethermodynamicpotentialinsubsection3.2.2,tormlyestablishthisdierence.Insubsection3.2.3,weobtainthenonanalyticTlnTterminthespecicheatusingthethirdorder(nonanalytic)self-energyandshowthatthistermispresentonlyforfermionswithspin.TheabsenceofthenonanalyticTlnTtermat2ndorderanditspresenceatthe3rdorder(forspinfulfermions)isconsistentwiththe

PAGE 110

renormalizationgrouptreatmentoftheSine-Gordonmodel,andthisisshowninsubsection3.2.4.Insection3.3,weshowthatthenonanalyticlnmax[Q;H;T],terminthestaticspinsusceptibilityispresentatsecondorderinperturbationtheorysimilartohigherdimensions.Wediscusspossibleexperimentalobservationsofthequantitiesstudiedinthischapterinsection3.4,andgiveourconclusionsinsection3.5. 28 ].Theyareinone-to-onecorrespondencewiththebareparticlesand,specically,carrythesamequantumnumbersandobeyFermi-Diracstatistics.ThefreeFermigas,thusisthesolvablemodelonwhichFermiliquidtheoryisbuilt.Theelectron-electroninteractionhasthreemainaects:(1)itrenormalizesthekinematicparametersofthequasi-particlessuchastheeectivemass,andthethermodynamicproperties(specicheat,spinsusceptibility),describedbytheLandauparametersFa;sn;(2)itgivesquasiparticlesanitelifetimewhichdiverges,however,as(EEF)2astheFermisurfaceisapproached,sothatthequasi-particlesarerobustagainstsmalldisplacementsawayfromEF;(3)itintroducesnewcollectivemodes.Theexistenceofquasi-particlesformallyshowsupthroughanitejumpzKF,ofthemomentumdistributionfunctionn(k)attheFermisurface,correspondingtoaniteresidueofthequasi-particlepoleintheelectron'sGreensfunction. Incontrast,thepropertiesoftheone-dimensionalinteractingsystem,theLuttingerliquid,arefundamentallydierentfromtwoorthree-dimensionalFermiliquids.Inparticular,theelementaryexcitationsarenotquasi-particlesbutratherbosoniccollectivechargeandspindensityuctuationsdispersingatdierentvelocities.Anincomingelectrondecaysintochargeandspinexcitationswhich

PAGE 111

thenspatiallyseparatewithtime(spin-chargeseparation)[ 82 ].Thecorrelationsbetweentheseexcitationsareanomalousandshowupasinteraction-dependentnon-universalpowerlawsinmanyphysicalquantitieswhereas,thoseofordinarymetals(FermiLiquids)arecharacterizedbyuniversal(interactionindependent)powers. Tobemorespecic,alistofsalientpropertiesofsuch1Dinteractingsystemsinclude:(1)acontinuousmomentumdistributionfunctionn(k)varyingasjkkFjwithaninteraction-dependentexponent,andapseudogapinthesingle-particledensityofstates/j!j,bothofwhicharetheconsequencesoftheoftheabsenceoffermionicquasi-particles;(2)similarpowerlawbehaviorinallcorrelationfunctions,specicallyinthoseforsuperconductingandspinorchargedensitywaveuctuations;(3)nitespinandchargeresponseatsmallwavevectors,andaniteDrudeweightintheconductivity;(4)spin-chargeseparation.Allthesepropertiescanbedescribedintermsofonlytwoeectiveparameters(K;uinEq. 3{2 )perdegreeoffreedom,(spinandcharge)whichplaytheroleofLandauparametersin1D. Thereasonforthesepeculiarproperties,istheveryspecialFermisurfacetopologyof1Dfermions,producingbothsingularparticle-holeresponsefunctionsandsevereconservationlaws.Ina1Dsystem,therearetwoFermi"points"kF,andonehasperfectnesting,namelyonecompleteFermipointcanbetranslatedintotheotherbyasinglewavevector2kF.Thisproducesasingularparticle-holeresponseat2kF.ThistypeofresponseisassumedniteinFermiliquidtheorybut,in1D,isdivergentforrepulsiveforwardscattering,leadingtoabreakdownoftheFermiliquiddescription.Inadditionwehave,asin3D,theBardeen-Cooper-Schrieer(BCS)singularityforattractiveinteractions.Infact,noneoftheseinstabilitiesoccur,thecompetitionbetweentheparticle-particle(BCS)andparticle-hole(at2kF)channelleadstoacriticallike(powerlaw)

PAGE 112

behaviorofthe1Dcorrelationfunctionsatzerotemperatures.Theone-dimensionalelectrongasisthusalwaysatthevergeofaninstabilitywithoutbeingabletoorder.Onetheotherhand,thedisjointFermisurfacegivesawelldeneddispersion,i.e.,particle-likecharactertothelowenergyparticle-holeexcitationsinafreesystem.Theseparticle-holeexcitationsarewelldened,meaningtheyhavewelldenedmomentaandenergy.Theynowcanbetakenasthebuildingblocksuponwhichonecanconstructadescriptionofthe1Dlow-energyphysics.Thedensityoperatorwhichisasuperpositionoftheparticle-holeexcitations((q)=Pkcyk+qckbq),isusedasabosonicbasisinwhichtheoriginalfourfermioninteractinghamiltonian(Eq. 3{1 )becomesquadraticandthereforeexactlysolvable.ThisistheessenceofthebosonizationtheorywhichwasusedbyMattisandLiebtosolvethe1DTomonaga-Luttingermodel[ 83 ].Thenotionofa\Luttingerliquid"wascoinedbyHaldanetodescribetheseuniversallowenergypropertiesofgapless1Dquantumsystemsandtoemphasizethatanasymptotic(!!0;q!0)descriptioncanbebasedontheLuttingermodelinmuchthesamewayastheFermiliquidtheoryin3DisbasedonthefreeFermigas. Figure3{1. Interactionvertices OurmodelhamiltonianforcalculatingthenonanalyticcorrectionstothespecicheatandspinsusceptibilitywillbethestandardTomonaga-Luttingermodel,extendedtoincludebackscatteringvertices[ 82 ], ^H=Xk;r=+;vF(rkkF)cyr;kcr;k+1 2Xr;k;k0;qV(q)cyr;k+qcyr;k0qcr;k0cr;k

PAGE 113

wherecyk(ck)istheelectroncreation(destruction)operatorandV(q)istheinteractionpotential.Thelinearizationofthespectrum(whichisessentialformakingtheparticle-holeexcitationswelldened)forcesonetointroducetwospeciesoffermions:rightmovers(r=+1)andleftmovers(r=1).OnehastokeepinmindthatthemostecientprocessesintheinteractionaretheoneswhichactsclosetotheFermisurface.Thefactthatinone-dimensiontheFermisurfaceisreducedtotwopoints(+pFandpF)allowsonetodecomposetheimportantlowenergyprocessesoftheinteractionintothreedierentsectors.ThesethreeinteractionprocessesareshowninFig. 3{1 ,wheresolidlinesrepresentrightmoversanddashedlinesdenoteleftmovers.Therstprocessg4couplesfermionsonthesamesideoftheFermisurface.Thesecondoneg2couplesfermionsfromdierentbranches.However,eachspeciesstaysonthesamesideoftheFermisurfaceaftertheinteraction(bothforwardscattering).Finally,thelastprocessg1correspondsto2kFscattering(backscattering)wherethefermionsexchangesides.Weassumethattheinteractionpotentialisniteranged(g16=g2),andforgeneralityweallowfordierentinteractionsbetweenleftandrightmovingfermions(g26=g4)butweneglectthemomentumdependenceoftheinteractioncoecients,treatingthemasconstants.TheinteractionpartoftheHamiltonianiswrittenintermsofoperatorsc+;k(c;k)denotingright(left)movingfermionsas[ 84 ],^Hint=1 2LXk1;k2;pX;g4k+g4?;(cy+;k1cy+;k2c+;k2+pc+;k1p++cy;k1cy;k2c;k2+pc;k1p):

PAGE 114

Forspinlessfermionswithonlyforwardscattering(g2andg4vertices,g1=0),theHamiltoniancanbebosonizedandtransformedtoaquadraticform[ 82 ]: 2ZdxhuK(r(x))2+u K(r(x))2i; whereandarethebosoniceldswhichsatisfythecanonicalcommutationrelations [(x1);r(x2)]=i(x2x1); anduandKaretheparametersrenormalizedbytheinteraction, 1+y4=2+y2=21=2; withy=g=(vF)beingadimensionlesscouplingconstant.Thusthephysicsoftheone-dimensionalinteractingspin-lessfermionicsystemiscompletelydescribedbyfreebosons.Theenergyspectrumischangedfrom(p)=vFjpj(forafreefermionicsystem)to(p)=ujpjfortheinteractingsystem.Thespecicheatis dT=d dTXp(p)fB((p))=u2 u(L=3): Thespecicheatislinearintemperatureevenforaninteractingsystem(forfreefermionsC(T)=T(L=3vF)).Forfermionswithoutspin,includingbackscatteringamountstoreplacingg2withg2g1.Allthepreviousresultsstillholdaftermakingthechangeg2!g2g1inuandK.Thespecicheatwillremainlinearintemperaturewithanewcoecient.Forfermionswithspin,includingbackscatteringwillleadtoasine-Gordonterminadditiontothequadraticterminthespinpartofthehamiltonian.Thechargepartretainsitsquadraticformbutwithnewcoecientsuc;Kc.Thespecicheatforthismodelisanalyzedindetailinsection3.2.4.

PAGE 115

Therearethreedistinctnon-analyticitiesinthebososnicresponsefunctions,inonedimensionsatT=0(asinanyotherdimensions[ 53 ]).Thesearethesingularitiesin(1)theparticle-holepolarizationbubbleforsmallmomentumandfrequencytransfers,in(2)theparticle-holepolarizationbubbleformomentumtransfernear2kFandin(3)theparticle-particlepolarizationoperatorforsmalltotalmomentum.ThefreeGreen'sfunctionforleft()andright(+)moversare Herek=ppFisthemomentumcountedfromthecorrespondingFermipoint.Theparticle-holepolarizationbubbleforleftmoversandsmallq;!is (q;i!)=Zdk andsimilarlyforrightmovers ++(q;i!)=Zdk wherevF=1.Boththepolarizationoperatorshavethesamesingularityinq:(q;!)! q,forqkFandq!.TheaboveformofthepolarizationoperatorindicatesLandaudamping:Collectiveexcitations(spinandchargedensitywaves)decayintoparticle-holepairs,thisdecayoccursonlywithintheparticle-holecontinuum,whichin1D,shrinkstoasingleline!=vFq.Asitwasshownintheintroduction,thissingularityintheparticle-holepolarizationbubbleresultedinnon-analytic,sub-leadingtermsintheselfenergyandthermodynamics.Wewillshowbelow(insubsection3.2.1),thattheforwardscatteringresponsefunctionsin1Dgivesnonanalytictermsintheself-energy,butthesedonotleadtoNACtoC(T)ors. ThedynamicalKohnanomaly,whichisthesingularityinthe(2)particle-holeresponsefunction(forq2kF),alsogivesnon-analyticsub-leadingtermsinthe

PAGE 116

self-energyandthermodynamics[ 53 ].In1D, 2kF+(q;i!)=Zdk 4lnj(q+i!)(qi!) (2)2j; whereistheultravioletcut-o.Finally,the(3)particle-particleorCooperbubble(forsmalltotalmomentum)hasthesamenon-analyticitybutwithanoppositesignasthe2kFparticle-holechannelin1D pp(q;i!)=Zdk 4lnj(q+i!)(qi!) (2)2j: Wewillshowthattheabovesingularitiesgiverisetononanalyticsub-leadingtermsintheone-dimensionalself-energy,fore.g.Im+R(";k=0)/j"jatsecondorderandIm+R(";k=0)/j"jlnj"j Fromthisfunction,onecandeterminetheentropybyusingthethermodynamicrelation@N @T=(@S @)T.FollowingthestepsofRef.[ 18 ]weisolatethe

PAGE 117

temperaturedependenceofthenumberdensityandobtainfortheentropy,S V=2@ @TTX"Zd~p V=2Zdp 2iTZ1d""@nF wherenFistheFermidistributionfunction.GR(GA)istheretarded(advanced)Green'sfunctionatzerotemperature.UsingDyson'sequation(G=1 @TS V=CFG+C(T)theinteractioncorrectiontothespecicheat(tolowestorderin)is[ 18 ]: @ @T1 whereR(A)istheretarded(advanced)selfenergyevaluatedatzerotemperature,andG0R(G0A)isthefreeretarded(advanced)Green'sfunction.Strictlyspeakingtheaboveformulaforthespecicheatisvalidonlyfortheleadingtemperaturedependence(seeadiscussionaboutthisinRef.[ 45 ]).However,wearejustiedinusingthezerotemperatureformalismin1D,sincethenon-analyticTlnTterminthespecicheatgrowsfasterthantheanalyticTtermforlowenoughtemperatures.Insub-section 3.2.1 ,weevaluatethespecicheatfromthesecondorderself-energy(atT=0)andndonlyaregular,linearinTcontribution.Insub-section3.2.2,weagainevaluatethespecicheat,onlythistimeusingthethermodynamicpotential(forT6=0)alsoatsecondorder,toverifytheabsenceofthenonanalytictemperaturedependenceatsecondorderinperturbationtheory.Thisisonemajordierencefromthehigherordernonanalyticites.Insubsection3.2.3,weshowthatthenon-analyticTlnTtermarisesonlyatthirdorderininteraction,andonlyforfermionswithspin.Inthissectionthenonanalyticcorrectiontothespecicheatisobtainedfromthethirdorderselfenergy

PAGE 118

evaluatedatzerotemperature.Weveriedour3rdorderresultbyperformingarenormalizationgroupanalysisofthesine-Gordontheoryinsubsection3.2.4. 3{2 .Thedashed(solid)linesrepresenttheGreen'sfunction,G(G+)forleft(right)movingfermionsandthewigglylinesdenotetheinteractionvertices.Therestofthesecondorderandrstorderselfenergydiagrams[ 18 ]areconstantandleadtoatrivialshiftofthechemicalpotentialandthusresultinalinearTdependenceforC(T). Figure3{2. Non-trivialsecondorderselfenergydiagramsforrightmovingfermions Thesingularitiesinthe2kFparticle-holepolarizationbubbleandtheparticle-particlechanneldonotaectthe2ndorderself-energy(andthisisthereasonwhywegetananalyticcontributionforthespecicheatin1Dat2ndorder),whichcanbesolelywrittenintermsoftheforwardscatteringpolarizationbubblesand++.Thusthereareonlytwodistinctcontributionsfromalloftheabovediagrams,theoneinFig. 3{2 (a)and 3{2 (c).ThediagramsinFig. 3{2 (b),(d)and(e)whichclearlyhaveabackscatteringparticle-holebubblecanbeshownequaluptoaconstantpre-factortothediagraminFig. 3{2 (a).ThediagraminFig. 3{2 (f)

PAGE 119

issameasthatofFig. 3{2 (c).Fortheself-energydiagraminFig. 3{2 (a)wehave,+ 32 3{14 oneneedstheimaginaryandrealpartoftheretardedselfenergy.ApplyingthespectralrepresentationfortheGreen'sfunctionandthepolarizationoperator[ 85 ],followedbyasimplepoleintegration(forT=0)andanalyticcontinuationtorealfrequencies,onegets ImR+ 32 Thisisthezerotemperatureform.Atnitetemperatures,onecansumoverthebosonicMatsubarafrequenciestoget ImR+ 32 However,itwillbeshownthatthenitetemperatureformdoesnotchangetheresultforthespecicheat.Now,substitutingImR(!;q)=q(!+q)=2fromEq. 3{8 ,andImG+R("!;kq)=("!k+q)toEq. 3{15 ,andperformingtheintegrations,weget ImR+ 32 Therealpartoftheself-energyisthenobtainedfromtheKramers-Kronigrelation ReR+ 32 whereisacut-o.Noticethattheselfenergy(realpart)iszeroonthemass-shell("=k)contrarytohigherdimensions.In1D,theentireself-energycomesfromtheprocesses!=q(becausetheparticle-holecontinuumhasshrunktoasingle

PAGE 120

line,!=q).Thisistheultimatecaseofforwardscattering,whoseprecursorsinhigherdimensionsleadtonon-analyticitiesinthespecicheat[ 44 ].Howeverin1D,thesenonnalyticitiesinthesecondorderselfenergywillleadtoalineartemperaturedependenceforthespecicheat.AlsonotethatImR(";k=0)/j"j,whichisindicativeofthepoorlydenedquasi-particlesin1D(ImRscaleswith"inthesamewayastheenergyofafreeexcitationabovetheFermilevel).InaconventionalFermiliquid,theconditionforwelldenedquasi-particleexcitationsis,ImR(")"2ReR".In1D,ReR(";k=0)"lnj"j,whichmeansthateectivemassdivergesas:m?lnj"jand,tothisorderthebehaviorisreminiscentofamarginalFermiliquid[ 86 ]. Theself-energydiagramofFig. 3{2 (c)is, ImR+ 32 SubstituteImR++(!;q)=1 2q(!q),fromEq. 3{9 andtheGreen'sfunctionandperformingtheintegrationsweobtain ImR+ 32 Weseethattheself-energydivergesonthemass-shell.Thisistheinfra-redcatastrophe[ 87 ]in1D.Theonedimensionalelectronscanemitinnitenumberofsoftbosons:quantaofdensityexcitations.Therealpartoftheself-energyisfoundagainfromtheKramers-Kronigrelation ReR+ 32 where(A=g42

PAGE 121

Whenthisformoftheself-energyissubstitutedintheGreen'sfunction,therearetwopoles,whichcorrespondstodispersionofthethespinandchargemode.This,essentiallynon-perturbativeand1Deectisthespin-chargeseparation,whichisconrmedbyanexactsolution(Dzyaloshinskii-Larkin[ 88 ]solutionofTomonaga-Luttingermodel).Therestoftheself-energydiagramsinFig. 3{2 canbereducedtoeitherofthetwoselfenergiesevaluatedabove(uptoapre-factor)byrelabelingthedummyvariables,e.g.,theselfenergyinFig. 3{2 (b)is (b)=g12ZqG(kq)2kF(q)=g12ZqG(kq)Zk0G+(k0)G(k0q)=g12Zk0G+(k0)ZqG(kq)G(k0q)=g12Zk0G+(k0)(kk0)=g12Zq0G+(kq0)(q0)=g12 (a) (f)=1 2 (c),thenegativesignisduetooneextraclosedloopandthefactoroftwoisduetospinsuminclosedloopofFig. 3{2 (c).Also (d);(e)=g1 (a).Sothenetself-energyatsecondorderininteractionis:+R(";k)g22g12+g1g21+g422;Im1=sgn(")("k)(j"jjkj);Re1=("k)lnjk2"2 3{14 .AlthoughthelogsingularityinRe1,suggestsanonanalyticterminC(T);however,acarefulanalysisshowsthatthisisnotthecase.NotethatinEq. 3{14 ,ReismultipliedbyImGR+(";k)whichis("k),sodoingthekintegration

PAGE 122

projectstheRe1(";k)onthemassshellwhere,itiszero(Re1(";k=")=0).ConsidernowthecontributionfromtoIm1.Themomentumintegrationgives,PZdk1 @T1 @"'Tg22g12+g1g2: ThelineartemperaturedependenceinEq. 3{23 ,canbeseenbyre-scaling"byTandbringingtheintegralinadimensionlessformwhichgivesanumberoforderone.Boththerealandimaginarypartof2willcontributeequallytothespecicheat.Considerthemomentumintegralfor2'scontributiontoC(T),ZdkImGRReR2+ReGRImR2=Zdk("k)Ak2 44 ]).Thelineartemperaturebehaviordoesnotchangeevenifweusethenitetemperatureformulafortheself-energy.TheresultforthediagraminFig. 3{2 (a)atnitetemperaturefromEq. 3{16 is, ImR (a)/("k)coth"k (3{24) ThetemperaturedependencecomessolelyfromtheMatsubarasum,thepolarizationoperatorisstillevaluatedatzerotemperature.ThemomentumintegrationinthespecicheatformulaEq. 3{14 givesalinear"dependence(thesameresultbothfor

PAGE 123

zeroandnitetemperature).PZdk"k "kcoth"k 3{2 (c)is ImR (c)(";k)/"2+2T2("k)=2(";k;T): UsingtheaboveformofthenitetemperatureselfenergythemomentumintegrationinEq. 3{14 givesatermproprotionalto",whichagainleadstoC(T)/TThus,1Disdierentfromhigherdimensionsbecausethespecicheatisanalyticatsecondorderininteraction.Beforeconsideringthethirdorderself-energycontributiontothespecicheat,wewillcalculatethespecicheatfromthethermodynamicpotentialatsecondorder(seeFig. 3{3 )andshowthatanapparentTlnTcontributiontothespecicheatgetscanceled,whenweconsiderthetemperaturedependenceofthepolarizationbubble. Secondorderdiagramsforthethermodynamicpotentialwithmaximumnumberofexplicitparticle-holebubbles

PAGE 124

AnotherwaytoobtainC(T)beyondtheleadingterminTistondthethermodynamicpotential(T)withintheLuttinger-Wardapproach[ 89 ],andthenusethethermodynamicrelationC(T)=T@2 85 ](itisanapproximatemethod,sinceonlyacertainsubsetofthediagramsaresummed).Hereinthissectionwewillevaluatethethermodynamicpotentialdiagramsdirectlytosecondorderinperturbationtheory,andverifythelinearspecicheatobtainedviatheselfenergycalculationoftheprevioussub-section.Fig. 3{3 showsthesecondorderdiagrams,whichhavemaximumnumberofexplicitparticle-holebubblesforthethermodynamicpotential.Onceagaindashed(solid)linesdenoteleft(right)movers.Wewillshowthattheforwardscatteringdiagrams[(Fig. 3{3 (a)and(c))]givealinearspecicheat.Usingthezerotemperatureformforthe2kFparticle-holepolarizationbubbleinthediagraminFig. 3{3 (b),onewouldgetaTlnTterminthespecicheat,however,suchatermgoesawaywhenweusethefullnitetemperatureresultforthebubble.Therstorderandtheothersecondorderdiagrams(nonRPAtype)forthethermodynamicpotentialgivealineartemperaturebehaviorforthespecicheat.ForthediagraminFig. 3{3 (c),=Lg22 WeomittheconstantfactorofL=2andsumoverthebosonicMatsubarafrequenciesusingacontourintegration[ 85 ]andget (c)=g22Zdq whereIm++RRshouldnowbeevaluatedatanitetemperature.Forforwardscatteringprocesses,boththenitetemperatureandzerotemperatureformsofthepolarizationoperator(++;)arethesame.Atnitetemperature

PAGE 125

fromEq. 3{8 ,(q;i!)=1 3{8 andEq. 3{9 toobtain,ImR++R=q (c)=g22 (c)=+g222T (c)(T)/T: NowforthediagraminFig. 3{3 (a), (a)=g42Zdq UsingEq. 3{9 ,onegets 2(!q); evenforniteT.Thusthethermodynamicpotentialbecomes, (a)=+4g42

PAGE 126

andonceagaintheentropyandthespecicheatarelinearintemperature. (a)=+g42T (a)(T)/T: ThebackscatteringdiagramofFig. 3{3 (b)is, (b)=g12Zdq Wenowshowthatusingthezerotemperatureformofthe2kFbubblefromEq. 3{10 ,wegetaTlnTterminthespecicheat;howeverthisnonanalytictermdropsoutwhenweusethenitetemperatureformofthebubble.AtT=0, (b)=g12 j! (b)=Tg12 j2 (b)(T)/TlnT Thisnon-analyticityisarticialandisremoved(exactlycanceled)whenwesubstitutethenitetemperatureIm2kF2inthethermodynamicpotential.Thatsuchacancelationmustoccurcanbeseeneasily:thediagraminFig. 3{3 (b)canbeshownequivalenttothediagraminFig. 3{3 (c),(uptoanoverallmultiplicativeconstant)bypairingdierentGreen'sfunctiontoformthebubble,interchangingtheorderofintegrationofthedummyvariablesandrelabelling.SincethediagramofFig. 3{3 (c)givesalinearspecicheat(seeEq. 3{27 ),thedoublebackscatteringdiagramofFig. 3{3 (b)mustgivealinear(regular)temperaturecorrectionforthespecicheataswell.Toresolvetheapparentcontradiction,wecalculateexplicitly

PAGE 127

thenite-Tformofthebackscatteringbubble Im22kF(q;!)R=+1 4n!q Thethermodynamicpotentialnowbecomes =APZd!coth! (2)2j22T2 (!q)2+1 (!+q)2; whereAisanumericalconstant.Evaluatingthespecicheat,wendthatthenonanalyticTlnTtermdropsoutandthespecicheatremainslinearintemperature.Thiscalculationistedious,sowedonotpresentitinthethesis.Thissectionveriestheresultobtainedinprevioussection,thatthespecicheatin1Disanalyticat2ndorderinperturbationtheoryunlikeinhigherdimensions(D=2;3)[ 44 47 ].Inthenextsectionweevaluatethespecicheatfromthethirdorderself-energyandobtainagenuinenonanalyticTlnTcontributionforspin-fullfermionsin1D. 3{4 .Thesediagramsexplicitlycontaintwoparticle-holepolarizationbubblesinthem.Therearefourdistinctpossibilities

PAGE 128

whichcanoccurindiagramswithtwopolarizationbubbles:(1)bothpolarizationbubblescanbebackscatteringones(2kF2)asinFig. 3{4 (a);(2)bothpolarizationbubblesareforwardscattering,withoneofthembeing++andtheotherasinFig. 3{4 (b);(3)bothpolarizationbubblesareforwardscatteringandasinFig. 3{4 (c);and,nally,(4)bothforwardscatteringpolarizationbubblesare++,asinFig. 3{4 (d).Allthirdorderself-energydiagrams,whichhavetwoparticle-holepolarizationbubblescanbeclassiedintotheabovefourcategories.Wewillexplicitlyevaluateallfourofthesediagramsandshowthatonlywhenboththeparticle-holepolarizationbubblesareofthebackscatteringtype(2kF2),onegetsanonanalyticTlnTdependenceforthespecicheat.Theparticle-particlechannelhasthesamenonanalyticmomentaandfrequencydependenceastheparticle-holebackscatteringbubble,soweexpectanonanalyticTlnTterminthespecicheatfromdiagramswhichhavetwoCooperbubblesaswell.Allthe3rdorderdiagramswhichgiveanon-analyticspecicheatareshowninFig. 3{9 .Inthissectionwewillbeusingthezerotemperatureformsofthebosonicresponsefunctionsinevaluatingtheselfenergyandthespecicheat.Weremindourselvesthattheg4couplesfermionsonthesamesideoftheFermisurfacewherasg2couplesfermionsfromdierentbranches.However,eachspeciesstaysonthesamesideoftheFermisurfaceaftertheinteraction(bothforwardscattering).Finally,theg1processcorrespondsto2kFscattering(backscattering)wherethefermionsexchangesides.Onceagain,solidlinesrepresentrightmoversanddashedlinesdenoteleftmovers. Figure3{4. Thedierentchoicesforthe3rdorderdiagram.

PAGE 129

(1)Twobackscatteringbubbles: + 34 (a)(k;i")=g13Zdq UsingthespectralrepresentationfortheGreen'sfunctionandthepolarizationoperator, ImR (a)=2g13 FromEq. 3{10 ,wegetIm2kF2(q;!)=1 8lnj!2q2 ImR (a)(k;")=g13 Therealpartoftheself-energyisobtainedfromKramers-Kronigrelationandonthemassshellitis ReR (a)=PZd!(!") (3{38) becausetheintegralisanoddfunctionof!.Thustherealpartoftheself-energydoesnotcontributetothespecicheat.ThecontributiontothespecicheatfromImRisnon-analytic,C(T) (a)=b2T @ @T1 2T @ @T1 j;

PAGE 130

one. (a)/g13Tln(T=): (2)Twoforwardscatteringbubbles,withonebubblebeingandtheother++; ImR (b)(k;")=2g22g4 Usingthezerotemperatureformsfortheaboveresponsefunctionsoneobtains, ImR (b)(k;")=+g22g4 Theaboveformfortheself-energyconsistsoftwopartseachofwhichwereearlierobtainedforthe2ndorderselfenergydiagramsinEq. 3{17 andEq. 3{20 .Thisisnotunexpectedbecausethis3rdorderdiagramismadeupofsecondorderpieces(seeFig. 3{4 (b)).Thenfromthesecondorderspecicheatanalysiswecansurelysaythattheaboveformoftheself-energygivesalinearspecicheat. (3)Twopolarizationbubbles,bothbeing, ImR (c)=2g22g4 Onceagaintheselfenergyisthesameasthatforthesecondorderdiagram,(seeEq. 3{17 ), ImR (c)(k;")=g22g4 Fromthe2ndorderanalysisweknowthatthisformoftheself-energydoesnotgiveanonanalyticcontributiontothespecicheat.

PAGE 131

(4)Twopolarizationbubbles,bothare++; ImR (d)=2g43 Theaboveformoftheself-energyisthesameasthatobtainedforthesecondorderdiagramwithamassshellsingularity(seeEq. 3{20 ).ThisalsoresultsinalinearinTcontributiontothespecicheat.ThereforewehaveshownthatthethreeforwardscatteringdiagramsofFig. 3{4 (b),(c)and(d)allgiveonlyalinearinTcontributiontospecicheat.TheonlydiagramwhichgivesanonanalyticTlnTcorrectiontothespecicheatistheonewithtwobackscatteringbubbles,Fig. 3{4 (a).Fromhereonwewillfocusonlyonthosediagramswhichgiveanonanalyticcontributiontothespecicheat. TheselfenergydiagramsinFig. 3{4 containstwoexplicitparticle-holebubbles.Thereareseveralother(seven)selfenergydiagrams(seeFig. 3{5 )whichdonotcontainexplicitparticle-holebubbles,buttheycanbeshownequivalenttotheonesshowninFig. 3{4 ,bytriviallyre-labelingthedummyvariables.Thesearetheselfenergydiagramswhichimplicitlycontaintwoparticle-holebubblesinthem,Fig. 3{5 (b)-(h).Thusallthirdorderselfenergydiagramswhichhavetwoparticleholebubbles(explicitorimplicit)fallintothefourcategoriesstudiedabove.Weshowthisnext,onlyforthecaseoftwobackscatteringbubblesbecausethenonanalyticTlnTtermarisesonlyfromtwo2kF,bubbles.ConsiderthediagraminFig. 3{5 (b),whichwewillshowisequal(uptoanumericalpre-factor)tothediagram

PAGE 132

inFig. 3{4 (a) (b)(k;i")=+g12g2ZdqZd!Zdk1Zd"1Zdq1Zd!1G(kq;i("!))G(k1;i"1)G+(k1+q;i("1+!))G+(k1+q+q1;i("1+!+!1))G(k1+q1;i("1+!1))=+g12g2ZdqZd!G(kq;i("!))Zdk1Zd"1G(k1;i"1)G+(k1+q;i("1+!))Zdq1Zd!1G+(k1+q+q1;i("1+!+!1))G(k1+q1;i("1+!1))=+g12g2ZdqZd!G(kq;i("!))[2kF(q;i!)]2: (b)=g2 (a),andthespecicheatisC(T) (b)/g12g2TlnT.Similarlyitcanbeshownthatthealltheself-energydiagramofFig. 3{5 withtwo2kFbubblesgiveaTlnTcontributiontothespecicheat.ThediagramsofFig. 3{5 (b)-(h),canalsobedrawnwithforwardscatteringbubbles(analogoustothediagramsinFig. 3{4 (b)-(d)).However,thesediagramsgivealinearTcorrectiontothespecicheatandwehaveomittedtheminthischapterforlackofspace.Thenonanalyticcontributionsfromallthoseselfenergydiagramswithtwoparticle-hole(backscattering)bubblesare: (a)!C(T)/g13TlnT (b)+ 35 (c)+ 35 (d)!C(T)/3g12g2TlnT (e)!C(T)/g23TlnT (f)+ 35 (g)+ 35 (h)!C(T)/+3g22g1TlnTThenetnonanalyticcontributionisC(T)/(g1g2)3TlnT.Thenon-analyticities

PAGE 133

Figure3{5. All3rdordersediagramsforrightmoverswhichhavetwo2kF. seemtoarisefromg1(exactbackscattering)andg2(exactforwardscattering)interactionvertices,similarto2D,asshowninRef.[ 44 ].Thisresultsuggeststhatevenifg1=0(longrangeinteractionpotential),thespecicheatremainsnonanalyticin1D.ThiscontradictsthebosonizationresultwhichstatesthatforaGaussiantheory(withg4andg2interaction,seeEq. 3{2 )thespecicheatremainslinearintemperature.Thereforeourresultcannotbecorrectandwemusthaveoverlookedsomediagramswhichmustcancel(atleast)theg2dependenceoftheNACtoC(T).Thesearetheparticle-particleorCooperdiagrams,whichhavethesamenonanalyticbehaviorasthe2kFparticle-holebubble(compareEq. 3{11 andEq. 3{10 ).ThereforeallthethirdorderselfenergydiagramswithtwoCooperbubblesinthem(explicitorimplicit),alsogiveaTlnTterminthespecicheatandmaycancel(someorall)thenonanalyticcontributionarisingfromthebackscatteringparticle-holebubbles.ThesediagramsareshowninFig. 3{6 .OnecanreducetheseCooperselfenergydiagramstoa2kFself-energydiagramtoobtainaTlnTcontributionforthespecicheat. Theself-energydiagraminFig. 3{6 (e)hastwoparticle-particlebubblesanditwillbeshowntobeequaltotheself-energydiagramwithtwobackscattering

PAGE 134

Figure3{6. AllthirdorderselfenergydiagramscontainingtwoCooperbubbles bubblesandthuswillgiverisetoaTlnTcontributiontothespecicheat. (e)(k;i")=g23ZdqZd!G(qk;i(!"))Zdk1Zd"1G+(k1+q;i("1+!))G(k1;i"1)Zdk2Zd"2G+(k2+q;i("2+!))G(k2;i"2);=g23ZdqZd!G(qk;i(!"))pp(q;i!)2;=g23ZdqZd!G(kq;i("!))[2kF(q;i!)]2= (e)=)C(T)/g32TlnT: 3{10 andEq. 3{11 ).Sincetheselfenergiesareequalandopposite,theTlnTcorrespondingtermsinthespecicheatareexactlycanceled.ThisconrmsourpreviousexpectationthattheNACtoC(T)fromtheCoopertypeselfenergydiagramswillcancelsome(orall)ofthenonanalyticcontributionfromthebackscatteringparticle-holeselfenergydiagrams. Inordertosystematicallylistallthethirdorderselfenergydiagrams,onemuststartfromthesecondorderselfenergydiagramsandreplaceoneoftheinteractionlineswithavertex(seeFig. 3{7 ).Allsecondorderverticeswithg2

PAGE 135

Figure3{7. Eectivethirdorderself-energydiagrams(thedoublelineisavertex). andg1interactionlinesareshowninFig. 3{8 .Wehaveleftouttheg4verticesastheydonotresultinanon-analyticself-energy.Thisisbecauseag4interactionlinecanonlybepairedwithanotherg4linetoformag42vertex.Therecannotbeag4g2org4g1vertexeither.Therefore,althougheachoftheverticesshowninFig. 3{8 ,canbedrawnwithg4lines,theycanbeonlybecombinedwithanotherg4lineintheself-energymakingtheoverallcoecientinfrontofthediagramg43.Thesediagramscanonlyhavetwoforwardscatteringparticle-holebubblesinthemandthuscannotleadtoanonanalyticity.AlsonoticethatthevertexinFig. 3{8 (f)couldhavebeendrawnwithtwog2processes;however,suchavertex,whenincludedinaselfenergydiagramcomeswithtwoforwardscatteringbubblesonebeingandtheother++andwehaveseenthatthiscombinationgivesalineartemperaturedependenceofthespecicheat.InFig. 3{9 weshowallthethirdorderself-energieswhicheitherhavetwop-h(2kF)bubblesortwop-pchannels,andallofthemgiveTlnTnonanalyticitytothespecicheat.However,forfermionswithoutspinthereisanexactcancelationamongthediagrams,makingthespecicheatlinearintemperature.ThenonanalyticTlnTtermsurvivesonlyforfermionswithspin. ThersteightdiagramsofFig. 3{9 ((a)..(h))arisewhenwereplacethevertex(doubleline)inFig. 3{7 (c)byeachoftheeightverticeslistedinFig. 3{8 .ThenexteightdiagramsofFig. 3{9 ((i)..(p))arisewhenwereplacethe(double)interactionlineinthesecondorderselfenergydiagramofFig. 3{7 (a)byeachofoftheeightverticesinFig. 3{8 .Nowletuswritethetotalnonanalyticspecicheat

PAGE 136

Figure3{8. Allg2andg1verticesat2ndorder. contributionfromallthediagramsinFig. 3{9 (a)+(d)+(e)!C(T)/3g22g1TlnT; (k)+(l)+(m)!C(T)/+3g12g2TlnT; (c)!C(T)/g13TlnT; (i)!C(T)/+g23TlnT; (b)+(g)+(h)!C(T)/+3g22g1TlnT; (f)+(p)+(o)!C(T)/3g12g2TlnT; (j)!C(T)/g23TlnT; (n)!C(T)/+g13TlnT: 3{9 (a)+(d)+(e)andthoseinFig. 3{9 (k)+(l)+(m)withtwoCooperbubblescancelwiththediagramsinFig. 3{9 (b)+(g)+(h)andFig. 3{9 (f)+(p)+(o);correspondingly.SimilarlythediagraminFig. 3{9 (i)cancelswiththeoneinFig. 3{9 (j)andthediagraminFig. 3{9 (c)cancelswiththeoneinFig. 3{9 (n).Thusthespecicheatisperfectly

PAGE 137

Figure3{9. Allthirdorderself-energydiagramswithtwoCooperbubblesortwo2kFbubbles. regularwithalineartemperaturedependenceevenatthe3rdorderininteraction.Thisisconsistentwiththebosonizationtreatmentsinceforfermionswithoutspin(evenwithbackscatteringvertexg1)onehasaquadratichamiltonian(seeEq. 3{2 )whereinK;uonehastoreplaceg2!g2g1. ForfermionswithspinthecancelationbetweenFig. 3{9 (c)andFig. 3{9 (n)isincompletebecauseofextraspinsuminthepolarizationbubble.Theparticle-particlediagramofFig. 3{9 (c)canonlyhaveg1kinteractionvertexsoC(T) (c)/g1k3TlnT,butthedoublebackscatteringparticle-holediagramofFig. 3{9 (n)canhavetwochoices.Itcan(1)haveallthreeg1kinteractionlines,forwhichC(T) (n)/g1k3TlnTwhichwillcancelthepreviouscontribution,butitcanalsohave(2)twog1?linesandoneg1kline,andsoC(T) (n)/g1?2g1kTlnT,anonanalyticcontributionwhichsurvives.Furthermore,ifweassume,g2k6=g2?,thenthethreediagramsinFig. 3{9 (f),(o),(p)andthecorrespondingonesinFig. 3{9 (k),(l),(m)donotcancel.IntherstsetFig. 3{9 (f),(o),(p),thecoecient

PAGE 138

infrontoftheTlnTtermcanbeeither(g1k)2g2kor(g1?)2g2kwhereasforthesecondsetFig. 3{9 (k),(l),(m),thecoecientscanbe(g1k)2g2k(whichcancelout)or(g1?)2g2?,whichdonotcancel.Alltherestofthediagramscanceloutcompletelyevenwithspin.Thusthemainresultofthissectionisthatthespecicheatatthirdorderininteractionis: (3{45) (3{46) Inthenextsectionweshowthatthisresultisconsistentwitharenormalizationgroupanalysisofthesine-Gordonmodelwhichariseswhenoneusesthebosonrepresentationtotreatfermionswithspinandwithbackscatteringinteractionvertexg1.Inthebosonizationdescription,theg1?term(backscatteringwithantiparallelspins)inthehamiltoniangivesrisetoacosterminspinpartoftheaction.Wewillshowthatthissine-Gordontermleadstoanonanalytic(TlnT),temperaturedependenceinthespecicheat,withthesamecoecient(g1?2g1k)wepredictedusingthediagrammaticanalysis. 82 ], where 2Zdxu 2Zdxu

PAGE 139

where=("+#)=p 2KZdxZ0d(@)2+(@x)2; Wehavesetu=1,becauseitis2ndorderinthecouplingconstants,whereasK,whichislinearinthecouplingconstantsiskeptnite.Below,wedropthesuxfortheelds,aswewillbesolelyconsideringthespinelds.Treatingthesine-Gordontermperturbatively(g1?1),onecanevaluatecorrectionstothespecicheat.ConsidertheFree-energy,F=TlnZ=TlnZDexp([S0+S1])=TlnZDexp(S0)Tln1+RDexp(S0)S12 2RDexp(S0)S21

PAGE 140

where,A(jxj;jj)=2g1? 82 ], 2h(Pj(Aj(rj)+Bj(rj)))2i: Wethenget, (1=T)2sinh2xT+sin2(T)!2K; byusingthenitetemperatureformofthecorrelationfunction[ 82 ], 4ln1 ToevaluatethetemperaturedependenceofthecorrectiontotheFreeenergywewillgototherelativeandcenterofmasscoordinatesystemforbothandx,andscaleoutthetemperaturedependencebybringingtheintegraltoadimensionlessform, (=)2sinh2(x=)+sin2(=)!2K; wherewehavesetK1=g1kg2k+g2?,forg1.NowC(T)=T@2F @T2.ThereforethereisanonanalyticTlnTterminthespecicheat,withthesame

PAGE 141

coecient(g1?2(g1kg2k+g2?))asobtainedusingthediagrammaticapproach. Weseethatthespecicheatin1Dacquiresnonanalytictemperaturedependencestartingatthirdorderininteraction(unliked=2;3wheretheyoccurevenat2ndorder).Thisnonanalyticpieceexistsonlyforfermionswithspin,becausethenthe1Dhamiltonianwithspinandbackscatteringinteractionhasasine-Gordonterminadditiontothegaussianterm.ThisresultwasearlierobtainedbyJaparidzeandNersesyan[ 49 ]byanexactsolutionoftheSU(2)Thirringmodel.Inarecentworkonthissubject,AleinerandEfetov[ 48 ]usedasupersymmetricapproachandobtainedthenonanalyticcorrectionstothespecicheatinarepulsiveFermigasinalldimensions.Howevertheirworkseemtosuggestthattheone-dimensionalNACtoC(T),startatfourthorderinperturbationtheorywhereaswejustshowed(usingtwodierentmethods:diagrammaticallyaswellasusingbosonization)thattheTlnTnonanalyticityin1Dshouldoccuratthirdorderininteraction. Next,weshowthatthesingularityinthebackscatteringparticle-holechannelcausesanon-analyticityinthespinsusceptibilityalreadypresentatthesecondorderininteractionjustasinhigherdimensions. ^H=hZdx1 2["(x)#(x)]=h p whereh=gBH,withHthemagneticeld,BtheBohrmagnetonandgistheLandefactor.Ifweassumethatg1?=0,thenthespinHamiltonianisquadratic(seeEq. 3{49 withg1?=0)andtheaboveelddependenttermcanbeabsorbedintothequadraticpartbyshiftingtheeldby,~=+hKx=p

PAGE 142

thespinsusceptibilityis@h("#)i=@his Aconstantspinsusceptibilitywithrenormalized(byinteractions)coecientsis"Fermiliquid"like.Aswesawinthecaseofthespecicheat,makingg1?nite,ledtoanonanalyticcorrectiontotheFermiliquidform(CFL(T)T)forthespecicheat,inthesamewayweexpectthespinsusceptibilitytoacquirenonanalyticcorrectionswhichareproportionaltog1?.Indeed,wendthatthesingularityinthebackscatteringparticle-holechannel(thedynamical-Kohnanomaly)isresponsibleforanonanalyticityinthespinsusceptibility,/lnmax(jQj;jHj;jTj)atsecondorderininteraction,andverifyanearlierresultofDzyaloshinskiiandLarkin[ 90 ].Thechargesectorisstillgaussian,hencechargesusceptibilityremainsanalytic. Wewillevaluatethethermodynamicpotentialinanitemagneticeld,(atsecondorderperturbationtheory)andthenobtainthespinsusceptibilityusingthethermodynamicrelation,(s(H)=@2=@H2).ThefreeGreen'sfunctionin1D(nowinthepresenceofthemagneticeld)is ThediagramsforthesecondorderthermodynamicpotentialwereconsideredbeforeinFig. 3{3 .HerefermionshavespinthereforeeachofthediagramsofFig. 3{3 ,willcomeinthreevarietiesdependingonwhetherthebubbleshaveparallelspins(eitherbothbubbleshaveGreen'sfunctionwithupspins,orbothbubbleshaveGreen'sfunctionwithdownspins)orifthebubbleshaveantiparallelspins(onebubblehasbothspinsupinitsGreen'sfunctionandtheotherbubblehasbothspinsdowninitsGreen'sfunction).AllthesediagramsareshowninFig. 3{10 .Wewillshowthattheforwardscatteringparticle-holepolarizationoperatorsdonot

PAGE 143

haveanexplicitelddependence(theycanonlyhaveananalyticmagneticelddependencethroughthedensityofstates).Therefore,allthosethermodynamicpotentialdiagramswhichhaveforwardscatteringpolarizationbubblesinthemfore.g.,diagramsinFig. 3{10 (a),(b),(c)andFig. 3{10 (g),(h),(i)cannotgiveanon-analyticcontributiontothespinsusceptibility. Figure3{10. Secondorderdiagramsforthethermodynamicpotential.

PAGE 144

++""(q;i!)=Zdk Thereforethereisnononanalyticcontributiontothespinsusceptibilityfromdiagramsin 3{10 (a),(b),(c),(g),(h),(i).Thebackscatteringpolarizationoperatorhasanexplicitnonanalyticelddependence. 2kF##(q;i!)=Zdk 4lnj(q+hi!)(q+h+i!) (2)2j: Henceweexpectallthreebackscatteringthermodynamicpotentialdiagrams(Fig. 3{10 (d),(e),(f))togiveanonanalyticcontributiontothespinsusceptibility.However,itturnsout,thatonlythethermodynamicpotentialdiagramwithantiparallelspins(Fig. 3{10 (f))inthetwobackscatteringparticle-holebubblewillgiveanonanalyticelddependenceandtheparallelspindiagrams(Fig. 3{10 (d),(e))canagaingiveananalyticcontributiontothespinsusceptibility.

PAGE 145

Thethermodynamicpotentialwithparallelspinsinthetwobackscatteringbubblesis, (e)=L=g1k2Zdq Noticethatinthemomentumintegrationonecanrelabelthedummyvariable,q+h=q0,andthentheelddependencedropsoutandthereforethisdiagramcanatthemost,makeananalyticcontribution.Thesameargumentappliestothediagramwherebothbackscatteringbubbleshavespin-up,whichalsocannotgiveanonanalyticterminthespinsusceptibility.Thereforethenon-analyticityinthespinsusceptibility,canonlyarisesfromasinglediagramatsecondorder;theonewhichhasantiparallelspinsinthetwobackscatteringparticle-holebubbles,Fig. 3{10 (f).Thisargumentalsoexplainswhytheremainingsecondorderdiagramsforthethermodynamicpotential(singleloopwithtwointeractionlines),andtherstorderdiagram(singleloopwithoneinteractionline),donotgiveanonanalyticcorrectiontothespinsusceptibility;theycannothaveantiparallelspinsastheinteractionlinecannotipthespinintheloop.Theonlynonanalyticcontributiontothespinsusceptibilityat2ndorderisfrom, (f)=L=Zdq

PAGE 146

Hererelabelingthedummymomentumdoesnotgetridoftheelddependence,sotherewillbeanitenonanalyticcontributionto.Performingthemomentumintegrationweget, (f)=4cZ10d!coth(!=2T)4!+2!lnj!2h2 !hj; wherec=(g1?)2 (f)(0)=4cZ10d!coth(!=2T)2!lnj!2h2 !hj: Noticethattheintegraldivergeslogarithmicallyattheupperlimit.Wecancutitat!=EFanddotherestofthecalculationtologaccuracy: Analyzingtheaboveintegralinthetwolimitsa)hTandb)hT,wendtheleadingcontributiontothethermodynamicpotentialtobe (3{67) and ThiscontributionarisessolelyfromasinglediagramtheoneinFig. 3{10 (f).TheissueofapreciseformofthefunctioninterpolatinginbetweenhandTunderthelogisoutsidethelogaccuracy.Thusweseethatthespinsusceptibilityisanon-analyticfunctionofhandTalreadyatsecondorderininteraction.The

PAGE 147

nonanalyticmomentumdependenceisnotmanifestintheabovemethod,(asoneintegratesoverthemomentumtogetthethermodynamicpotential).Onecanshowthatthespinsusceptibilityisanonanalyticfunctionofthebososnicmomentum,magneticeldandtemperaturein1D[ 44 90 ].s/g1?2lnEF 54 ]aswellasbulkHe3[ 55 ].AnonanalyticT2,terminthespecicheatin2DhasbeenobservedrecentlyonmonolayersofHe3adsorbedonsolidsubstrate[ 56 ].However,tothebestofmyknowledge,theTlnTnonanalyticityintheone-dimensionalspecicheathasnotbeenobservedinexperiments.Themainreasonforthiscouldbetheinherentdicultyassociatedwithmakingspecicheatmeasurementsonreal1Dsystemsfore.g.,quantumwiresandcarbonnanotubeswhichareextremelysmall(mesoscopic)structures.Onewaytoavoidthisdicultymightbetomeasurethethermalexpansioncoecientofacarbonnanotube.TheGruneisenlawstatesthattheratioofthethermalexpansioncoecienttothespecicheatstaysconstantinthelimitT!0.Atlowtemperatures,thespecicheatisdetermined,mostly,byelectrons,therefore=Cel=const..Measuringthethermalexpansioncoecientofacarbonnanotube,onecantrytodetecttheTlnTbehavior.AdenitivetestofourtheorywouldbetoseethepolarizationdependenceoftheTlnTterm(whichshouldvanishforcompletespinpolarization).Graphiteintheultra-quantumlimit(whichclearly

PAGE 148

showssignaturesof1Dlocalization,chapter2)mightbeapossiblecandidateforobservingtheTlnTterminC(T).

PAGE 149

TheLandauFermiliquid(FL)theorystatesthatthelow-energypropertiesofaninteractingfermionicsystemaredeterminedbystatesinthevicinityoftheFermisurface,andaresimilartothatofanidealFermigas.Atthelowesttemperatures,whenthedecayofquasiparticlescanbeneglected,thespecicheatC(T),scaleslinearlywithTandspinsusceptibilitys(T),approachesaconstantvalue,astheydoinaFermigas,theonlydierencebeingtherenormalizationsoftheeectivemassandgfactor[ 18 ].However,thislowtemperaturelimitoftheFLtheory,consideredbyLandau,cannottellwhetherthesub-leadingtermsinTareanalyticornot,andwhethertheycomeonlyfromlow-energystates(andarethereforedescribedbytheFLtheory)orfromthestatesfarawayfromtheFermisurface. Fornoninteractingfermions,thesub-leadingtermsinC(T)=Tands(T)scaleasT2(comefromSommerfeldexpansion)andcomefromhigh-energystates.However,itwasfoundbackinthe1960sthatin3Dsystems,theleadingcorrectiontoC(T)=Tduetointeractionwitheitherphonons[ 37 ]orparamagnons[ 38 ]isnonanalyticinTandcomesfromthestatesinthevicinityoftheFermisurface.Thesameresultwaslatershowntoholdfortheelectron-electroninteractions[ 36 40 47 48 ].Morerecently,itwasshownbyvariousgroups[ 41 { 46 80 ]thatthetemperaturedependenceofC(T)=Tisalsononanalyticin2Dandstartswithalinear-in-Tterm.Furthermore,itwasshowninRef.[ 44 45 ]thatthenonanalytictermsinthespecicheatin2Doccursexclusivelyfromonedimensionalscatteringprocesses(wheretheincomingfermionmomentaareanti-parallel,andmomentum 138

PAGE 150

transfersareeither0or2kF)and,foragenericFermi-liquid,thecoecientinfrontofthenonanalytic(T)correction,isexpressedintermsofthespinandchargecomponentsofthescatteringamplitudeatthescatteringangle=(backscatteringamplitude).Howeverin3D,both1Dandnon-1DscatteringprocessescontributetotheT2lnT,correctiontoC(T)=TforagenericFL[ 47 ],andthecoecientinfrontcannotbeexpressedsolelyintermsofthebackscatteringamplitude.Therearecontributionsfromtheangularaverages(andnotjust=)ofthescatteringamplitude(orLandaufunction).Thenonanalyticcorrectionstothespecicheathavebeenobservedexperimentallybothin3D(heavyfermionmaterialslikeUPt3)aswellasin2D(monolayersofHe3adsorbedonasolidsubstrate). Inthischapterwewillbestudyingthenonanalyticcorrectionstothespinsusceptibilityinboth2Dand3D.Untilrecently,theprevailingopinionhadbeenthatthenonanalytic,T3lnTterminthespecicheatisnotparalleledbyasimilarnonanalyticityinsin3D.CrucialevidenceforthisviewwasprovidedbytheresultsofCarneiroandPethick[ 36 ]andBeal-Monodetal.,[ 91 ]whofoundthattheleadingterminthespinsusceptibilityscalesasT2in3D.However,inanimportantpaperBelitzetal.[ 42 ]demonstratedthattheapparentanalytictemperaturedependenceofsmaybemisleading.Theyperformedaperturbativecalculationofthemomentumdependentspinsusceptibilitys(Q;T=0)atsmallQandfoundanonanalyticQ2lnQbehavior.Later,itwasfound[ 92 ]thatthemagnetic-elddependenceofanonlinearspinsusceptibilityparallelstheQdependence,i.e.,s(Q=0;T=0;H)/H2lnjHjwhichnegatedanearlierresultofBeal-Monod[ 93 ],whofoundonlyananalyticmagneticelddependences/H2in3D. Nonanalyticityofthespinsusceptibilitywasalsofoundfor2DsystemsbyMillisandChitov[ 43 ]and,later,byChubukovandMaslov,[ 44 ],Galitski

PAGE 151

etal.[ 46 ]andBetourasetal.[ 92 ].Theseauthorsshowed,(usingsecondorderperturbationtheory),thats(T;Q;H)scaleslinearlywiththelargestoutofthethreeparameters(inproperunits).FurthermoreChubukovandMaslov[ 44 45 ]andGalitskietal.[ 46 ]haveshownthatin2D,thenonanalyticterminscanbesolelyexpressedintermsofthebackscatteringamplitude.Wearegoingshowthatthisresultisvalidonlyifthebackscatteringamplitudeissmall.Ingeneralthenonanalyticcorrectiontothespinsusceptibilityin2Dacquirecontributionsfromtheangularaveragesofthescatteringamplitude(andnotjust=,aswouldhavebeenthecaseforbackscatteringamplitude)(section4.3).Therefore,thereisanimportantdierencebetweenthenonanalyticcorrectionstothespecicheatandthespinsusceptibilityin2D.Theformersolelycomesfrom1Dscattering,expressedintermsofbackscatteringamplitudewhereasthelattergetscontributionsfromboth1Daswellasnon-1Dscattering. AsitwasmentionedintheIntroduction,thesignofthenonanalyticdependenceofs(H;Q)isimportantinunderstandingthenatureofthephasetransitiontotheferromagneticstate.Alltheknownresultsforthenonanalyticdependence(bothinD=2;3)givesanincreaseofs(Q;H)withQ;Hwhichpointstowardsametamagnetic(rstorder)transitionororderingatniteQ.Theseresultswereforthesecondorderininteraction.Weshow(insection4.1and4.2)thatthethirdorderininteractiongivesadecreaseofs(H)withH(oppositeto2ndorder)whichfavors,Hertz'ssecondorderphasetransitionpicture.Howeverthesesignsoscillateateveryorderinperturbationandingeneral,itisimpossibletodeterminetheorder(rstorsecond)ofthephasetransitionfromthesignofafewloworderinperturbationtheory.Toresolvethisissue,wecalculates(H)atthecriticalpointusingthespin-Fermionmodelandshowthatitisofthemetamagneticsign.Evenexperimentally,thesituationisnotclear,TheferromagneticmetallicalloyswithlowCurieTemperatures(\weakferromagnets")

PAGE 152

doseemtoshowavarietyofbehaviors:insomeofthem,theQCPisoftherstorder,e.g.,MnSi[ 94 ]andUGe2,whereasother,e.g.,NixPd1x[ 95 ]showasecondordertransitiontothelowesttemperaturemeasured. Inordertokeepthisdiscussionfocusedwewillonlytalkaboutthenonanalyticmagnetic-elddependences(H)inD=2;3.Thischapterisorganizedasfollows.Wewillobtainthenonanalyticcorrectionstos(H),in2Dinsection4.1.atbothsecondandthirdorderinperturbationtheory.Section4.2isdevotedtothenonanalyticcorrectionsins(H),in3D.Insection4.3,weobtains(H)foragenericFermiliquidin2D,andshowthatitcannotbeexpressedonlyintermsofthebackscatteringamplitude.Thesameargumentcanbeextendedtoshowtheimportanceofany-anglescatteringin3D(andnotjust=);however,duetolackofspacewedonotpresentthe3Dcalculationhere.Insection4.4,weanalyzethebehaviorofthenonanalytictermsinthespinsusceptibilitynearthequantumcriticalpointusingthelowenergyeectivespin-fermionmodel.Weconcludeinsection4.5. Wewillbeonlyinterestedinthespineectofthemagneticeld,butnottheorbitaleect.AmagneticeldsplitstheFermisurfacesforfermionswithspinsparallelandanti-paralleltotheeld.Wewillseethatthissplittingdoesnotaectthe! qnonanalyticity(seeIntroduction,section1.2)ofthepolarizationbubbleatsmallq,ifaparticleandholehavethesamespins(inthiscaseamagneticeldjust

PAGE 153

shiftsthechemicalpotentialwhichcanatmostgiveananalyticH2contribution).However,ithasanontrivialeectonabubblecomposedofaparticleandholeofoppositespins,"#(q;i!;H).Atsmallmomentumtransfer(q0),partoftheparticle-holepolarizationoperatorhasanexplicitmagneticelddependenceonlyifthespinsareantiparallel.Toseethis,rstconsiderthecaseofparallelspins""(q;i!)=Zd2k ""(q;i!)=FZdkZd Toarriveatthelastexpressionweperformastandardcontourintegration.Weseethattheelddependencedropsout.Similaranalysisshowsthat##(q;i!)isalsogivenbyEq. 4{2 .Thereforeallthosethermodynamicpotentialdiagramswhichcontainstheq0particle-holebubbleswithparallelspinsdonotgiveanynonanalyticelddependenceinthespinsusceptibility.Letusnowconsiderthesebubblesforanti-parallelspins."#(q;i!)=Zd2k

PAGE 154

Wenowgetanexplicitelddependence.FollowingthesamestepsusedtoobtainEq. 4{2 ,weget(atT=0) "#(q;i!)=FZ20d Wewillusethisresponsefunctiontoevaluatethethermodynamicpotentialatsecondandthirdorderininteraction. Figure4{1. Particle-holetypesecondorderdiagramforthethermodynamicpotential. ConsiderthesecondorderdiagramforthethermodynamicpotentialshowninFig. 4{1 .InthisdiagramonepairsaGreen'sfunctionfromthetopbubblewithaGreen'sfunctioninthelowerbubbletoform"#(q;i!).(H)=g2 (H)=g2F2

PAGE 155

wherewehaveintroducedultravioletcut-ostoregularizetheformallydivergentintegrals.Thenonanalytictermsarisefromthelowerlimitsoftheintegralsandarecut-oindependent. (H)=BjHj3; whereB=16(gF)2B3 Weseethatsincreasesasafunctionoftheeld.NowconsiderthethirdordercorrectionshowninFig. 4{2 Figure4{2. Particle-holetypethirdorderdiagramforthethermodynamicpotential. (H)=g3

PAGE 156

Keepingthenonanalyticpartoftheresponsefunction(Eq. 4{3 .)andrestrictingourcalculationstozerotemperature,weobtain (H)=(gF)3 (x2+(!ih)2)3=2;=(gF)3 where:::standforanalytictermsinh.Thenonanalyticcorrectiontothespinsusceptibility(at3rdorder)is, Herethespinsusceptibilitydecreases,astheeldisincreasedoppositetothebehavioratsecondorder.HoweverateveryhigherorderininteractiononegetsanindependentnonanalyticjHj3termsinthethermodynamicpotential.Thereforeonecannotpredictthenatureofthephasetransitionbylookingatrstfewordersinperturbationtheory.SincethenonanalyticcontributionoccursforqvF!,(seemomentumintegrationatbothsecondatthirdorder),theycomefromanyanglescatteringevents,(inordertohave1Dscatteringacts,thenecessaryconditionwasqvF!,i.e.,deepinsidetheparticle-holecontinuum). "#(q;i!)=Zd3k =FZdkZ11d(cos) 2nF(~k+h=2)nF(~k+~qh=2)

PAGE 157

Performingthesamemanipulationsasin2Dcase,wegetfortheparticle-holebubblein3DatT=0, "#(q;i!)=FZ11d(cos) 2(h+vFqcos) i!vFq+h): Usingthisformofthebubblewegetanonanalyticcontributionforthethermodynamicpotentialateveryorderininteraction,startingatsecondorder(seeFig. 4{1 ).(H)=g2 i!q+h)2; q=y;h q=~H)andwritetheintegrandinadimensionlessform(H)=(gF)2 (H)=AZEF0dqq3~H4Z1dy(6y5tan1(1=y)+3y46y3tan1(1=y)+4y2 Theyintegralconverges,sothatwecanextendtheintegrationlimitstoinnity,uponwhichitgivesanumber.Theremainingqintegralislog-divergent.Aswehadperformedanexpansionin~H=h=q1,itislegitimatetocutthedivergentintegralatq=h.Tologarithmicaccuracythearbitrarinessinchoosingthe

PAGE 158

numericalprefactorinthecut-odoesnotaecttheresult. (H)=2A whereA=(gF)2=(2vF)3.Thenon-analyticcorrectiontothespinsusceptibilityis, Followingtheaboveprocedureonecanevaluatethethirdorderthermodynamicpotential(seeFig. 4{2 ),(H)=+2(gF)3 i!q+h)3; Notethatthethirdordersignisoppositetothesecondordersign.However,ateveryhigherorder4th,5th,therewillbeanindependentnonanalyticcontributionwhosesignwilloscillateandthusonecannotdenitelypredictthesignofthenonanalytictermbylookingatfewlowordersinperturbationtheory.Furthermoreinrealsystemsinteractionsarenotweakandonecannotterminatetheperturbationseriesexpansiontothelowestorders.Tocircumventthisinherentproblemwithperturbativecalculationsandtomakepredictionsforrealisticsystems(e.g.,He3),weobtainthenonanalyticelddependenceofthespinsusceptibilityforagenericFermiliquidin2D. 89 ] =Tr(ln[G01]+G)+skel

PAGE 159

withbeingtheexactself-energy,andskelisthesetofskeletondiagramsforallinteractioncorrectionstothatarenotaccountedforbythersttwoterms.Thetraceistakenoverspace,time,andspinvariables.TheskeletondiagramforthethermodynamicpotentialisshowninFig. 4{3 ,wherethebareinteractionlinesU(q),arereplacedwithfullydressedvertices,(~p~k).Thevertex(~p~k),hasanexpansionintermsofparticle-holebubbles[ 18 ],whichtolowestordergivesadiagramwhichistheparamagnondiagram(at2ndorder,seeFig. 4{1 ),exceptwithmomentumdependentinteractionlines,theanalyticexpressionforwhichis (H)=TXi!Zd2~q (4{16) Figure4{3. Theskeletondiagramforthethermodynamicpotential. Anext-to-leadingorderexpansionofthevertex,givesadiagramwhichisthethirdorderparamagnondiagram,againwithmomentumdependentinteractionlines. (H)=TXi!Zd2~q (4{17)

PAGE 160

Belowweevaluatethespinsusceptibilityarisingfromtheselowestorderexpansionoftheskeletondiagrams.ForthesecondorderdiagramgivenbyEq. 4{16 ,weget(H)=(F)2Z1d! vFq vFq)Z20dk vFq vFq)2(pF(kp)) NowwewillexpandtheangledependentvertexinaFourierseries,(kp)=1Xn=nein(kp): (4{18) (4{19) Usingtherelation(cospib)1=sgn(b)iZ10deisgn(b)(cospib)

PAGE 161

UsingtheBesselfunctionproperty,Z20d Letusrstevaluate1(H).Itisconvenienttosplitthesumovertheintegersn;mintotwosums,onewithn+m>0andtheotherwithn+m<01(H)=(F 69 ], (4{20)

PAGE 162

Wepulloutthecommondenominatorfromboththesumswhichyields1(H)=(F vFq)2+1(!+ih vFq)2n+2m+1Xn;m=n+m<0(1)2n2ms vFq)2+1(!+ih vFq)2n2mnm(1)(n+m) 1(H)=(F EF)2BS wherethebackscatteringamplitude(scatteringamplitudeatthescatteringangle=)isdenedas,BS=Pn(1)nn.Similarlyonecanshowthat,2(H)=(F EF)2BS (H)=(F wherewehavemultipliedwithanoverallfactorof1=2,whichisthesecondordercombinatorialcoecient.Therefore,thesecondorderskeletondiagramforthespinsusceptibilitycanbeexpressedsolelyintermsofthebackscatteringamplitude.

PAGE 163

Thisresultissimilartothatforthespecicheat[ 45 47 ], wherecisapositivenumber.However,weshowthatthenext-to-lowestorderexpansionofthevertex,intheskeletondiagram,acquirescontributionswhichcannotbeexpressedintermsofthebackscatteringamplitude.Thissuggeststheimportanceofnon-one-dimensionalscatteringprocesses.Thethirdorderskeletondiagramreads(H)=TXi!Zd2~q Thenonanalyticpartatthirdordercomesfrom(H)=(F)3Z1d! vFq vFq)Z20dk vFq vFq)Z20dk1 vFq vFq)1Xn;m;l=nmlein(pk)+im(k1p)+il(kk1) (H)=1(H)+2(H); where, 1(H)=(F)3

PAGE 164

where,Anml=[(mn)(p 2(H)=(F)3 where,Bnml=[(mn)(1)nm(p 4{25 andEq. 4{26 ,hasapowerlawsingularity,soonecannotcutitatqvF>!+ihandsetqvF!+ihinAnmlandBnml,aswedidinsecondordercasewheretherewasalogsingularity.Thisistheessentialdierencebetweenthesecondorderandthirdordercalculationswhichmakesitextremelydiculttoarriveataclosedform(forarbitraryn;m;l)atthirdorder.WewillobtainthecoecientinfrontofthejHjterminthespinsusceptibility,forthefewlowestharmonics(0n;m;l1)andshowthatitisnotpossibletowritethethirdorderresultsolelyintermsofthebackscatteringamplitude.Severalcasesneedtobeconsideredseparately.Case(a):

PAGE 165

000(H)=(F0)3 wherethesuperscriptdenotestheharmonics. Case(b):m=1;n=0;l=0.ThenA100=(1)(p q2[q2+(!+ih)2]3=2(p 100(H)=3(F)3120 Case(e):m=1;n=1;l=0.ThenA110=(1)(p

PAGE 166

pre-factoris210.Theextrafactorof3arisesduetothecase(f):m=1;n=0;l=1andthecase(g):m=0;n=1;l=1whichgivesthesamecontribution. 110(H)=3(F)3021 Case(h):m=1;n=1;l=1,hereA111=B111=1.Theintegralsareidenticaltocase(a)and 111(H)=+(F1)3 Addingtheresultfortherstfewharmonicswegetforthespinsusceptibility, Notethatthenumericalfactorof2ln(2)(andalsothesigns)makesitimpossibletorepresentthe3rdorderresultintermsofbackscatteringamplitude.Ifthethirdorderresulthadanexpansionintermsof3BS,thenourlowestharmonicsresultswouldhavebeen(Pn(1)nn)3=303201+321031whichisdierentfromwhatweobtained. Parameters0,1,canbeestimatedusingtherelationbetweentheharmonicsoftheamplitudesandtheLandaufunctionsofa2DFermiliquid[ 47 ] n=Fn TheLandauparameterscanbemeasuredfromtherenormalizationsoftheleading(analytic)termsinthermodynamicquantities.InbulkHe3, (4{33) (4{34)

PAGE 167

inawideintervalofparameters[ 55 ].AssumingthattheFermi-liquidparametersarethesameinbulkHe3andina2Dlayer,wecanestimate0and1tobe 0=2:3 (4{35) 1=0:76 (4{36) (4{37) Inthisapproximation,thesquareofthebackscatteringamplitude 2BS=(Xn()nn)22:5 (4{38) whereasthecoecientofthethirdorderwhichwefoundis3=0:05.Therefore,inthisapproximationthebackscatteringamplitudestilldominatestheresult. 62 ], (H)=FG+1 2ZdD~q 2ln((s;0)1+g2); whereFGisthethermodynamicpotentialfortheFermigas,gisthespin-Fermioncouplingconstant,ands;0isthebarespinsusceptibility,whichisanalyticatqpFandisgivenbytheOrnstein-Zernickeform 2+q2: NoticethatA2istheq=0susceptibility,whichdivergesatthesecond-orderQCP.istheparticleholepolarizationoperator,inanitemagneticeldorinasystemwithnitemagnetization (q;i)=2ZdD~k

PAGE 168

where(G),arethedressedGreen'sfunctioncontainingthefermionself-energies.s;0isaphenomenologicalinputofthetheory.Itisassumedthatthehighenergystates(awayfromtheFermisurface)arealreadyintegratedout,andtheireectistodrivethesystemtothevicinityoftheQCP.TheSpin-Fermionmodelprovidesanaccuratedescriptionofthefeedbackfromthelow-energystatesonthepropertiesoftheQCP. Onecanperformthecalculationseitherinanitemagneticeld,inwhichcaseG!G"#,oronecanworkwithasystemwithnitepolarizationinazeromagneticeldassumingspinupandspin-downelectronshavedierentdensitiesnanddierentchemicalpotentials Thefermionself-energies(seeFig. 4{4 )arecalculatedwithintheEliashbergapproximation,whereoneassumesamomentumindependentself-energyandneglectsthevertexcorrections.ThedoublelineindicatesthedressedGreen'sfunctionandthedoublewavylineindicatesthespinuctuationpropagator,renormalizedbythebosonicself-energy.ItcanbeshownthattheEliashbergapproximationworksaslongas1[ 62 ],whereisthedenesasfollows,(!)=i~(!),with ~(!)=!; where(themassrenormalizationfactor,m?1)takesdierentfunctionalformsintheFermi-liquid(farfromtheQCP)andthenon-Fermiliquid(neartheQCP)regimes,andalsodependingonwhetheroneisin2Dor3D.In2D[ 62 ], (4{44) (4{45)

PAGE 169

whereg2A=EFisasmallparameter,isthecorrelationlengthwhichisassumedtobemuchlargerthantheinteratomicdistanceand!0=2EFisaconstant.In3D, wherecisasmallparameter,and0EFisagainaconstant. Figure4{4. Fermionself-energy(a)andBosonicself-energy(b). 4{41 in2D,wehave(q;i)=2F

PAGE 170

Inthenon-perturbativeregime(1)wehave~(!)!,sotheMatsubarafrequenciesi!andi!+iintheGreen'sfunctioncanbeneglectedcomparedto~(!)and~(!+).andPerformingasimplepoleintegrationink,wearriveat(for>0)(q;i)=2Fi (q;i)=2FZmax[0;]min[;0]d! Changingthesignof,isthesameasm!m,whichisequivalenttocomplexconjugation.Thenthethermodynamicpotential(Eq. 4{39 ,nonFermi-gaspart),becomes (m)=2BReZEF0dqqZEF0dlnq2 whereB=1=2(2vF)2isconstantpre-factorandwherewehavescaledoutthevFdependencebyrelabelingvFq!q.TherstterminsidethelogcomesfromtheOrnstein-Zernickeformofthebaresusceptibilityandwehaveset!1. First,weanalyzethenonanalyticpartofthethermodynamicpotentialintheFermi-liquidregime,where~(!)=!with=(pF).Then,=,andthe!integralistrivial.Keepingjustthedynamicpartofthebubbleweget FL(m)=2BReZEF0dqqZEF0dlnjg1

PAGE 171

whereg1=2g2F.Re-scaling,=1weget FL(m)=2B ReZEF0d1ZEF0dqqln1 ReZEF0d1(1im)2lnj1im wherewehavedroppedtheanalyticinmterms.Weseethatthenonanalyticterminthethermodynamicpotentialcomeswithanegativepre-factorwhichindicatesthepossibilityofarstordertransition.TheFreeenergyisgivenby, Wewillnowperformathermodynamicanalysisandndthedependenceonthemagneticeld.TheGibbsfreeenergyis and wherebandcareconstantsandwherewehavekeptthenextanalytic(m4)termsinF,whichstabilizesthephase.TheunitsarechoseninsuchawaythatFandmanddimensionless.TheminimumofG(m;H)withrespecttothemagnetizationgivestheequationofstate, @m=m 2cmjmj Solvingthisequation 5HjHjb8H3+:::

PAGE 172

Thereforethespinsusceptibilityisgivenby @H=2+c 5jHj3b8H2+::: whichdivergesattheQCPas!1. NextweanalyzethestabilityoftheQCP,wheretheelddependence(ofs),changestojHj3=2.ToanalyzethebehaviorneartheQCPwestartfromEq. 4{49 .Noticethatifweusethenon-FLformoftheself-energyin2D(~(!+)sgn(!+)j!+j2=3),in,wegetthefollowingscalingbehavior 2=3m; neartheQCP.Also,sinceinthequantumcriticalregime;()isafunctionof!,onecannotdothe!integrationtrivially.InEq. 4{49 ,thenonanalyticelddependencearisesfromthedynamicbubblesothatonecanscaleoutthestaticpart,(m)=2BReZEF0dqqZEF0dln1+2g2FA q3Z0d! (m)=2BReZEF0dqqZEF0dln[g q3Z0d!(11 2(im q)2)] (4{59) whereg=2g2FA, (m)=2BReZEF0dqqZEF0dln[g (4{60) Onceagainwescaleoutthersttermwhichdoesnotdependonthemagneticeld.Deninganaverageself-energysquare,((;m))2=1R0d!(

PAGE 173

(m)=2BReZEF0dZEFdqqln[1+()2 wherewewilltakethelimit!0intheend.Theaverageself-energysquareis, 2((;m))2=1 Z0d![(!0)1=3sgn(!+)j!+j2=3(!0)1=3sgn(!)j!j2=3im]2=4=3(!0)2=3Z10dx[(1x)2=3+x2=3ia]2 wherea=m=(2=3(!0)1=3).SubstitutingEq. 4{62 ,inEq. 4{61 ,weperformascalinganalysistoget (m)jmj7=2=)s(H)jHj3=2; WeseethattheFree-energy(thermodynamicpotential)becomenegative,whichsignalsaninstabilityofthesecondorderquantumcriticalpoint. 4{41 ),=F cos(m+i

PAGE 174

wherewehaveperformedthesamemanipulationsasin2D.Performingtheangularintegrationwhichgivesalog,andintegratingbypartsweobtainfortheleadingterminthepolarizationoperator,(>0), (q;i)=g2Fi (4{64) Usingtheaboveformofthepolarizationoperator(andalsoconsideringthecase<0),theformulaforthethermodynamicpotential,(nonFermigaspart),(Eq. 4{39 ). (m)=1 (2vF)32ReZEF0dqq2ZEF0dln[g2F(i wherewehavekeptjustthedynamicpartandhavedroppedthestaticpart(comingfromtheOrnstein-Zernickeformofthebaresusceptibility).IntheFLregime,~()=,where,(=cln(kF),isaconstant.Therefore,FL(m)=1 (2vF)32ReZEF0dqq2ZEF0dln[g2F(i Nowonecanre-scale=1toget, FL(m)=1 (4{66) Writingtheintegrandintermsofdimension-lessvariables,1 q=dandTaylorexpandingtheintegrandford1,andextractingthecoecientofthed4term(similartowhatwedidinperturbationtheory)weget, FL(m)=1 (4{67) q(42

PAGE 175

wherec1isapositiveconstant.Thereforeperformingthesamethermodynamicanalysisaswasdonefor2D,onegets wherewehavesubstituted,(ln).AsoneapproachestheQCP,becomesfrequencydependentandthatchangestheelddependenceatthecriticalpointto,sQCP(H)/H2ln(ln(jHj)).Toanalyzetheelddependenceinthenon-FLregime,westartfromEq. 4{65 anduse~()=cln(0 nFL(m)=2BReZEF0dqq2ZEF0dln[i (4{70) whereonceagainwehavere-scaledthedummyvariablesas,y==q;d=m=qanda=0=qandB1=(2vF)3isaconstantpre-factor.Toobtainascalingformforthethermodynamicpotentialweneglecttheiyln(y)termsinsidethelogarithm.nFL(m)2BReZEF0dqq3ZEF0dyln[iyln(1+d+iyln(a) whichisofthesameformasEq. 4{67 .OnceagainTaylorexpandford1andintegrate(overy1)thecoecientofthem4term. nFL(m)Bm4ZEFmdq qln(0=q)(42 (4{71) wherebisapositivenumber.ThesecondorderQCPisunstableevenin3D,withatendencytowardsrstordertransition.ThespinsusceptibilityattheQCPissnFL(H)/H2ln(ln(0=H)).

PAGE 177

Inthisworkwehaveinvestigatedtheroleofone-dimensionalelectroniccorrelationsinthetransportandthermodynamicpropertiesofhigherdimensional(D=2;3)systems. Intherstpartofthisworkwepresentedastudyoftransportpropertiesofathree-dimensionalmetalsubjectedtoastrongmagneticeldthatconnestheelectronstothelowestLandaulevel(UQL).Weshowedthatthenatureofelectrontransportisonedimensionalduetothereducedeectivedimensionalityinducedbytheeld.TherstsignofthisisthatthelocalizationcorrectionstotheconductivitywasoftheorderofthebareDrudeconductvityitself.Thereforeperturbationtheorybreaksdownjustasitdoesin1D.However,unlikeinthe1Dcase,weshowedthattheconductivityremainsniteatzerotemperature.Therefore,wecallthisregimeintermediatelocalization.Thesecondimportantmanifestationofelectroniccorrelationsandlowerdimensionalityisthattherstorderinteractioncorrectiontotheconductivityislog.divergentintemperature,justasfor1Dsystems.Thephysicalreasonforsuchabehavioroftheconductivityisanearly1DformoftheFriedeloscillationaroundanimpurityinthestrongmagneticeld.Arenormalizationgroupcalculationofthetransmissionamplitudethroughasinglebarrierallowedforasummationofaseriesofthemostdivergentlog-correctionsatallordersintheinteraction.Justasin1Dthissummationinourcaseledtopowerlaw(temperature)scalingbehaviorfortheconductivity.Somerecenttransportmeasurementsingraphitewerecomparedwiththeabovetheoreticalndingsandshowntodisagree.Toresolvethedisagreement,we 166

PAGE 178

invokedamodelwithlongranged-disorderandphononinduceddephasing(1Dphenomenon)toexplaintheexperimentalobservations. Previousworkonthethermodynamicpropertiesofhigherdimensionalsystemshadindicatedthespecialroleplayedbyone-dimensionalscatteringevents,inthenonanalyticcorrectionstothespecicheatandspinsusceptibility.Inthesecondpartofthiswork,wehaveshownthatthenext-to-leadingtermsinthespecicheatandspinsusceptibilityin1Darenonanalytic,inthesamewayastheyareinhigherdimensions(D=2;3).Thus,eventhoughthelowenergytheorywhichdescribesaone-dimensionalinteractingsystem(Luttingerliquidtheory)isdierentfromthehigherdimensionallowenergytheory(Fermiliquid),thesub-leadingtermsinthethermodynamicpropertiesgetnonanalyticcorrectionswhicharisesfromthesamesourcesinalldimensions.Theonlydierenceisthatthenon-analyticcorrectiontothespecicheatin1Dispresentonlyforfermionswithspin,anditoccursat3rdorderininteraction;C(T)/g1?2(g1kg2k+g2?)TlnT,whereasinhigherdimensionstheyoccurevenforspin-lessfermionsandstartat2ndorder.Thenon-analyticcorrectionstothespinsusceptibilityoccuratsecondorderininteractioninalldimensionsD=1;2;3.Thus,wehaveshownthat1Dsystemsaresimilartohigherdimensionalsystems,atleastinthecontextofnonanalyticcorrectionstothermodynamics. Inthethirdpartofthiswork,weperformedadetailedinvestigationofthenonanalyticmagneticelddependenceofthespinsusceptibility,inhigherdimensionalsystems,andshowedthattheyarisefromboth1D,aswellasnon-1Dprocesses.Weobtaineds(H)foragenericFermi-liquidin2D.Ourresultcanbecomparedwiththeexperimentalresultsonatwo-dimensionalHe3layer.Wealsostudiedthespinsusceptibilityinthevicinityoftheferromagneticquantumcriticalpointusingthespin-Fermionmodelandshowedthatthesecondordercriticalpointisunstablebothin2Dand3Dwithatendencytowardsrstordertransition.

PAGE 179

[1] S.A.Brazovskii,Zh.Eksp.Teor.Fiz.61,2401(1971)[Sov.Phys.JETP34,1286(1972)]. [2] H.Fukuyama,SolidStateComm.26,783(1978). [3] V.M.Yakovenko,Phys.Rev.B47,8851(1993). [4] Y.Iye,P.M.Tedrow,G.Timp,M.Shayegan,M.S.Dresselhaus,G.Dresselhaus,A.FurukawaandS.Tanuma,Phys.Rev.B25,5478(1982);Y.IyeandG.Dresselhaus,Phys.Rev.Lett.54,1182(1985);H.YaguchiandJ.Singleton,Phys.Rev.Lett.81,5193(1998). [5] C.Biagini,D.L.Maslov,M.Yu.ReizerandL.I.Glazman,Europhys.Lett.55,383(2001). [6] S.-W.Tsai,D.L.MaslovandL.I.Glazman,Phys.Rev.B65,241102(2002). [7] I.E.DzyaloshinskiiandA.I.Larkin,Zh.Eksp.Teor.Fiz.65,411(1973)[Sov.Phys.JETP38,202(1974)]. [8] B.L.AltshulerandA.G.AronovinElectron-ElectronInteractionsinDisor-deredSystems,eds.A.L.EfrosandM.Pollak(North-Holland),Amsterdam,1985). [9] P.A.LeeandT.V.Ramakrishnan,Rev.Mod.Phys.57,287(1985). [10] G.Bergman,Phys.Rep.107,1(1984). [11] I.L.Aleiner,B.L.AltshulerandM.E.Gershenson,WavesRandomMedia,9,201(1999). [12] L.P.Gorkov,A.I.Larkin,andD.E.Khmel'nitskii,Zh.Eksp.Teor.Fiz.30,248(1979)[Sov.Phys.JETPLett.30,228(1979)]. [13] P.W.Anderson,Phys.Rev.109,1492(1958). [14] N.F.MottandW.D.Twose,Adv.Phys.10,107(1961). [15] V.L.Berezinskii,Zh.Eksp.Teor.Fiz.65,1251(1973). [16] E.Abrahams,P.W.Anderson,D.C.LicciardelloandT.V.Ramakrishnan,Phys.Rev.Lett.42,673(1979). 168

PAGE 180

[17] B.L.Altshuler,D.E.Khmel'nitskii,A.I.LarkinandP.A.Lee,Phys.Rev.B22,5142(1980). [18] A.A.Abrikosov,L.P.Gorkov,andI.E.Dzyaloshinski,MethodsofQuantumFieldTheoryinStatisticalPhysics(Prentice-Hall,Inc.,EnglewoodClis,NewJersey,1963). [19] G.Zala,B.N.NarozhnyandI.L.Aleiner,Phys.Rev.B64,214204(2001). [20] S.V.Kravchenko,G.V.Kravchenko,J.E.Furneaux,V.M.PudalovandM.D'Iorio,Phys.Rev.B50,8093(1994). [21] D.Yue,L.I.GlazmanandK.A.Matveev,Phys.Rev.B49,1966(1994). [22] L.D.LandauandE.M.Lifshitz,QuantumMechanics(Pergamon,NewYork,1977). [23] A.Sergeev,M.Yu.ReizerandV.Mitin,Phys.Rev.B69,075310(2004). [24] C.L.KaneandM.A.P.Fisher,Phys.Rev.Lett.68,1220(1992). [25] B.L.Altshuler,A.G.AronovandP.A.Lee,Phys.Rev.Lett.44,1288(1980). [26] B.L.Altshuler,A.G.AronovandD.E.Khmel'nitskii,J.Phys.C15,7367(1982). [27] I.V.Gornyi,A.D.MirlinandD.G.Polyakov,Phys.Rev.Lett.95,046404(2005). [28] L.D.Landau,Zh.Eksp.Teor.Fiz.35,97(1958)[Sov.Phys.JETP8,70(1959)],andreferencesthereintoearlierpapers. [29] P.W.Anderson,Phys.Rev.Lett.66,3226(1991)andreferencestherein. [30] R.Shankar,Rev.Mod.Phys.66,129(1994). [31] W.Metzner,C.Castellani,andC.diCastro,Adv.Phys.47,317(1998). [32] C.NayakandF.Wilczek,Nucl.Phys.B430,534(1994). [33] ForrecentreviewsseeT.TimsukandB.Statt,Rep.Prog.Phys.62,61(1999);M.NormanandC.Pepin,cond-mat/0302347(unpublished);S.Sachdev,QuantumPhaseTransitions(CambridgeUniversityPress,1999);A.Abanov,A.V.ChubukovandJ.Schmalian,Adv.Phys.52,119(2003). [34] See,e.g.,G.R.Stewart,Rev.Mod.Phys.73,797(2001),andreferencestherein. [35] D.PinesandP.Nozieres,TheTheoryofQuantumLiquids(Addison-Wesley,RedwoodCity,1966).

PAGE 181

[36] C.J.PethickandG.M.Carneiro,Phys.Rev.A7,304(1973). [37] G.M.Eliashberg,Sov.Phys.JETP,16,780(1963). [38] S.DoniachandS.Engelsberg,Phys.Rev.Lett.17,750(1966). [39] W.F.BrinkmanandS.Engelsberg,Phys.Rev.169,417(1968). [40] D.J.Amit,J.W.KaneandH.Wagner,Phys.Rev.175,313(1968);175,326(1968). [41] D.CoeyandK.S.Bedell,Phys.Rev.Lett.71,1043(1993). [42] D.Belitz,T.R.KirkpatrickandT.Vojta,Phys.Rev.B55,9542(1997). [43] G.Y.ChitovandA.J.Millis,Phys.Rev.Lett.86,5337(2001). [44] a)A.V.ChubukovandD.L.Maslov,Phys.Rev.B68,155113(2003);b)ibid69,121102(2004). [45] A.V.Chubukov,D.L.Maslov,S.Gangadharaiah,andL.I.Glazman,Phys.Rev.B71,205112(2005);S.Gangadharaiah,D.L.Maslov,A.V.ChubukovandL.I.Glazman,Phys.Rev.Lett.94,156407(2005). [46] V.M.Galitski,A.V.ChubukovandS.DasSarma,Phys.Rev.B71,201302(2005). [47] A.V.Chubukov,D.L.MaslovandA.J.Millis,Phys.Rev.B,73,045128(2006). [48] I.L.AleinerandK.B.Efetov,cond-mat/0602309(unpublished). [49] G.I.JaparidzeandA.A.Nersesyan,Phys.Lett.94A,224(1983). [50] I.E.DzyloshinskiiandA.I.Larkin,Zh.Eksp.Teor.Fiz.61,791(1971)[Sov.Phys.JETP34,422(1972)]. [51] T.holstein,R.E.Norman,andP.Pincus,Phys.Rev.B8,2649(1973);M.Reizer,ibid.40,11571(9189). [52] P.A.Lee,Phys.Rev.Lett.63,680(1989). [53] D.L.Maslov,inNanophysics:CoherenceandTransport,LesHouchesLXXXI(eds.H.Bouchiat,Y.Gefen,S.Gueron,G.MontambauxandJ.Dalibard),(Elsevier,Amsterdam,2005),p.1;cond-mat/0506035. [54] G.R.Stewart,Rev.Mod.Phys.86,755(1984). [55] D.S.Greywall,Phys.Rev.B27,2747(1983)andreferencestherein.

PAGE 182

[56] A.Casey,H.Patel,J.Nyeki,B.P.Cowan,andJ.Saunders,Phys.Rev.Lett.90,115301(2003). [57] H.Fukuyama,T.M.Rice,C.M.VarmaandB.I.Halperin,Phys.Rev.B10,3775(1974). [58] J.A.Hertz,Phys.Rev.B14,1165(1976). [59] A.Millis,Phys.Rev.B48,7183(1993). [60] T.Moriya,SpinFluctuationsinItinerantElectronMagnetism(springer-Verlag,Berlin,NewYork,1985). [61] See,e.g.,G.R.Stewart,Rev.Mod.Phys.73,797(2001),andreferencestherein. [62] A.V.Chubukov,C.PepinandJ.Rech,Phys.Rev.Lett.92,147003(2004). [63] T.Vojta,D.Beltiz,R.NarayananandT.R.Kirkpatrick,Z.Phys.B103,451(1997). [64] B.Altshuler,L.B.Ioe,andA.J.Millis,Phys.Rev.B52,5563(1995). [65] A.Chubukov,A.Finkelstein,R.Haslinger,andD.Morr,Phys.Rev.Lett.90,077002(2003). [66] A.A.AbrikosovandI.A.Ryzhkin,Adv.inPhys.,27,147(1978). [67] D.G.Polyakov,inProceedingsofthe20thInternationalConferenceonthePhysicsofSemiconductors(Greece,1990)p.2321. [68] A.A.Abrikosov,Zh.Eksp.Teor.Fiz.56,1391(1969)[Sov.Phys.JETP29,746(1969)]. [69] I.S.GradshteynandI.M.Ryzhik,TableofIntegrals,Series,andProducts(AcademicPress,SanDiego,1994). [70] A.Ya.BlankandE.A.Kaner,Zh.Eksp.Teor.Fiz.50,1013(1966)[Sov.Phys.JETP23,673(1966)]. [71] D.G.Polyakov,Zh.Eksp.Teor.Fiz.83,61(1982)[Sov.Phys.JETP56,33(1982)]. [72] S.-W.Tsai,D.L.MaslovandL.I.Glazman,PhysicaB312-312,586(2002). [73] S.S.Murzin,Usp.Fiz.Nauk.170,387(2000)[Sov.Phys.Usp.43,349(2000). [74] S.S.MurzinandN.I.Golovko,Pis'maZh.Eksp.Teor.Fiz.54,166(1991)[JETPLett.54,551(1991)].

PAGE 183

[75] F.A.EgorovandS.S.Murzin,Zh.Eksp.Teor.Fiz.94,315(1988)[Sov.Phys.JETP67,1045(1988)]. [76] X.Du,S.-W.Tsai,D.L.Maslov,andA.F.Hebard,Phys.Rev.Lett.94,166601(2005). [77] N.W.AshcroftandN.D.Mermin,SolidStatePhysics(Holt,RinehartandWinston,NewYork,1976). [78] A.A.Gogolin,V.I.Mel'nikovandE.I.Rashba,Sov.Phys.JETP42,(1975). [79] V.V.Afonin,Yu.M.GalperinandV.L.Gurevich,Zh.Eksp.teor.Fiz.88,1906(1985)[Sov.Phys.JETP61,51985]. [80] M.A.Baranov,M.Yu.Kagan,andM.S.Mar'enko,JETPLett.58,709(1993). [81] V.M.Galitskii,Zh.Eksp.Teor.Fiz.34,151(1958)[Sov.Phys.JETP7,104(1958)]. [82] T.Giamarchi,QuantumPhysicsinOneDimension,(InternationalSeriesofMonographsonPhysics,ClarendonPress,Oxford,2004). [83] D.C.MattisandE.H.Lieb,J.Math.Phys.6,304(1965). [84] J.Soloyom,Adv.Phys.28,209(1979). [85] G.D.Mahan,Many-ParticlePhysics,(KluwerAcademic/PlenumPublishers,NewYork,2000). [86] C.M.Varma,P.B.Littlewood,S.Schmitt-Rink,E.Abrahams,andA.E.Ruckenstein,Phys.Rev.Lett.63,1996(1989). [87] Yu.A.Bychkov,L.P.Gor'kovandI.E.Dzyaloshinskii,Zh.Eksp.teor.Fiz.50,738(1996)[Sov.Phys.JETP23,489(1966)]. [88] I.E.DzyaloshinskiiandA.I.Larkin,Zh.Eksp.Teor.Fiz.65,411(1973)[Sov.Phys.JETP38,202(1974)]. [89] J.M.LuttingerandJ.C.Ward,Phys.Rev.118,1417(1960). [90] I.E.DzyaloshinskiiandA.I.Larkin,Zh.Eksp.Teor.Fiz.61,791(1971)[Sov.Phys.JETP34,422(1972)]. [91] M.T.Beal-Monod,S.-K.Ma,andD.R.Fredkin,Phys.Rev.Lett.20,929(1968). [92] J.Betouras,D.Efremov,andA.Chubukov,Phys.Rev.B72,115112(2005). [93] M.T.Beal-MonodandE.Daniel,Phys.Rev.B27,4467(1983).

PAGE 184

[94] C.Peiderer,S.R.Julian,andG.G.Lonzarich,Nature414,427(2001). [95] M.Nicklas,M.Brando,G.Knebel,F.Mayr,W.TrinklandA.Loidl,Phys.Rev.Lett.82,4268(1999).

PAGE 185

RonojoySahawasbornonJuly17,1978,inCalcutta,India.HespenttheearlyyearsofhischildhoodinCalcutta,andafewyearsinnorthBengal,beforemovingtoNewDelhiforhishighschool.Aftercompletinghighschoolin1995,hecamebacktoCalcuttaandjoinedthePhysicsDepartmentofPresidencyCollegeforhisundergraduatestudies.DuringhisundergraduatedaysinPresidencyCollege,hemethisfuturewifeSreya.HereceivedhisBachelorofSciencedegreein1998andmovedtoNewDelhitopursuehismaster'sinphysicsattheJawaharlalNehruUniversity(JNU),whichhecompletedinsummerof2000.Duringhismaster'satJNUhedevelopedakeeninterestintheoreticalphysics.InFall2000,hejoinedthegraduateprograminphysicsattheUniversityofFlorida.SinceFall2001,hehasworkedwithProfessorDmitriiMaslovonvariousproblemsinstronglycorrelatedelectronsystems.HereceivedhisPh.D.inAugust2006. 174


Permanent Link: http://ufdc.ufl.edu/UFE0015302/00001

Material Information

Title: Manifestations of One-Dimensional Electronic Correlations in Higher-Dimensional Systems
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015302:00001

Permanent Link: http://ufdc.ufl.edu/UFE0015302/00001

Material Information

Title: Manifestations of One-Dimensional Electronic Correlations in Higher-Dimensional Systems
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015302:00001


This item has the following downloads:


Full Text











MANIFESTATIONS OF ONE-DIMENSIONAL ELECTRONIC CORRELATIONS
IN HIGHER-DIMENSIONAL SYSTF:\ IS
















By

RONOJOY SAHA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Ronojoy Saha



































To my family















ACKNOWLEDGMENTS

First and formeost I would like to thank my research supervisor, Professor

Dmitrii Maslov, for his constant encouragement and guidance throughout the

entire course of my research. His enthusiasm, dedication, and optimism towards

physics research have been extremely infectious. The countless hours I have spent

discussing physics with him were highly productive and intellectually stimulating.

I would like to thank Professor Jim Dufty, Professor Arthur Hebard and

Professor Pradeep Kumar, who were ahv-- willing and open to discuss any physics

related questions. I am honored and grateful to Professor Russell Bowers, Professor

Adrian Roitberg, Professor Khandker Muttalib, Professor Sergei Obukhov and

Professor Arthur Hebard for serving on my supervisory committee.

My thanks go to the Physics Department secretaries, Ms. Balkcom, Ms.

Latimer, Ms. Nichola and Mr. Williams; and to my friends Partho, Vidya, Suhas,

Aditi, Aparna and Karthik for their help and support.

I would like to thank my wife and best friend Sreya for being my source of

strength and inspiration through all these years.

I would like to thank my family for the unconditional love, support and

encouragement they provided through the years.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF FIGURES ................... ......... vii

ABSTRACT ... .. .. .. ... .. .. .. .. ... .. .. .. ... .. .. x

CHAPTER

1 INTRODUCTION .................... ....... 1

1.1 Transport in Ultra Strong Magnetic Fields ...... ........ 2
1.1.1 Weak Localization QCC ................. .. .. 5
1.1.2 Interaction Correction to the Conductivity-Altshuler Aronov
Corrections (class I) ..... . . .... 10
1.1.3 Corrections to WL QCC due to Electron-Electron Interactions:
Dephasing (class II) ....... . . .... 17
1.2 Non-Fermi Liquid Features of Fermi Liquids: 1D Physics in Higher
Dimensions ................... . . 19
1.3 Spin Susceptibility near a Ferromagnetic Quantum Critical Point
in Itinerant Two and Three Dimensional Systems. . ... 34
1.3.1 Hertz's LGW Functional ............ .. .. .. 36

2 CORRELATED ELECTRONS IN ULTRA-HIGH MAGNETIC FIELD:
TRANSPORT PROPERTIES .................. ...... 43

2.1 Localization in the Ultra Quantum Limit . . ...... 45
2.1.1 Diagrammatic Calculation for the Conductivity ...... ..46
2.1.2 Quantum Interference Correction to the Conductivity . 50
2.2 Conductivity of Interacting Electrons in the Ultra-Quantum Limit:
Diagrammatic Approach .................. .... .. 58
2.2.1 Self-Energy Diagrams .................. ..... 61
2.2.1.1 Diagram Fig. 2-10(a) ............... .. 64
2.2.1.2 Diagram Fig. 2-11(a). .............. 67
2.2.2 Vertex Corrections .................. .. 69
2.2.2.1 Diagram 2-10(b) .................. .. 69
2.2.2.2 Diagram Fig. 2-11(b) ............... .. 71
2.2.3 Sub-Leading Diagrams .................. .. 72
2.2.4 Correction to the Conductivity ................ 73
2.2.5 Effective Impurity Potential ................ 73









2.3 Impurity Scattering Cross-Section for Interacting Electrons . 75
2.3.1 Non-Interacting Case ..... ........... ...... 76
2.3.2 Interacting Case ................ ... ... .. 78
2.4 Experiments ............... ......... .. 84
2.5 Conclusions ............... .......... .. 93

3 SINGULAR CORRECTIONS TO THERMODYNAMICS FOR A ONE
DIMENSIONAL INTERACTING SYSTEM: EVOLUTION OF THE
NONANALYTIC CORRECTIONS TO THE FERMI LIQUID BEHAVIOR 95

3.1 One-Dimensional Model ............... .. .. .. 99
3.2 Specific Heat ..... . . ............. 105
3.2.1 Specific Heat from the Second Order Self Energy ....... 107
3.2.2 Specific Heat from the Thermodynamic Potential at Second
Order . . . . .... 112
3.2.3 Specific Heat from Third Order Self Energy . ... 116
3.2.4 Specific Heat from the Sine-Gordon Model . .... 127
3.3 Spin Susceptibility .................. ........ 130
3.4 Experiments .................. ............ 136
3.5 Conclusion. .................. ........... 137

4 SPIN SUSCEPTIBILITY NEAR A FERROMAGNETIC QUANTUM
CRITICAL POINT IN ITINERANT TWO AND THREE DIMENSIONAL
SYSTEMS ................... .............. 138

4.1 Spin Susceptibility X,(H), in 2D .............. .. 141
4.2 Spin Susceptibility Xs(H), in 3D .............. .. 145
4.3 Spin Susceptibility for a Fermi Liquid in 2D . . 147
4.4 Spin Susceptibility near the Quantum Critical Point ......... 156
4.4.1 2D ........... ... ....... ........ 158
4.4.2 3D ........... ... ....... ........ 162
4.5 Conclusions . . . . . . . 165

5 CONCLUSIONS .................. .......... 166

REFERENCES .............. ........ . .. 168

BIOGRAPHICAL SKETCH ............. . . .... 174















LIST OF FIGURES
Figure page

1-1 Weak localization corrections .............. ... 5

1-2 Ladder diagram for M (diffuson) and C (Cooperon). .. . 7

1-3 Quantum corrections to conductivity for noninteracting electrons . 7

1-4 Scattering by Friedel oscillations. ................ ..... 12

1-5 Self-energy at first order in interaction with a bosonic field . ... 23

1-6 Kinematics of scattering. (a) "Any-angle" scattering leading to regular
FL terms in self-energy; (b) Dynamical forward scattering; (c) Dynamical
backscattering. Processes (b) and (c) are responsible for nonanalytic terms
in the self-energy .................. ............ .. 25

1-7 Non trivial second order diagrams for the self-energy .......... ..26

1-8 Scattering processes responsible for divergent and/or nonanalytic corrections
to the self-energy in 2D. (a) "Forward scattering -an analog of the g4
process in 1D (b) "Forward scattering with anti-parallel momenta-an
analog of the g2 process in 1D (c) "backscattering; with antiparallel momenta-
an analog of the gl process in 1D .............. ...... 29

1-9 Typical trajectories of two interacting fermions . ..... 31

2-1 Diagram (a) is the leading contribution to the self energy at fourth order 48

2-2 Dyson's series ............... ............. .. 49

2-3 Drude conductivity ............... ........... .. 49

2-4 Third and second order fan diagram. ................ .... 50

2-5 Cooperon sequence for 3D electrons in the UQL. Unlike in 1D, each term
in the series comes with a different coefficient c.. . . 54

2-6 First and second order diffuson ............... .... 55

2-7 Interference correction to conductivity .............. .. .. 56









2-8 Crossed diffuson diagrams. Left, a double-diffuson diagram, which also
acquires a mass. Right, a third-order non-cooperon diagram which, up
to a number, gives the same contribution as the third order fan diagram. 58

2-9 First order interaction corrections to the conductivity where effects of
impurities appear only in the disorder-averaged Green's functions. 61

2-10 Exchange diagrams that are first order in the interaction and with a single
extra impurity line. The Green's functions are disorder-averaged. Diagrams
(a) and (b) give In T correction to the conductivity and exchange diagrams
(c), (d) and (e) give sub-leading corrections to the conductivity. . 62

2-11 Hartree diagrams that are first order in the interaction and with a single
extra impurity line. The Green's functions are disorder-averaged. Both
diagrams give In T correction to the conductivity. . ..... 63

2-12 The self-energy correction contained in diagram 2-10(a), denoted in the
y(2--12)
text as 12) . . . . .. .. . . 64

2-13 The self-energy correction contained in diagram 2-11(a), denoted in the
text as 213) ................... ............ ..67

2-14 Diagram 2-10(b) vs diagram 2-11(b). .................. 71

2-15 Effective impurity potential .................. ..... .. 74

2-16 The handle diagram corresponds to diagrams 2-10(a) and 2-11(a) and
the crossing diagram corresponds to 2-10(b) and 2-11(b). . ... 75

2-17 Profile of the Friedel oscillations around a point impurity in a 3D metal
in the UQL. The oscillations decay as 1/z along the magnetic field direction
and have a Gaussian envelope in the transverse direction. . ... 79

2-18 Renormalized conductivities parallel (az) and perpendicular (a,,) to
the direction of the applied magnetic field. Power-law behavior is expected
in the temperature region 1/7 < T < W. ....... . ... 82

2-19 Temperature dependence of the ab-plane resistivity pxx for a graphite
crystal at the c-axis magnetic fields indicated in the legend . ... 86

2-20 Temperature dependence of the c-axis conductivity az for a graphite
crystal in a magnetic field parallel to the c axis. The magnetic field values
are indicated on the plot, with the field increasing downwards, the lowest
plot corresponds to the highest field .................. .. 87

2-21 Temperature dependence (log-log scale) of the ab-plane resistivity scaled
with the field p,,/B2 for a graphite crystal at the c-axis magnetic fields
indicated in the legend ............ . .. 88









2-22 Temperature dependence (on a log-log scale) of the ab-plane resistivity
px/B2 at the highest attained c-axis magnetic field of 17.5T for the same
graphite crystal .................. ............. .. 89

2-23 Phase breaking rate vs T due to electron-phonon scattering ...... ..92

3-1 Interaction vertices ............. . . ... 101

3-2 Non-trivial second order self energy diagrams for right moving fermions 107

3-3 Second order diagrams for the thermodynamic potential with maximum
number of explicit particle-hole bubbles .................. .. 112

3-4 The different choices for the 3rd order diagram. ............. .117

3-5 All 3rd order se diagrams for right movers which have two II2kp ... 122

3-6 All third order self energy diagrams containing two Cooper bubbles 123

3-7 Effective third order self-energy diagrams (the double line is a vertex). 124

3-8 All g2 and gi vertices at 2nd order. ............. .. 125

3-9 All third order self-energy diagrams with two Cooper bubbles or two II2ka
bubbles .................. ................. .. 126

3-10 Second order diagrams for the thermodynamic potential. . ... 132

4-1 Particle-hole type second order diagram for the thermodynamic potential. 143

4-2 Particle-hole type third order diagram for the thermodynamic potential. 144

4-3 The skeleton diagram for the thermodynamic potential. . ... 148

4-4 Fermion self-energy (a) and Bosonic self-energy (b) . . ... 158















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MANIFESTATIONS OF ONE-DIMENSIONAL ELECTRONIC CORRELATIONS
IN HIGHER-DIMENSIONAL SYSTF:\ IS

By

Ronojoy Saha

August 2006

C('! ,i: Dmitrii L. Maslov
Major Department: Physics

In this work we have studied the fundamental aspects of transport and

thermodynamic properties of a one-dimensional (ID) electron system, and

have shown that these 1D correlations pl iv an important role in understanding

the physics of higher-dimensional systems. The first system we studied is a

three-dimensional (3D) metal subjected to a strong magnetic field that confines the

electrons to the lowest Landau level. We investigated the effect of dilute impurities

in the transport properties of this system. We showed that the nature of electron

transport is one dimensional due to the reduced effective dimensionality induced

by the magnetic field. The localization behavior in this system was shown to be

intermediate, between a 1D and a 3D system. The interaction corrections to the

conductivity exhibit power law 1, 1iwi;- a oc T" with a field dependent exponent.

Next we studied the thermodynamic properties of a one-dimensional

interacting system, where we showed that the next-to-leading terms in the specific

heat and spin susceptibility are nonanalytic, in the same way as they are for

higher-dimensional (D = 2, 3) systems. We obtained the nonanalytic, TlnT term

in the specific heat in 1D and showed that although the nonanalytic corrections









in all dimensions arise from the same source, there are subtle differences in the

magnitude of the effect in different dimensions.

In the final part of this work we analyzed the nonanalytic corrections to

the spin susceptibility (Xs(H)) in higher dimensional systems. We showed that,

although there were contributions from non-1D scattering in these nonanalytic

terms, the dominant contribution came from 1D scattering. We also showed that

the second order ferromagnetic quantum phase transition is unstable both in 2D

and 3D, with a tendency towards a first order transition.















CHAPTER 1
INTRODUCTION

One-dimensional interacting systems (Luttinger-liquids) exhibit many features

which appear distinct from their higher-dimensional counterparts (Fermi-liquids).

Our goal in this thesis is to highlight the similarities between higher-D and 1D

systems. The progress in understanding of ID systems has been greatly facilitated

by the availability of exact or .,-vmptotically exact methods (Bethe Ansatz,

bosonization, conformal field theory), which typically do not work very well above

1D. The downside of this progress is that 1D effects, being studied by specifically

1D methods, look somewhat special and not really related to higher dimensions.

We are going to argue that this is not true. 1t ,ii: effects which are viewed as

the hallmarks of 1D physics, e.g., the suppression of tunneling conductance

by the electron-electron interaction, do have higher dimensional counterparts

and stem from essentially the same physics. In particular, scattering at Friedel

oscillations caused by tunneling barriers and impurities is responsible for zero-bias

tunneling anomalies in all dimensions. The difference lies in the magnitude of the

effect, not in its qualitative nature. We illustrate this similarity by showing that

1D correlations p1 iv an important role in understanding the physics of higher

dimensional systems. We studied three seemingly different problems, but as we will

show, all three of them are connected by the common feature of 1D correlations.

Our goal in the introduction is to provide a background for the physics discussed

in the three chapters of this dissertation. We have set h and kB equal to unity

everywhere.









1.1 Transport in Ultra Strong Magnetic Fields

The behavior of an isotropic three-dimensional (3D) metal in a high magnetic

field has been a focus of attention of the condensed matter community for many

decades. Due to Landau quantization of orbits, the energy of an electron in this

system depends only on the momentum along the magnetic field,


E= Pz 2 c
j+(2m 1) (11)

where wc = eH/mc is the cyclotron frequency and n is the Landau level. Thus

the system exhibits effects characteristic of one-dimensional (1D) metals, while

being intrinsically a 3D system. This reduction of effective dimensionality of

charge carriers from 3D to 1D is most pronounced in the ultra-quantum limit

(n = 0, when only the lowest Landau level remains populated) and is expected

to result in a number of unusual phases. It is well known that the ground state

of repulsively interacting electrons in the UQL is unstable with respect to the

formation of a charge density wave [1-3], which has been observed experimentally

in magneto-resistance measurements on graphite in high magnetic fields [4].

The most complete a i i ,-i; of the CDW instability for the case of short range

interactions was performed in Ref. [3], by solving the renormalization-group (RG)

equations for the interaction vertex. On the other hand, it has recently been

shown that for the case of long-range (Coulomb) interactions between electrons, a

3D metal in UQL exhibits Luttinger-liquid like (1D) behavior at energies higher

than the CDW gap [5, 6]. Biagini et al. [5] and Tsai et al. [6] showed that in the

UQL, the tunneling conductance has a power law anomaly (nonlinearities in I-V

characteristics at small biases), which is typical for a one dimensional interacting

system (Luttinger liquid). The magnetic-field-induced Luttinger liquid phase

can be anticipated from the following simplified picture. In a strong magnetic

field, electron trajectories are helices spiraling around the field lines. A bundle of









such trajectories with a common center of orbit can be viewed as a 1D conductor

( V.--i '). In the presence of electron-electron interactions, each vi..-.," considered

separately, is in the LL state. Interactions with small momentum transfers among

electrons on different i.--i -' do not change the LL nature of a single wire [7]. In

chapter 2 of this dissertation we study the transport properties of a disordered

3D metal in the UQL, both with and without electron-electron interactions. Both

the localization and interaction corrections to the conductivity show signatures

typical for one-dimensional systems. Before we get into the details of our study, we

will briefly review the physics of the interplay between the interaction effects and

disorder induced localization in diffusive systems of low dimensionality.

At low temperatures, the conductivity of disordered conductors (normal metals

and semiconductors) is determined by scattering of electrons off quenched disorder

(e.g., impurities and defects). The residual conductivity is given by the Drude
formula,

o- = (12)


where n is the electron concentration, e is the electron charge, 'r is the transport

mean free time, and m is the effective mass. The Drude formula neglects

interference between electron waves scattered by different impurities, which

occur as corrections to Eq.1-2, in the parameter (kFI)- 1 < 1 (where kF is the

Fermi momentum and f is mean free path). In low dimensions (d < 2), these

(interference) quantum corrections to the conductivity (QCC) diverge when the

temperature T decreases and eventually, drive the system to the insulating regime.

The quantum corrections to the conductivity are of substantial importance even

for conductors that are far from the strong localization regime: in a wide range

of parameters QCC, though smaller than the conductivity, determine all the

temperature and field dependence of the conductivity. The systematic study of









QCC started almost three decades ago. A comprehensive review of the status of

the problem from both theoretical and experimental viewpoints can be found in

several papers [8-11].

According to their physical origin, QCC can be divided into two distinct

groups. The correction of the first type, known as the weak localization (WL)

correction, is caused by the quantum interference effect on the diffusive motion of

a single electron. For low-dimensional (d = 1,2) infinite systems the WL QCC

diverge at T 0; this divergence is regularized either by a magnetic field or by

some other dephasing (inelastic scattering) mechanism. We will elaborate on this

type of QCC in section 1.1.1 below and also see how it changes for a 3D metal in

UQL in chapter 2.

The second type of QCC, usually referred to as the interaction effects, is

absent in the one-particle approximation; they are entirely due to interaction

between electrons. These corrections can be interpreted as the elastic scattering

of an electron off the inhomogeneous distribution of the density of the rest

of the electrons. One can attribute this inhomogeneous distribution to the

Friedel oscillations produced by each impurity. The role of the electron-electron

interactions in this type of QCC is to produce a static self-consistent (and

temperature dependent) potential which renormalizes the single particle density of

states and the conductivity. Such a potential does not lead to any real transitions

between single-electron quantum states (those require real inelastic scattering).

Therefore, it does not break the time reversal invariance of the system and neither

does it affect nor regularize the WL corrections. We will elaborate on this type of

QCC in section 1.1.2, and also study it for our case of 3D metal in UQL in chapter

2.

However the interaction between electrons is by no means irrelevant to the

WL QCC. Indeed, these interactions cause phase relaxation of the single electron






5


states, and thus result in the cut-off of the divergences in the WL corrections.

This dephasing (described by by the phase breaking time 7r(T)) requires real

inelastic collisions between the electrons and can be obtained experimentally from

the temperature dependence of magneto-resistance measurements. We will discuss

the phase breaking time due to electron-electron interactions in section 1.1.3, of

this introduction. Therefore there are two classes of interaction contribution to

the conductivity: the genuine interaction corrections (elastic scattering of Friedel

oscillation: Altshuler-Aronov corrections)-Class I, and corrections to WL QCC due

to interactions (inelastic scattering-dephasing)-Class II.

1.1.1 Weak Localization QCC

These corrections are caused by the quantum interference effect in the diffusive

motion of a single electron. In going from point A to point B a particle can travel














Figure 1-1. Weak localization corrections.


along different trajectories (Fig.1 1). The total probability W for a transfer from

point A to point B is


w I= A 1 A |-2 + Y AA. (1-3)
i i iyj

The first term in Eq.1-3 describes the sum of the probabilities for each path and

the second term corresponds to interference of various amplitudes. The interference









term drops out when averaging over many paths because of its oscillatory nature.

However, there exists special type of trajectories, i.e., the self- intersecting ones,

for which interference cannot be neglected (see Fig.1 1). If A1 is the amplitude for

the clockwise motion around the loop and A2 is the amplitude for the anticlockwise

motion, then the probability to reach point O is


W = |Ai 2 + A2 + 2+ReATA2 41A12. (1 4)


i.e., twice the value we would have obtained by neglecting interference. Enhanced

probability to find the particle at a point of origin means reduced probability

to find it at final point (B). Therefore this effect leads to a decrease in the

conductivity (increase in resistivity) induced by interference.

The relative magnitude of weak localization QCC, 6a/a, is proportional to the

probability to form a loop trajectory

6 dP dt V (1-5)
a (Dt)d/2

leading to a ~ -T 4(2-d)/2 (ln r for d=2), which diverges as T is lowered

for d < 2, leading to strong Anderson localization. Here v is the electron

velocity, D is the diffusion coefficient, A is the electron wavelength and 7r(T) is

the phase breaking time. Phase coherence is destroyed by inelastic scattering

(electron-electron, electron-phonon) or by magnetic and a.c electric fields. The

temperature dependence of the WL correction is determined by 'rT (T). Typically,

rT ~ T-P, where the exponent p depends on the inelastic scattering mechanism

(electron-electron, electron-phonon) and dimensionality. Interference effects occur

for 7- < r- (T) i.e., at low temperatures.

In the language of Feynman diagrams, the WL QCC [12] is obtained by

including the maximally crossed ladder diagram, the Cooperon (see Fig.1-2), in

the conductivity diagram. The other type of ladder (vertex ) diagram, the Diffuson










(Fig. -2), when included in the conductivity diagram changes the elastic scattering

time to the transport time. In the field-theoretic language, the Weak Localization

p+q;E+w


+
i i


C(Q;w)


Il



'K


+ Y


+


+ 'I


+ ...


+ ...


Figure 1-2. Ladder diagram for M (diffuson) and C (Cooperon).


correction, which arises due to interference of time reversed paths is determined by

the "Cooperon" mode C (Q; w), i.e., the particle-particle diffusion propagator,


c(Q;w)


27rvT (


1rDQ2
iwr + DQ27r'


(1-6)


to first order in (kft)-1. Calculation of the singular contributions to conductivity

(interference effect) at small w, Q should include diagrams containing as an internal

block the graphs which yield after summation C(Q;wG) (1-3). The WL QCC is

P q-P

< : = < + < : :> +

P q-p






Figure 1-3. Quantum corrections to conductivity for noninteracting electrons.


obtained from


De2 /
D2 J j (dQ) C (Q; w),
6~WL (127


(1-7)


So-WL









which gives

6--WL (1D) (1 8)
a \- aPT
SIn( 1 n(( 1 ) (2D), (1-9)
kpf r hpf UJT
1 1T

~ ( )2 +( )2 (3D) (1 10)

in different dimensions. Perturbation theory breaks down in one- and two-dimensions

(for w < 1, in the diffusive limit or at low temperatures, T7 ~ T-P) which -,-..-. -1i

strong localization in reduced dimensions. Anderson [13] had first shown that

at sufficiently high impurity concentration, electronic states become localized

and the system becomes an insulator. Mott and Twose [14] had predicted that

the conductivity for a one dimensional system should vanish in the limit of low

frequencies (Mott's law) which was later rigourously proved by Berezinskii [15],

who showed that electron states in 1D are strongly localized and there is no

diffusive regime in 1D. The localization length in 1D is of the order of the mean

free path (f), therefore in ID for length scales shorter than the electron motion is

ballistic and for lengths longer than i then electron motion is localized. 2D systems

are also strongly localized but the localization length is very large (Lloc ~ fekf) as

compared to 1D (Loc r~ ). Thus in 2D, the ballistic regime (L < f) crosses over to

the diffusive regime ( for i < L < Loc) and then to the localized regime (L,,o < L).

Scaling theory of localization proposed by Abrahams et al. [16] describes

Localization in higher dimensions. This theory is based on the assumption that the

only parameter that determines the behavior of the system under renormalization

is the dimensionless conductance, g (in units of e2/h). The variation of g with the

system size obeys the Gell-Mann Low equation

ding
dlnL 3g).









In the metallic regime g > 1, the conductance shows ohmic behavior for which

(g) = d 2. Corrections to 3 (g) in the metallic regime are obtained by
perturbation theory in -. For g < 1 (insulating regime), f (g) is obtained from

the simple argument that g must decrease exponentially with the system size in

this regime. In 1D and 2D, g decreases with increasing system size, which means

that the electron states are ah--iv- localized. In 3D there is a continuous phase

transition between metallic and insulating phases. This transition happens when

kff 1 (Anderson transition).

Effect of magnetic field on WL QCC. If the system is placed in a

magnetic field H, the amplitude for a particle to pass the loop clockwise and

anticlockwise (Fig.1 1) acquire additional phase factors,

(i I itHS
A1 -+ Alexp i dlA Ale o

A2 A2zexp e dlA) A2e 0o



where Oo = hc/2e is the flux quantum. The phase difference between waves passing

the loop clockwise and anticlockwise is 6 = 27r/0o, = HS is the flux through

the loop with cross-section S. Thus the magnetic field destroys interference,

reducing the probability for a particle to return to a given point, and hence reduces

the resistivity. This mechanism is responsible for negative magneto-resistance [17].

The characteristic time scale for phase breaking is tH lH2/D where IH c/eH

is the magnetic length. The typical magnetic field involved is H ~ c/eD7r. At this

field, the product wu,- satisfies wu-T ~ (EFT)-1 < 1 where EF is the Fermi energy

and w~ is the cyclotron frequency. Thus at the phase breaking field the classical

magneto-resistance, determined by the value of w,- is still small.

A weak magnetic field destroys phase coherence and increases the conductivity.

If the field is increased further, we reach the classical magneto-resistance regime,









where the conductivity decreases with the field. What happens at even higher fields

when Landau quantization becomes important? We address this issue in chapter 2

of this dissertation. We show that a three-dimensional disordered conductor in the

Ultra quantum limit, where only the lowest landau level is populated, exhibits a

new phenomenon: intermediate localization. The quantum interference correction

6a is of the order of the Drude conductivity UD (as in 1D) which indicates a

breakdown of perturbation theory. However, the conductivity remains finite at T

- 0 (as in 3D). It is demonstrated that the particle-particle correlator (Cooperon)

is massive. Its mass (in units of the scattering rate) is of the order of the impurity

scattering rate.

1.1.2 Interaction Correction to the Conductivity-Altshuler Aronov
Corrections (class I)

The effect of electron-electron interaction in disordered systems makes

it drastically different from that of pure metals, where the interaction at low

temperatures manifests itself only in the renormalization of the electron spectral

parameters [18] (the wave function renormalization Z, effective mass m*, etc.).

First we note that within the transport equation, electron-electron collisions can

in no way affect the conductivity in the case of a simple band structure and in

the absence of Umklapp processes, since electron-electron collisions conserve the

total momentum of the electron system. Inclusion of the Fermi liquid corrections

renormalizes the residual resistivity while not resulting in any dependence of

the conductivity on the temperature and frequency. However one frequently

encounters the situation that the resistivity scales as T2. This dependence is often

interpreted as the ." i i"-liquid" effect, arising from electron-electron scattering

with characteristic time T', oc T-2. In fact, the resistivity is due to Umklapp

scattering. In good metals, normal processes (which conserve the total electron

moment) and Umklapp processes (which conserve the moment up to a reciprocal









lattice vector) are equally probable and the Umklapp scattering rate entering

the resistivity also scales as T2. Note that at low temperatures this resistivity

due to electron-electron scattering (Umklapp) gives the dominant contribution

because the electron-phonon contribution to the resistivity scales as T5 (Bloch's

law, Te-ph 1/T5).

As was mentioned previously, taking into account the interference of elastic

scattering by impurities with the electron-electron interaction produces non trivial

temperature and frequency dependence of the conductivity. This correction

arises from coherent scattering of an electron from an impurity and the Friedel

oscillation it creates [19]. We will first study this correction to the conductivity in

the ballistic limit, (TT- > 1, where 7r is the elastic scattering lifetime) and then in

the diffusive regime (TT
collisions with impurities before it scatters from another electron, whereas in the

ballistic limit the electron-electron collision rate is faster than electron-impurity

rate, thus single impurity effects are important in the ballistic limit. In chapter

2 of this dissertation we will evaluate this interaction QCC in the ballistic limit

in a 3D metal in the UQL. There has been a recent renewal of interest in the

interaction QCC, (class I) due to the metal to insulator transition observed in

two-dimensional (high mobility) Si-MOSFET samples [20]. The qualitative features

of this transition was understood by Zala, Narozhny and Aleiner [19] who showed

that the insulating (logarithmic upturn in the resistivity) behavior in the diffusive

regime and metallic (linear rise in temperature) behavior of the resistivity in the

ballistic limit (2D), are due to coherent scattering at Friedel oscillations. Below,

we first outline their simple quantum mechanical scattering theory approach to

show how temperature dependent corrections to conductivity arise for scattering at

Friedel oscillations, and then extend their analysis to obtain the interaction QCC in









3D ballistic limit. In chapter 2 we evaluate this correction for a 3D system in the

UQL.





__^__--------^ --- ---





A


Figure 1-4. Scattering by Friedel oscillations.


Scattering at Friedel Oscillations Friedel oscillations in the electron

density are created due to standing waves formed as a result of interference

between incoming and backscattered electron waves (Fig.1-4). Consider an

impurity at the origin; its potential Uimp(r induces a modulation of electron

density around the impurity. In the Born approximation one can find the

oscillating correction, 6n(r) = n(r) no to the electron density n(r = Yk "' 2 ()12:

n(r) v sin(2kFr)
n(r) ~ -gv (1-11)

Here r is the distance from the impurity, kF is the Fermi momentum, g =

f Uimp (Idrj is the matrix element for impurity scattering and no is the electron

density in the absence of impurities and d is dimensionality. Taking into account

electron-electron interactions Vo(r5 r7) one finds additional scattering potential

due to the Friedel oscillations Eq.1-11. This potential can be presented as a sum of

the direct (Hartree) and exchange (Fock) terms [21]


6V(7, r2) VH(t1)6( r-) V(rl r),


(1-12)










VH(i) dWVo( )6p(7), (1-13)

1 _
VF(, 2) V Vo(f -2)Jn(T, 2), (1-14)
2

where by p(r) we denote diagonal elements of the one electron density matrix,


n(Tr, r-) W*k^ 1r) k(T2). (1-15)
k

As a function of the distance from the impurity, the Hartree-Fock energy 6V

oscillates similarly to Eq.1-11. The leading correction to conductivity is a result

of interference between two semi-classical paths shown in fig. 4. If an electron

follows path "A," it scatters off the Friedel oscillation created by the impurity

and path "B" corresponds to scattering by the impurity itself. Interference is

most important for scattering angles close to 7 (or for backscattering), since the

extra phase factor accumulated by the electron on path "A" (ei2kR) relative to

path "B" is canceled by the phase of the Friedel oscillation e-i2kFR, so that the

amplitude corresponding to the two paths are coherent. As a result, the probability

of backscattering is greater than the classical expectation (taken into account

the Drude conductivity). Therefore, accounting for interference effects lead to

a correction to the conductivity. We note that the interference persists to large

distances, limited by temperature R ~ Ik kF|-1 < vF/T. Thus there is a

possibility for the correction to have nontrivial temperature dependence. The

sign of the correction depends on the sign of the effective coupling constant that

describes electron-electron interaction. First, we will study the contribution arising

from the Hartree potential. Consider a scattering problem in the potential given in

Eq.1-13. The particle's wave function is a sum of the incoming plane wave and the

outgoing spherical wave (3D),

ikr
'I' e"' + f(O)
r









where f(0) is the scattering amplitude, which we will determine in the Born

approximation. For the impurity potential itself the amplitude f(0) weakly depends

on the angle. At zero temperature it determines the Drude conductivity oD,

while the leading temperature correction is T2 (when the scattering time energy

dependent), as is usual for Fermi systems. We now show that this is not the

case for the potential in Eq.1-12. In fact, taking into account Eq.1-12 leads to

enhanced backscattering and thus to the conductivity correction which depends

on temperature as 6a oc T2 lnT (in 3D), 6a ~ T (in 2D) and, as we will see later

6 T2" (in 3D UQL, a is the interaction parameter) all for the ballistic limit.

Far from the scatterer the wave function of a particle can be found in the first

order of perturbation theory as = eikd. + 6q(rl, where the correction is given by

[22]

(r =1 /drVH 1i) r (116)

Substituting the form of the Hartree potential from Eq: -13, and introducing

the Fourier transform of the electron-electron interaction Vo(q), we obtain for the

scattering amplitude (at large distances from the impurity)


f(0) = Vo() di (r)e. (1-17)
27 j

where q = kF/r and | 2k sin(0/2). We see that the scattering amplitude

depends on the scattering angle (0), as well as the electron's energy (e = k2/2m).

The density oscillation in 3D, with hard wall boundary condition at the origin

(impenetrable impurity), is

6n(r) 1d(k)[Tk 2 -_ 0k 2]

-2kF sin(2kF(r a)) sin(2kFr)
2 2k(r a) 2kpr '









where a is the size of the impurity and f(k) is the Fermi distribution function. We

make the s wave scattering approximation (slow particles, kFa < 1) to obtain

(2kp)2a cos(2kpr) sin (2kr) ( 8)
r2 2kFr (2kFr)2 "

Substituting the density from Eq.1-18 in Eq.1-17, we obtain for the scattering

amplitude

-2mVo(2kF sin())2kaF 1 01 sin( ) 2 0
f(0) = sin(-) + In I |Cos2 19()
sin() L 2 4 1 + sin() 2

In the limit 0 wr + x where x < 1, the scattering amplitude behaves as

f (x) Vo(2kF)[ Iln x]. The transport scattering cross section is now

At, dO sin(0) dQ(1 -cos(0)) fo + f(0) 2, (1-20)
0 Jo

where fo is the amplitude for scattering at the impurity itself (which does not

depend on 0 in the Born limit and gives a constant (T independent) value for the

Drude conductivity). The leading energy dependence comes from the interference

(cross term), which is proportional to f(0). The main contribution to the integral
comes from 0 7 backscatteringg). Expanding near 7, i.e., 0 = 7 + 01 where 01 is

small [19], 01 ~ \k k/k ~ c/Ep, we obtain for the scattering cross section

and transport rate ((rt,)- oc niVpAt, ~ 6p)

oc vVo(2k)(e)21 n(e). (1-21)
7tr, W

Then one obtains the interaction QCC from the Hartree channel [23] in 3D (using

6a/aD = -Sp/pD)
-vVo(2kF) )n( ). (1-22)
JD EF T
One obtains a similar contribution from the exchange (Fock) potential, except

now the coupling constant in front of the T2 In T term is Vo(0). The Hartree and

exchange contribution come with opposite signs. In 2D the interaction QCC is









linear in temperature [19]


-- v-[2Vo(2kF) Vo(0) (1-23)
JD EF

In 1D Yue, Glazman and Matveev [21] used the same approach and calculated the

correction to the transmission coefficient due to scattering at the Friedel oscillation

and obtained a logarithmic temperature correction at the lowest order


t -to In T |, (1-24)


where a [Vo 2V2kFvpF. Using a poor man renormalization group procedure,

they showed that the first order logarithmic correction is in fact a weak coupling

expansion of the more general power law scaling form of the transmission

coefficient,


t to )


where W is the band width. The transmission coefficient is related to the

conductance using the Landauer formula G ~ |t|2, which gives in 1D


G G ) (1-25)


This result was also obtained independently (via bosnization) by Kane and Fisher

[24]. Eq.1-22, 1-23, and Eq.1-25 give the interaction QCC in the ballistic limit in

3D, 2D and ID systems respectively. In chapter 2 of this dissertation we show that

in 3D UQL, this interaction correction to the conductivity behaves similar to that

of a true 1D system.

The interaction correction to QCC in the diffusive limit also arises from the

same physics (namely scattering at friedel oscillations) but now one has average

over many impurities diffusivee motion). This correction to the conductivity was

evaluated by Altshuler and Aronov in 3D [8] and by Altshuler, Aronov and Lee in









2D [25].

6 (2 2F) ln(TT), (2D) (1-26)

where F is the depends on the strength of the interaction, and

4 3F T
S-( ( T(3D). (1-27)
3 2 D

Scattering at the Friedel oscillations also results in a singular energy (temperature)

dependence of the local density of states which can be observed as a zero bias

anomaly in tunneling. The local DOS can be obtained from the electrons Green's

function using 6v(c) -Im f dfiGR((p), e). The correction to the Green's

function can be evaluated the same way as we evaluated the correction to the wave

function due to Friedel oscillation or it can also be evaluated diagrammatically by

calculating the electron's self energy in the presence of disorder and interaction [8].

1.1.3 Corrections to WL QCC due to Electron-Electron Interactions:
Dephasing (class II)

The basic feature underlying the quasi-particle description of electrons in

metals and semiconductors is the small width of the one electron states. The

minimum width of a wave packet and, hence, the minimum decay of a state are

determined by the wave function phase relaxation time 7r. For strongly inelastic

processes this time coincides with the out-relaxation time. In degenerate Fermi

systems where the energy transferred in each collision is of the order c, i.e., of

the order of the excitation energy measured from the Fermi level, the inverse

excitation decay time is of the order of c2/Ep and thus, is smaller than the

excitation energy, which is c. These considerations does not depend on the specific

details of the electron interaction and originate from the fact that scattering of

quasiparticles by one another is governed by large momentum transfers. Therefore

the decay is determined only by the phase volume of final states. It was believed

by analogy with the Fermi liquid, that in the case of weak disorder, kgt > 1, the









excitation decay should likewise be proportional to e2. It turns out, however, that

excitation in disordered systems decays faster, which raises the question of validity

of quasiparticle description of disordered conductors in low dimensional systems.

Apart from being important in the development of the theory, the decay time

for one electron excitations (the phase relaxation time), governs the temperature

dependence of the WL QCC.

It was shown by Altshuler and Aronov that for 3D disordered systems, the

phase relaxation time Tr is governed by large energy transfer processes and in

this regime T'r Tee (where Tee is the out relaxation time). The out relaxation

time can be calculated from the Bolztmann equation (with diffusive dynamics for

the electrons). This gives- ~ (c)3/2 in 3D, [8]. However in lower dimensions

(d = 1, 2) electron-electron collisions with small energy transfers is the dominant

mechanism for dephasing. The Bolzmann approach (which is good for large energy

transfers) fails in 2D and 1D case. Technically, there would be divergences for

small energy transfers [8] both in 2D (logarithmic) and quasi-1D (power law)

in the Bolztmann-equation result for the out relaxation rate. These divergences

must be regularized in a self-consistent manner. The phase breaking time in

lower dimensions can also be obtained by solving the equation of motion for

the particle-particle (Cooperon) propagator in the presence of space and time

dependent fluctuating electromagnetic fields which model the small energy transfer

processes [26]. This gives (rT)-'1 ~ T (in 2D) and (rT)-'1 ~ T2/3 (in quasi-ID).

In true one-dimensional systems, this subject is controversial as true 1D

systems do not have a diffusive regime (the ballistic limit crosses over to the

localized regime) and the quasiparticle description breaks down for an interacting

1D system which is in the Luttinger liquid state. As a result one cannot define ,ee.

In a recent work on this subject [27], it was shown that even for a 1D disordered

Luttinger liquid, there exists a weak localization correction to the conductivity









whose temperature dependence is governed by the phase relaxation rate, (,r)-1 oc

VT (for spinless electrons in 1D) and (r0)-1 oc T (for electrons with spin) and the

WL QCC behaves as,

( )2
WL ND ( nj), (1t28)


where aD 2= e2vvF2T is the Drude conductivity in 1D, which depends on T through

a renormalization of static disorder, 7-0/ = (Ep/T)2". Here To is non-interacting

scattering time and 7 is the renormalized (by Friedel oscillation) scattering time

and a characterizes the interaction.

At present there are no theoretical predictions for Tr- in 3D UQL. The Fermi

liquid approaches for calculating the phase breaking time are not expected to

work here because the Cooperon is not a singular diagram (it acquires a mass in

3D UQL as shown in chapter 2) and, once again, there are no single particle like

excitations as the ground state is a charge-density-wave and excitations above the

ground state are Luttinger liquid like. However in chapter 2 we will show that some

recent magneto-resistance measurements on graphite in UQL qualitatively agree

with predictions of 7T- due to electron-phonon interactions in 1D.

1.2 Non-Fermi Liquid Features of Fermi Liquids: 1D Physics in Higher
Dimensions

The universal features of Fermi liquids and their physical consequences

continue to attract the attention of the condensed-matter community for almost

50 years after the Fermi-liquid theory was developed by Landau [28]. A search

for stability conditions of a Fermi liquid and deviations from a Fermi liquid

behavior, [29-32] particularly near quantum critical points, intensified in recent

years mostly due to the non-Fermi-liquid features of the normal state of high T,

superconductors[33] and heavy fermion mat(i i-[;4].

The similarity between the Fermi-liquid and a Fermi gas holds only for the

leading terms in the expansion of the thermodynamic quantities (specific heat









C(T), spin susceptibility Xs) in the energy (temperature) or spatial (momentum)

scales. Next-to-leading terms are singular (nonanalytic) and, upon a deeper look,

reveal a rich physics of essentially ID scattering processes, embedded into higher

dimensional phase space.

In this introduction, we will discuss the difference between the i 5,il ,

processes which lead to the leading Fermi-liquid forms of thermodynamic quantities

and i ,i. ID processes which are responsible for the nonanalytic (non-Fermi

liquid) behavior. We will see that the role of these rare processes increases as the

dimensionality is reduced and, eventually, the rare processes become normal in ID,

where the Fermi-liquid description breaks down.

In a Fermi gas, thermodynamic quantities form regular, analytic series as a

function of either temperature T, or the inverse spatial scale q of an inhomogeneous

magnetic field. For T < EF and q < kF,


C(T)/T = +aT2+bT4 +..., (1-29)

X,(T,q 0) = s(0) + cT2 + dT +..., (30)

X(T 0,q) = o(0)+eq2 +f4+..., (1 31)

where 7 = 7r2 F/3, X,0 = gB2 F and vF ~ mkpD-2 is the density of states

(DOS) on the Fermi surface, g is the Lande factor and pB is the Bohr magneton

and a... f are some constants. Even powers of T occur because of the approximate

particle-hole symmetry of the Fermi function around the Fermi energy. The above

expressions are valid in all dimensions, except D = 2. This is because the DOS is

constant in 2D, the leading correction to the 7T term in C(T) is exponential in

Ep/T and X, does not depend on q for q < 2kg. However this anomaly is removed

as soon as we take into account a finite bandwidth of the electron spectrum, upon

which the universal (T2" and q2") behavior is restored.









An interacting Fermi system is described by Landau's Fermi-liquid theory,

according to which the leading terms in C(T) and X, are same as that of the Fermi

gas with renormalized parameters (replace bare mass by effective mass m*, bare g

factor by effective g-factor g* in the above Fermi gas results),


C(T)/T = 7* 7o( + (cos0F,)), (1-32)
1 +(cos OFe)
Xs(T,q) = Xs*(O) = s( + (cos (1-33)
1 + (F8,)

where Fe, F, are charge and spin harmonics of the Landau interaction function:

F(Jf,l) = F(O)I + F,(0)a.a', where 5, are the Pauli matrices. The Fermi-liquid

theory is an ..i-mptotically low-energy theory by construction, and it is really

suitable only for extracting the leading terms, corresponding to the first terms in

the Fermi gas expression. Indeed, the free energy of the Fermi-liquid of an ensemble

of quasiparticles interacting in a pairwise manner can be written as [35]


F Fo k + fk,kl'' 'nk' + (0(3k),
k k,k'

where F0 is the ground state energy, 6nk is the deviation of the fermion occupation

number from its ground state value, and fk,k' is the Landau interaction function.

As 6nk is of the order of T/EF, the free-energy is at most quadratic in T, and

so the corresponding C(T) is at most linear in T. Consequently the Fermi-liquid

(FL) theory (within the conventional formulation) does not -v- anything about the

higher order terms.

Strictly -I'" i1:ii a nonanalytic dependence of fk,k' on the deviations from

the Fermi surface k kF, accounts for the non-analytic T dependence of C(T)

[36]. Higher order terms in T or q can be obtained within microscopic models

which specify particular interaction and employ perturbation theory. Such an

approach is complimentary to the FL: the former works for weak interactions but

at arbitrary temperatures whereas FL works both for weak and strong interactions,









but only in the limit of lowest temperatures. Microscopic models (Fermi gas with

weak repulsion, electron-phonon interaction, paramagnon model, etc.) show that

the higher order terms in the specific heat and spin susceptibility are nonanalytic

functions of T and q [37-48]. For example,

C(T)/T = 7- a3T21n(Ep/T)(3D), (1-34)

C(T)/T = 72- 2T(2D), (1-35)

Xs(q) = Xs(O) + 3q2 1n(k/ql|)(3D), (1-36)

Xs(q) = Xs(0) + 2|q(2D), (1-37)

where all coefficients are positive for the case of electron-electron interaction.

As seen from the above equations the nonanalyticity becomes stronger as the

dimensionality is reduced. The strongest nonanalyticity occurs is 1D, where-at least

as long as single particle properties are concerned-the FL breaks down [49, 50]:

C(T)/T = i- ailn(EF/T)(ID), (1-38)

Xs(q) = Xs(0) 3iln(k F/ql)(1D). (1-39)

These nonanalytic corrections to the specific heat and spin susceptibility in 1D are

obtained in chapter 3. It turns out that the evolution of the non-analytic behavior

with the dimensionality reflects an increasing role of special, almost 1D scattering

processes in higher dimensions. Thus non-analyticities in higher dimensions can be

viewed as precursors of 1D physics for D > 1.

We will first study the necessary condition to obtain a FL description and then

see how relaxing these conditions lead to the nonanalytic form for the self-energy

and thermodynamic properties. Within the Fermi liquid

ReZE(R, k) -A + Bk + ... (1-40)

ImZE(R k) C(2 + 2T2) + ... (1 41)









Landau's argument for the E2 (or T2) behavior of ImER requires two conditions: (1)

quasiparticles must obey Fermi statistics, i.e., the temperature is smaller than the

degeneracy temperature TF = kFVF*, where vF* is the renormalized Fermi velocity,

(2) inter-particle scattering is dominated by processes with large (generally, of order

kF) momentum transfers. Once these two conditions were satisfied, the self-energy

assumes a universal form, Eq.1-40 and Eq.1-41, regardless of a specific type of

interaction (electron-electron, electron-phonon) and dimensionality. Consider the

self-energy of an electron (1st order) as it interacts with some boson (see Fig. 1-5

). The wavy line can be, e.g., a dynamic Coulomb interaction, phonon propagator,

etc. On the mass shell (E = k; where k = k2/2m- kF2/2m) at T = 0 and for E > 0


a) q,






k,e k-q, e-m k,e

Figure 1-5. Self-energy at first order in interaction with a bosonic field




ImER(E) w~ du dDqmGR(E u, k qImVR(w, q) (1-42)

The constraint on energy transfers (0 < u < E) is a direct manifestation of the

Pauli principle. The potential term V(r, t) is a propagator of some field which has

a classical limit, so V(r, t) is real, thus ImV(q, w) is an odd function of w and we

write it explicitly as


ImVR(w, q) = wF( w, q).


(1-43)









As a function of q, F has at least two characteristic scales. One is provided by
the internal structure of the interaction (screening wave vector for the Coulomb
potential) or by kF whichever is smaller. This scale, Q, does not depend on w and
provides the ultra-violet cutoff in the theory. In addition there is a second scale

I w/vF, and, since w is bounded from above by E and for low energies (E 0), one
can assume Q > IWu/vF. Thus in a dimensionless form

ImV"(w, q) =- U (( Q). (1

The angular integration over ImGR yields on the mass shell

S/'liG -T dOR(dO (- vF. + q2/2m) = 1 AD( 2/2), (1 45)
J J vFq vpq

where the subscript D stands for the dimensionality, and

A3(x) 0(1 Ix),
(I IXl)
A2(x) 0(1x
1 X2

The function AD primarily serves to impose a lower cutoff q > |w|/vF and we can
ignore the specific functional form. Using Eq.1 45 and Eq.1 44 into Eq.1 42, one
obtains

ImZR() j dW dq d U/2U( WQ (1-46)
JO q>II/VF Q VFQ

Now if the momentum integral is dominated by large moment of the order of
Q, then the function U to leading order can be considered to be independent of
frequency (since Q > IwI/vF), and one can set w = 0 in U, and also replace the
lower limit of the q integral by zero. The momentum and frequency integrals then
decouple, (the momentum integral gives a pre-factor and the frequency integral
gives E2), and one obtains an analytic E2 dependence for ImE. Then the linear
in E term in ReE can be obtained by using the Kramers-Kronig relation. Thus








we see that large momentum (and energy independent) transfers and decoupling
of the momentum and frequency integral are essential to obtain a FL behavior.
The E2 result seems to be quite general under the assumptions made. When and
why are these assumptions violated? Long-range interaction, associated with

Q-A

Im Occ W: ( I Q- co



Non-analytic part of Imi

Q-w /vF IQ-2kF, aI/vF


/ /4 Q-^ rQo




Figure 1-6. Kinematics of scattering. (a) "Any-angle" scattering leading to
regular FL terms in self-energy; (b) Dynamical forward ., i. li i-.
(c) Dynamical backscattering. Processes (b) and (c) are responsible for
nonanalytic terms in the self-energy

small-angle scattering, is known to destroy the FL. For example, transverse long
range (current-current [51] or gauge [52]) interactions which, unlike the Coulomb
interaction are not screened, lead to the breakdown of the Fermi-liquid. But these
interactions occur under special circumstances (e.g., near half-filling for gauge
interactions). For a more generic case, it turns out that even if the bare interaction
is of the most benign form, e.g., a delta-function in real space, there are deviations
from a FL behavior. These deviations get amplified as the dimensionality is
reduced, and, eventually, lead to a complete breakdown of the FL in ID. Already
for the simplest case of a point-like interaction, the second order self-energy shows









p

a) p-q b)


k k+q k k p p+q k+q k

Figure 1-7. Non trivial second order diagrams for the self-energy


a nontrivial frequency dependence. For a contact interaction the two self-energy

diagrams of Fig. can be lumped together (the second diagram is -1/2 the first

one). Two given fermions interact via polarizing the medium consisting of other

fermions. Hence the effective interaction at the second order is proportional to the

polarization bubble, which just shows how polarizable the medium is,


ImVR(w, q)= -U2Im R(w, q). (1 47)


For small angle scattering q < 2kg, w < EF, the particle-hole polarization bubble

has the same scaling form in all three dimensions [53],


ImHR (q; u vD BD (RD (1-48)
vpq vFq

where D = aDrmkFD-2 is the density of states in D dimensions (a3 r-2, a2

-1, al = 2/7) and BD is a dimensionless function whose main role is to impose

a constraint w < vpq in 2D and 3D, and u = vFq in 1D. The above form of the

polarization operator indicates Landau damping: Collective excitations (spin and

charge density waves) decay into particle-hole pairs, this decay occurs only within

the particle-hole continuum whose boundary for D > 1 is at u = vFq for small U, q,

therefore, decay occurs for w < vFq. Using the polarization operator in Eq.1-42 one









gets in 3D,

ImER(E) U2 d dqq 2 duu ],
0 Jiwl/VF VFq Vq JO wo VF
FL
beyond FL
~ a2- bl3, (149)

where the first term originates from the large momentum transfer regime and is

the Fermi-liquid result whereas the sub-leading second term originates from the

small-momentum-transfer regime and is nonanalytic. The fraction of phase space

for small angle scattering is small: most of the self-energy comes from large-angle

scattering events (q ~ Q), but we already start to see the importance for small

angle processes. Applying Kramers-Kronig transformation to the non-analytic part

(I 13) in ImER, we get a corresponding non-analytic contribution to the real part
as (ReER)non-an OC 3 In II and, finally, using the specific heat formula (see Eq.3-14

in chapter 3) we get a nonanalytic T3 In T contribution which has been observed

experimentally both in metals [54] (mostly heavy fermion materials) and He3 [55].

Similarly in 2D

ImER() ~ U2 Edw F dqq ~ ri 2 n In (1-50)
Jo wl/J F UqVFq Vp

and ReER (E) oc xE| and this results in the T2 non-analyticity for the specific heat

which has been observed in recent experiments on .i .ii,. i rs of He3 adsorbed on

solid substrate [56].

In 1D, as we show in chapter 3, the situation is slightly different. Even though

the same power counting arguments lead to ImER oc iE and ReER oc Eln F1 for

the second order self-energy, C(T) is linear (analytic) in T at second order and

the nonanalytic TIn T shows up only at third order in interaction and only for

fermions with spin. This difference is due to the fact that in 1D, small momentum

transfers (here particle-hole continuum shrinks to a single line w = vFq, so decay of









collective excitations is possible only on this line) do not lead to the specific heat

nonanalyticity which occurs solely from the nonanalyticity of the backscattering

(at q ~ 2kF) particle-hole bubble or the Kohn anomaly. Thus, we have the

same singular behavior of the bubble (response functions) and the results for the

self-energy differs because the phase volume qD is more effective in suppressing the

singularity in higher dimensions than in lower ones.

In addition to the forward scattering nonanalyticity, there is also a nonanalytic

contribution to the self-energy and thermodynamics arising from q w 2kF, part

of the response function, i.e., the Kohn anomaly. Usually, the Kohn anomaly is

associated with the 2kF nonanalyticity of the static particle-hole bubble and its

most familiar manifestation is the Friedel oscillation in electron density produced

by a static impurity (see section 1.1.2, of this thesis). Here the static Kohn

anomaly is of no interest as we are dealing with dynamical processes. However, the

dynamical bubble is also singular near 2kF, e.g., in 2D


ImHR(q ~ 2kF, ) ox O(2kF q). (1 51)
IkF(2kF q)

Due to the one-sided singularity in ImHR as a function of q, the 2kF effective

interaction oscillates and falls off as a power law in real space. By power counting,

since the static Friedel oscillation falls off as SlD then the dynamical one

behaves as:


Ssin(2kFr) (1 52)
U oc (1-52)
TD-1

Dynamical Kohn anomaly results in the same kind of non-analyticity in the

self-energy (and thermodynamics) as the forward scattering. The singularity

now comes from \q 2kF ~ wU/vF, i.e., dynamic backscattering. Therefore

the nonanalytic term in the self-energy is sensitive only to strictly forward or









backscattering events, whereas processes with intermediate momentum transfers

contribute to the analytic part of the self-energy.

(a) (b) (c)
k, ca ki k l CC ki k ik

k2" U(O)P k k B U(2k) p k2 ki U(2kFp k2

Figure 1-8. Scattering processes responsible for divergent and/or nonanalytic
corrections to the self-energy in 2D. (a) 1. i v ,ird scattering -an analog
of the g4 process in 1D (b) "Forward scattering with anti-parallel
momenta-an analog of the g2 process in 1D (c) ., I.:-, ii I, ui, with
antiparallel momenta- an analog of the gi process in 1D


We will now perform a kinematic a iJll~i ; and show that the nonanalytic terms

in the self-energy and specific heat in 2D comes from only 1D scattering processes.

Consider the self-energy diagram of Fig. 1-7.(a). The nonanalytic E2 In E term in the

self-energy came from two q-1 singularities: one from the angular average of ImGR

and the other one from the dynamic, U/vFq part of the particle-hole bubble. This

form of the bubble arises only in the limit u < vpq,


ImIR (, q) ~ Im dDidEG(F u,- q0G(E ,pq G dOS(cos 0 ),

~ -(foraw vpq). (1-53)
Vpq 1 Vpq
VFq IF 2 VFq
U2 V q2

From the delta function, cos = wo/vpq < 1, which means that the angle between f

and q is 0 7r/2 or j and q are perpendicular to each other. Similarly the angular

averaging of ImGR(k a,) also pins the angle between k and q to 90 degrees.


(ImGR(k ,E)),j ~ d016(e qvp cos0i)

S cos 01 .- < 1 81 ~ 7/2
vUq Vpq

Thus f; and k (the two incoming moment of the fermions) are almost perpendicular

to the same vector j. In 2D, this means that they are either almost parallel to each

other or anti-parallel to each other, and since the momentum transfer is either









small, q 0 or near 2kF, i.e., Iq 2kF I 0, we essentially have three 1D scattering

processes (see Fig.1-8 ) which are responsible for the nonanalytic corrections to

the self-energy. These three processes are (a) two fermions with almost parallel

moment (k ~ kC) collide and transfer a small momentum (q ~ 0)and leave

with outgoing momentum which are almost parallel to each other (k1' ~ k2') and

parallel to their incoming moment (k1' ~ k, k2 ~ k2I): analogous to the "94"

scattering mechanism in 1D (see Fig.1-8 (a) and chapter 3) (b) two fermions with

almost anti-parallel moment collide (ki ~ -k2) and transfer a small momentum

(q ~ 0) and leave with outgoing momentum which are almost anti-parallel to each

other (kl' ~ 2') but parallel to their incoming momenta(ki' k', kc' ~ k2):

analogous to the "g2" scattering mechanism in 1D (see Fig.1-8(b) and chapter 3),

(c) two fermions with anti-almost parallel moment collide (ki ~ -k2) and transfer

a large momentum q ~ 2kF and leave with outgoing momentum which are almost

anti-parallel to each other (kl' ~ -k2') and also anti-parallel to their incoming

moment (kI' ~ -ki, k2' ~ -kI): analogous to the "g1" scattering mechanism in

1D (see Fig.1-8(c) and chapter 3). Therefore the nonanalytic 2 InE term in the

self-energy in 2D comes from 1D scattering events, the only difference is that 2D

trajectories do have some angular spread, which is of the order of UIJ/EF. It turns

out (Ref. [44]), that out of the three 1D processes, the g2 process and gi process

are directly responsible for nonanalytic corrections (NAC) to C(T) in 2D and only

the gl process leads to NAC to C(T) in 1D. The g4 process although leads to a

mass-shell singularity in the self-energy in both 2D and 1D, but does not give any

NAC to thermodynamics.

In 3D the situation is slightly different, f I q and k _I mean that both

; and k lie in the same plane. However, it is still possible to show that for the

thermodynamic potential, j and k are either parallel or anti-parallel to each

other. Hence, the nonanalytic term in C(T) also comes from the 1D processes. In








addition, the dynamic forward scattering events (marked with a star in Fig.1-9.)
which, although not being 1D in nature, does lead to a nonanalyticity in 3D.
Thus the T3 InT anomaly in C(T) comes from both 1D and non-1D processes
[47]. The difference is that the former start already at the second order in
interaction whereas the latter occur only at third order. In 2D, the entire T2
nonanalyticity in C(T) comes from 1D processes. The nonanalytic correction to the
spin susceptibility will be the subject of discussion in chapter 4 of this thesis, where
we will show that the nonanalyticity in Xs, both in 2D and 3D comes from both 1D
and non 1D scattering processes.








"g4"






d c "any-angle" scattering event
dynamic forward scattering Regular (FL) contribution
+ I D dynamic
forward or backscattering Q r Q-2kF 0)




Figure 1-9. Typical trajectories of two interacting fermions

Our kinematic arguments can be summarized in the following pictorial way.
Suppose we follow the trajectories of two fermions, as shown in Fig. -9. There









are several types of scattering processes. First, there is a "any-angle" scattering

which, in our particular example, occurs at a third fermion whose trajectory is not

shown. This scattering contributes a analytic, FL terms both to the self-energy

and thermodynamics. Second, there are dynamic forward scattering events, when

q ~ wIUJ/vF. These are non-1D processes, as the fermions enter the interaction

region at an arbitrary angle to each other. In 3D, a third order in such a process

leads to a T3 n T term in C(T). In 2D dynamic forward scattering does not lead

to a nonanalyticity. Finally there are 1D scattering processes marked with a Sirius

star where fermions conspire to align their moment either parallel or anti-parallel

to each other. These processes determine the nonanalytic part of E and C(T) in

2D and 1D.

Therefore the nonanalytic terms in the two-dimensional self-energy and

thermodynamics are completely determined by 1D processes, 2D scattering does

not pl. i any role in the nonanalytic terms. As a result, if the bare interaction has

some q dependence, only two Fourier components matter: U(O) and U(2kF) e.g., in

2D

ImZR(E) oc [U2(O) + U2(2) U(O)U(2kF)E21 n (1-54)

ReR (E) oc [U2(0) + U2(2kF) U(O)U(2kFp)]EE|, (1-55)

C(T)/T = a[U2(0) + U2(2k) U(0)U(2kF)]T, (1-56)

Xs(Q, T) = Xs*(O) + bU2(2kF)max[vFQ, T, H], (1-57)

where a and b are some coefficients. These perturbative results can be generalized

for the Fermi-liquid case, when interaction is not weak. Then the vertices U(O)

and U(2kF), occurring in the perturbative expressions are replaced by scattering

amplitude (F) at angle 0 = 7,


F(j, ) F (0)1 + F,(0).a',


(1-58)









where c and s refer to the charge and spin sectors respectively. Thus in 2D [45],

T
C(T)/T = -a[F2() + 32,2). (1-59)

The additional (In T)2 factor in the denominator comes from the Cooper channel

renormalization of the backscattering amplitude [47, 48]. In 3D, the Ts lnT

nonanalyticity in the specific heat arises from both 1D (excitation of a single

particle-hole pair) and non-1D (excitation of three particle-hole pairs) scattering

processes [47].


C(T) 3 n T + FaF2a,0 + F3l + ...' (1 60)
(1 + g InT)2 V
ID, one p-h pair non ID, three p-h pairs

where subscript a = c, s and 0,1, 2... indicate the harmonics of the expansion.

Again, the additional (1 + g In T)2 factor in the denominator comes from the

Cooper channel renormalization of the backscattering amplitude [47, 48].

We saw that the nonanalytic corrections to the specific heat in D = 2, 3, arise

from one dimensional scattering processes, (and they show up at second order in

perturbation theory), and the degree of nonanalyticity increases with decrease in

dimensionality. This predicts that the strongest nonanalyticity in the specific heat

should occur in 1D. However, it was shown in Ref.[57], that the specific heat in

1D is linear in T, at least in second order in perturbation theory. In addition, the

bosonization solution of a one-dimensional interacting system, predicts that the

C(T) is linear in T. We resolve this paradox by showing (in chapter 3) that the

general argument for nonanalyticity in D > 1 at the second order in interaction

breaks down in 1D, due to a subtle cancelation and the nonanalytic T InT term

in the specific heat occurs at third order and only for electrons with spin. This is

verified by considering the RG flow of the marginally irrelevant operator in the

Sine-Gordon theory (which appears in the bosonization scheme for fermions with

spin). For spinless electrons we show that the nonanalyticities in particle-particle









and particle-hole channels completely cancel out and the resulting specific

heat is linear in T (the bosonized theory is gaussian). The singularity in the

particle-hole channel results in a nonanalytic behavior for the spin-susceptibility

Xs oc lnmax[|Q |HI, T], present at the second order.

The spin susceptibility both in 2D and 3D gets nonanalytic contributions from

both 1D and non-1D processes. These corrections will be described in detail in

C'i lpter 4 of this thesis where we also study the nonanalytic corrections near a

ferromagnetic quantum critical point.

1.3 Spin Susceptibility near a Ferromagnetic Quantum Critical Point in
Itinerant Two and Three Dimensional Systems.

The physics of quantum phase transitions has been a subject of great interest

lately. In contrast to the usual classical (thermal) phase transitions, quantum

phase transitions occur at zero temperature as a function of some non-thermal

control parameter (e.g., pressure or doping), and the fluctuations that drive the

transition are quantum rather than thermal. Among the transitions that have been

investigated are various metal-insulator transitions, the superconductor-insulator

transition in thin metal films, and (the first one to be studied in detail and the

subject of this thesis), the ferromagnetic transition of itinerant electrons that

occurs as a function of the exchange coupling between the electron spins. In a

pioneering paper, Hertz [58] derived a Landau-Ginzburg-Wilson (LGW) functional

for this transition by considering a simple model of itinerant electrons that interact

only via the exchange interaction. Hertz analyzed this LGW functional by means

of the renormalization group (RG) methods that generalize the Wilson's treatment

of classical phase transitions. He concluded that the ferromagnetic order in an

itinerant system sets in via a continuous (or 2nd order) quantum phase transition

and the resulting state is spatially uniform. Furthermore, he showed that the

critical behavior in the physical dimensions d = 3 and d = 2 is mean-field-like,









since the dynamical critical exponent z = 3, (which arises due to the coupling

between statics and dynamics in a quantum problem), decreases the upper critical

dimension from d+c = 4 for the classical case to d+c = 1 in the quantum case.

Hertz's theory which was later extended by Millis [59] and Moriya [60], (it is

commonly referred as the Hertz Millis Moriya (HMM) theory), is believed to

explain the quantum critical behavior in a number of materials [61]; however, there

are other systems which do not agree with the HMM predictions and show a first

order transition, (e.g., UGe2), to the ordered state. This contradiction motivated

the theorists to re-examine the assumptions made in the HMM theory.

The HMM scenario of a ferromagnetic quantum phase transition is based on

the assumption that fermions can be integrated out so that the effective action

involves only fluctuations of the order parameter. This assumption has recently

been questioned, as microscopic calculations reveal non-analytic dependence of

the spin susceptibility on the momentum (q), magnetic field (H), and for D / 3,

temperature (T) [42, 44] both away and near the quantum critical point (see the

discussion in section 1.2). For example, in 2D


xs(H,Q,T) = const. + max(HI, IQI, T), (1-61)

and in 3D


Xs(H, Q) = const. + (q2, H2) ln[max(lHI, |Q)], (1-62)

where H, q and T are measured in appropriate units. The dependence on T is

nonanalytic in the sense that the Sommerfeld expansion for the Fermi gas can only

generate even powers of T. Of particular importance is the sign of the nonanalytic

dependence: ,s(H, Q) increases both as a function of H and q (at 2nd order in

perturbation theory) for small H,q. As ,s(H, Q) must definitely decrease for H

and q exceeding the atomic scale, the natural conclusion is then it has a maximum









at finite H and q. This means that the system shows a tendency either to a first

order transition to a uniform ferromagnetic state (the metamagnetic transition as

a function of the field), or ordering at finite q, (to a spiral state). The choice of the

particular scenario is determined by an interplay of the microscopic parameters.

In C'! lpter 4 of this thesis, we will obtain the nonanalytic corrections to X,(H)

in second and third order in perturbation theory and show that these corrections

oscillate between positive at 2nd order, (which points towards a metamagnetic

transition), and negative at 3rd order (which points towards a continuous second

order phase transition) values. Thus it is impossible to predict the nature of the

phase transition by investigating the nonanalytic terms at the lowest order in

perturbation theory. Furthermore, in real systems interactions are not weak and

one cannot terminate the perturbation theory to a few low orders. To circumvent

this inherent problem with perturbative calculations and to make predictions

for realistic systems (e.g., He3), we obtain the nonanalytic field dependence for

a generic Fermi liquid by expressing our result in terms of the lowest harmonics

of the Landau interaction parameters. We also describe the nonanalytic field

dependence near the quantum critical point using the self-consistent spin-fermion

model, and show that the sign of the corrections is metamagnetic. Here, in the

introduction, we briefly review Hertz's theory of the second order phase transition.

1.3.1 Hertz's LGW Functional

Hertz considered the Hubbard model with the lagrangian L given by


L -i,,0 -It)= t1-P-t1,7/C
i,c l,l',

+ (hT + n11)2 1 1( n1)2 (1-63)
4 4

The partition function is obtained by performing a Hubbard-Stratonovich

transformation to decouple the four-fermion interaction in the charge and spin

channel. The charge channel is assumed to be non-critical and is thus discarded,









whereas the partition function for the spin channel takes the following form;


Z J= DDCtDCe- fo dL(,Ct,C)


where


L(p, Ct, C)


Y Cti, (, po)C,C t1 -Ct1,C1',
i,, 1,l',
U +U
4+ 2 ni il).
1 1


The field Q' is the conjugate field to the ni nil, which can be considered as the

magnetic field acting on the fermions. Performing the functional integration over

the fermion operators (C, Ct) he arrived at the partition function

Z = D Se-s(ff)


with the effective action;


Seff (0)


U drT 27) Trln[(O, t- t1_, + ].
4 0 2


(1-66)


The Mean-field-theory would correspond to the saddle point approximation to the

functional integration with respect to 4. To deduce an effective (LGW) functional,

one expands the interaction term (Tr n term) in Q'. The matrix (M) in the Tr In

term in Eq.1-66 in the Fourier space becomes


(M )(k, iLL, ) ; k', ium, a')


S[(-ILWn + (k))n,wm kk
crU -
+ (k k',i0 ikm).
2Q17


(1-64)


(1-65)


(1-67)









The first term on the right hand side of the above equation is the inverse Green's

function for free fermions (-G- o), the second term is the ,l'- I I I- ,i" (V). Then


Trln[M] Tr -G- G )] Trln[ ( V) n[-G-lo] + Trln[ GoV]

Trln[-G-lo] Tr(GoV).
n=

Expanding up to fourth order in V the effective action is


1

+4QV > v4Uq (q, iw j Ii2,i, Ii3)

4 4
x(4,wy iw4)J i)K K ). (1-68)
i= 1 i 1

The coefficients v, in Eq.1-68 are the irreducible bare m-point vertices in

the diagrammatic perturbation theory language. The quadratic coefficient is

v2(q iUi) = 1 Uxo(q, i1l), where xo(q, iui) is the free electron susceptibility given

by the Lindhard function (Polarization bubble), which at small q and small u/qvp

behaves as


ol, iumm) = -V- YGo'(E in)Go( + i+ iumrn),
k,iEs

F[1 t)2- _Q q...]. (1-69)
3 2kF 2 \qvF

Hertz assumed the all the higher order coefficients v, starting with v4 can be

approximated as constants as they vary on the scale of q ~ 2kF and w ~ EF. In

appropriate units Hertz's form of the effective LGW functional is


q,2Wm qiL;,1i
x (q3, I3) (-qT i -iL iw 2 iL3) (1-70)









where ro = 1 UVF -= -2 ( is the correlation length which diverges at the phase

transition), is the distance from the critical point and Uo = U4v"F/12 is a constant.

Thus, Hertz's effective action is almost of the same form as the classical LGW

functional (for the 44 theory), except for the presence of the frequency dependent

term in v2 which contains the essential information about the dynamics. The

action therefore describes a set of int( ,il ii:- weakly Landau-damped (due to the

:.,. I/qvp term) excitations: paramagnons.

Hertz then applied Wilson's momentum shell renormalization group transformation

to the above quantum functional. Here, q and w have to be re-scaled differently.

This is due to the fact that in the paramagnon propagator (v2-1), q and uWm\

appear in a non-symmetric way. Therefore, the system is anisotropic in the

time and space directions. As a result it becomes necessary to introduce a new

parameter, the dynamical critical exponent z for scaling


S~ NqZ. (1-71)


For the quantum ferromagnetic transition which we study here, z = 3. In the RG

procedure consists of the following steps (a) high energy states (with q and w) in

the "outer shell" (A > q > A/b; A > w > A/b; b > 1, A is a cut-off) are integrated

out; (b) variables q and w, are re-scaled as q' = qe and / = ..' with I being

infinitesimal. (c) fields Q are also re-scaled so that in terms of the new fields and

re-scaled q and a, the q2 and wul/q terms in the quadratic part of the action looks

like those in the original functional. Performing all these steps, Hertz obtained the

following RG equations

dr
d = 2r + 12uf2, (1-72)
du
u 18u2f4, (1 73)
dl









where e = 4 (d + z) and the expressions for f2 and f4 can be found in Ref. [58].

The second RG equation shows that the Gaussian fixed point, with u 0, is stable

if c is negative, that is, if d > 4 z. For z = 3, we should therefore expect a stable

Gaussian fixed point and Landau exponents in d = 2, 3.

The two main assumptions that Hertz made in arriving at his LGW functional

(Eq. 1-68 and 1-70) were: (1)the coefficients Vn,mn>4 are nonsingular and can

be approximated by constants and (2) the static spin susceptibility has regular

q2 momentum dependence. For the 2D ferromagnetic transition, nonanalytic

terms in v, were found by Chubukov et al., [62], however, the authors claimed

that these nonanalyticities do not give rise to an anomalous exponent in the spin

susceptibility and therefore were not dangerous. In chapter 4 of this dissertation

we examine the second assumption (2) more carefully. The reasoning behind

Hertz's second assumption was the belief that in itinerant ferromagnets the q

dependence of the 02 term comes solely from fermions with high energies, of

the order of EF or bandwidth, in which case the expansion in powers of (q/p)2

should generally hold for q < pp. This reasoning was disputed in Refs. [42, 44].

These authors considered a static spin susceptibility ,s(q) in a weakly interacting

Fermi liquid, i.e., far away from a quantum ferromagnetic transition, and argued

that for D < 3 and arbitrary small interaction, the small q expansion of Xs(q)

begins with a nonanalytic Iqd-1 term, with an extra logarithm in D = 3. This

nonanalyticity originates from the 2pF singularity in the particle-hole polarization

bubble [42-44] and comes from low energy fermions (in the vicinity of the Fermi

surface), with energies of the order of vFq < Ep. These nonanalytic terms

arise when one considers the reference action So as the one which contains the

particle-hole spin singlet channel interaction (charge channel) and the Cooper

channel interaction, which were neglected in the Hertz model (Hertz's reference

action was just the noninteracting one). Furthermore, the pre-factor of this









term turns out to be negative, which indicates the breakdown of the continuous

transition to ferromagnetism. Thus according to Ref. [42, 63] the modified effective

action near the critical point should be

q2 t i( (1D1 2 74)
Seff) 2 (o -q +D- + 2 )(; 2 +b44+... (174)
q,

with an extra logarithm in D = 3. The weak point of this argument is that

within the RPA, one assumes that fermionic excitations remain coherent at the

quantum-critical point (QCP). Meanwhile, it is known [64] that upon approaching

the QCP, the fermionic effective mass m* diverges as In in D = 3 and (3-D

in smaller dimensions. It can be shown that m/m* appears as a prefactor of the

Iq D-1 term; which would mean that the nonanalytic term vanishes at the QCP.

This still does not imply that Eq.1-70 is valid at the transition because, as we show

in chapter 4, the divergence in m* does not completely eliminate the nonanalytic

term, it just makes it weaker than away from the QCP.

Our approach will be to use the low-energy effective spin-Fermion hamiltonian,

which is obtained by integrating the fermions with energies between the fermionic

bandwidth W and a lower cut-off A (with A < W), out of the partition function

[64, 65]:


H VF(p- pF)ctp,acpa+ Xs,o-l(q)SqS-q + g Ctp+q,a7O, iSq. (1-75)
p,a q p,q

Here Sq describe the collective bosonic degrees of freedom in the spin channel,

and g is residual spin-fermion coupling. In Hertz's approach, all fermions were

integrated out, whereas in the Spin-Fermion model only the high-energy fermions

are integrated out while keeping the low-energy ones. This will turn out to be

important because the spin fluctuation propagator is renormalized by the fermions,

and the fermion self energy is renormalized by interaction with bosons. This model

has to be solved self-consistently as it takes into account the low-energy (mass)






42


renormalization of the spin fluctuation propagator. In chapter 4 of this dissertation

we use this model to obtain the magnetic field dependence of the spin susceptibility

near the quantum critical point, and analyze the stability of the second order

quantum phase transition.















CHAPTER 2
CORRELATED ELECTRONS IN ULTRA-HIGH MAGNETIC FIELD:
TRANSPORT PROPERTIES

One-dimensional systems exhibit unique physical properties which reflect the

influence of strong correlations. The effective dimensionality of charge carriers in

a bulk metal may be reduced from 3D to 1D by applying a strong magnetic field.

It has recently been shown that this reduction leads to formation of a strongly

correlated state, which belongs to the universality class of a Luttinger liquid [5].

The tunneling density of states exhibits a characteristic scaling behavior for the

case of long-range repulsive interaction [5, 6]. This effect is most pronounced in the

ultra-quantum limit (UQL), when only the lowest Landau level remains occupied.

Here, in this chapter we investigate the effect of dilute impurities on the transport

properties of the system. For good metals, the quantizing field is too high (of

the order of 104 Tesla), but semi-metals and doped semiconductors have a low

carrier density and quantizing fields of the order of 1 10 Tesla and allow for a

experimental test of the theoretical predictions made here.

In section 2.1 we discuss localization effects for non-interacting electrons in

the UQL. We find that the localization behavior is intermediate between 1D (

6a ~ CD: strong localization) and 3D (6a < CD: weak localization). We show that

the particle-particle correlator (Cooperon) is massive in the strong magnetic field

limit. It's i: i--" (in units of the scattering rate) is of the order of the impurity

scattering rate. Therefore, localization in the strong-field limit proceeds as if a

strong phase-breaking process is operating as frequently as impurity scattering.

Even at T= 0, this phase-breaking exists as it is provided by the magnetic field

and as a result complete localization never occurs in 3D UQL. On the other









hand, the particle-hole correlator (the diffuson) remains massless, which means

that normal quasi-classical diffusion takes place. Our findings are in agreement

with previous work on this subject [66, 67], where the localization problem was

analyzed for the case of long ranged disorder, whereas in our study we have

analyzed the case of short ranged disorder. Our result for conductivity in the UQL

is t5coop + mDrude = UDrude/2.

In section 2.2 we calculate the corrections to the conductivity due to

electron-electron interactions using finite-temperature diagrammatic technique

where disorder is treated in the ballistic limit. Due to this reduced effective

dimensionality, to first order in interaction, the leading corrections are logarithmic

in temperature. Another way of obtaining the conductivity is to calculate the

interaction correction to the scattering cross-section through an impurity (in a

Hartree-Fock approximation) and use a Drude relation between the cross-section

and the conductivity. We show in section 2.3 that, to first order in the interaction,

the two approaches are equivalent. This is important since, while a higher order

calculation using the diagrammatic technique would be extremely lengthy, the

interaction correction to the cross-section is obtained to all orders via an exact

mapping on to a 1D problem of tunneling conductance of interacting electrons

through a barrier [21]. We find that the Drude conductivities parallel (r = +1)

and perpendicular (r = -1) to the magnetic field exhibit the scaling laws

aJ oc T"2", where a is a function of the magnetic field. The physical reason for

such a behavior of the conductivity is a nearly 1D form of the Friedel oscillation

around an impurity in the strong magnetic field.

The ground state of repulsively interacting electrons in the UQL is known to

be unstable to the formation of a charge-density wave (CDW) [1-3]. This has been

confirmed, for example, by experiments on graphite in high magnetic fields [4].

Both the Hartree-Fock and the diagrammatic calculations presented here are done









without taking into account renormalization corrections for the interaction vertices

themselves. This is justified at energies much larger than the CDW gap but breaks

down at low enough energies. In order for our results to hold, there should exist an

intermediate energy interval in which the renormalization of the interaction vertices

due to CDW fluctuations is not yet important but the power-law renormalization

of the scattering cross-section is already significant. That such an interval exists

for the case of long-range electron-electron interaction was shown by solving the

full RG equations for the vertices and for the cross-section. We have not included

this discussion here for brevity. We discuss possible experimental verification of our

results in section 2.4 and conclude in Section 2.5.

2.1 Localization in the Ultra Quantum Limit

In this section we analyze the localization effects for electrons in the UQL.

As an external magnetic field is applied to the system, a question is whether the

reduction of the effective dimensionality leads to re-entrance of interference effects,

which were initially suppressed by a weak magnetic field. It seems plausible (and

was indeed -, .-.- -1. I by some authors in the past) that the application of a strong

magnetic field may result in a strong localization of carriers, similarly to what

happens in a truly 1D system. A physical argument rules out this possibility

[66, 68], at least for short-range impurities. In this case, while scattering at an

impurity, an electron moves transverse to the field by a distance of the order of the

magnetic length 1H = 1/eH. The flux captured by the electron's trajectory is

then of the order of the flux quantum, and thus interference is destroyed. However,

an application of the standard cross-technique to the calculation of the conductivity

in the UQL fails for the same reason that it does in 1D: all diagrams which go

beyond the Boltzmann equation level give contributions of the order of the Drude

conductivity aD itself. Therefore, perturbation theory breaks down. In 1D, a

similar breakdown is a signal but not the proof of strong localization. In the









following subsection we illustrate the different scenario which arises for 3D electrons

in the UQL.

2.1.1 Diagrammatic Calculation for the Conductivity

In this section we will use the zero temperature diagrammatic formalism to

evaluate first the conductivity and then the WL type quantum corrections. We

choose to work in the symmetric gauge, in which the eigenfunction of a three

dimensional electron gas in UQL is [22]


',m (p, 0, z .) = ezz Rm (p) ,
i27r

where m(pz) is the angular (linear) momentum quantum number in the direction of

the field. In the UQL the allowed values of m are m = 0, 1,2, 3... and


Rm (p) pmexp (-p2 /41

The single particle Green's function, in the Matsubara representation is


G (r,r'; i) (r') (r),


where p = (pz2 pF2)/2m + w/2,(wc is the cyclotron frequency). The Green's

function in the mixed representation (momentum and position representation

because the magnetic field breaks translational invariance) separates into a 1D and

a transverse part

1 e -im(4p-4')
G (pz,p, p'O ; I) 2= m-(p) R (Pp) (2-1)
P m=0
SGld(pz; i) GI (ri, r'), (2-2)

where (r = p, Q). The reason for this separability is the degeneracy of the Landau

level; the energy does not depend on the transverse quantum numbers. The 1D

part, GID, is in the momentum space and the transverse part, G, has been

kept in real space. The disorder averaged Green's function is obtained by doing









perturbation theory in the impurity potential U (for weak disorder kF >> 1, so
that the small parameter is 1/kF) and employing standard cross technique [18] for

disorder averaging. The perturbative (in U) solution of the Schrodinger equation
for the Green's function is


G (r, r'; iE)


Go (r,'; ) + i d3r1Go (r, r1; i) U (r) Go (r r'; i)


+ ff d3rd3r2Go (r, ri; i) U (rl)Go (ri, r2; iE) U (r2)Go (r2, r'; i) ...

where G(Go) is the full (bare) Green's function, U(r) is the random impurity
potential which is chosen to have a 6 correlated gaussian distribution with

(U (r)) = 0 and (U (r) U (r')) = r' ,,,23 (r r'). The number density of impurities
is ni, and uo is the impurity strength (for short ranged point like impurities).
The leading contribution to the self-energy comes from the second order

diagram. The first and third order corrections are zero as (U (r)) = 0. We work in
the Born limit, neglecting processes where an electron scatters from more than two

(same) impurities. The second order correction is


G2 (r, r'; i))

which gives


S 3 ri d3r2Go (r r; ;i) Go (ri, r2;i) Go (r2, r';i) {U


(r) U(r2)),


(G2 (p, p p'O, P iE))


[-ii (UO)2 VHHisgn (E)] IF x_


2x Rm (p) R (p') ,
27m
rn0


and the fourth order correction gives


(G4 Pz, p, p ; iE))


[-n (UO)2 VH7jSgn (E)] 2 IF-w


S-- Rm2 (p) Rm (p')
27e^









where the transverse part in each of these expressions is simply Gi(ri, r'). At

fourth order, there are three diagrams, the rainbow diagram (Fig.2-l,(b)) and

the intersecting diagram (Fig.2-1, (c)) are small by a factor l/kFY compared to

the leading one ((G4)) (Fig.2-1,(a)) for short-range weak disorder. In the Born





b



Figure 2-1. Diagram (a) is the leading contribution to the self energy at fourth
order

approximation, the scattering rate in a magnetic field, is 1/7 = 2,' 1,,,2vH, where

VH = 1/(27r2VF1H2) is the 3D density of states in the presence of a magnetic field,

and the self-energy is E = -isgn (E) /27. The full Dyson's series (Fig. 2-2) can be

summed to give:


(p,,rr, ri) 1 )G i(rr) 1 + + -- 2

S ) Gi(ri, r') (2-3)


Therefore, the effect of impurity scattering enters only in the 1D part of the

Green's function. Using the above form of the Green's function and the Kubo

formula we now evaluate the Drude conductivity. The Kubo formula for the

longitudinal d.c. conductivity (E || H I| z) in the kinetic equation approximation is

1 e2 d dp
azz lim e 2dt i (G i (pz, rI, r; IE
wo a m2 2 j 2w

(G (p, r i vit i si s2

The diagram for the Drude conductivity is shown in (Fig. 2-3). Due to the






49


/ -\ I \ / x
+ -<--- + +



G = Go + dG



dG = + +


Figure 2-2. Dyson's series






Figure 2-3. Drude conductivity


factorizability of the Green's function, the conductivity is also separable into a 1D

and a transverse part: uzz = -1-d x al. The 1D part is in the standard form and

gives the famous Einstein's relation for the d.c conductivity a1d = e2vD = e2/r,

where D = vFr is the diffusion coefficient and vl = l/(rvF) is the density of

states in ID. Using the orthonormality: f dpp [Rm, (p)] 2 1 and completeness:

E, [R. (p)]2 = /(1H2) properties of the wave function, the transverse part can be
shown to be equal to 1 with the degeneracy factor of the lowest landau level.

SC-m(p c4>) in(p'-) R
Sdp d 21F Rm (p) R () 2(p R, (p') R (p) ,
m,n= O
21


Thus az e2Dv1/(27rl) = e2DvH is the Drude result in UQL. The Diffuson

(or particle-hole correlator) corrections to the conductivity can be shown to be

zero for a delta function impurity potential. In the next sub-section we show

that the particle-particle correlator, or the Cooperon acquires a mass in a strong










magnetic field and evaluate the Cooperon correction to the conductivity. The

Diffuson remains massless which means normal quasiclassical diffusion occurs in the

particle-hole channel. Implications of these results on electron localization will be

discussed.

2.1.2 Quantum Interference Correction to the Conductivity

Every diagram for the conductivity, even higher order ones, can be split

into a 1D and a transverse part. The transverse part is ahv-- a number, cn,

multiplied by the Landau level degeneracy factor (1/2r12)"'+1, where n denotes

the order of the diagram (the number of dashed lines in the vertex diagrams), so

that 6Jz = 6(1D x or and ar = n x (1/27w1l)'+1 We show that for the cooperon

type diagrams, = 1/(2'-1) and for the diffuson type C, = 1. In 1D all quantum

corrections to conductivity (QCC) are of the same order as the Drude value,

16aiD JD. This indicates the breakdown of perturbation theory in 1D. Similarly,

for 3D electrons in the UQL, 16ao-z ~ D, because the transverse numbers Cn are

of order one. Therefore perturbation theory also fails in the UQL. However, we will

see below that these transverse numbers Cn make strong localization impossible in a

strong field. In technical terms, they are responsible for generating a finite mass for

the Cooperon.

p, r 2 r P
Pq qp r p qp ,
r qs p rr


Sp2 q p P q
r qp q +P



r q 1 q


Figure 2-4. Third and second order fan diagram.


We calculate these coefficients for the lowest order Cooperon diagrams (2nd

and 3rd order fan diagram shown in Fig. 2-4) explicitly and then state the general

argument by which these numbers can be obtained for all higher order diagrams.









For the second order fan diagram,

S1 2 e2 fdE fdr idpz f dq, f dpi
6jzz = hm (r I,, ) Im dr' dri dr2
w- v2 727 27 27i 27
Pz (qz pz) G (pz, i-, rl_; iE) G (plz, ril, r72; iE+) G (qz pr2, r i; iE+)

x G (q\ pY, ,, rl; iE_) G (qZ pl, r1_, r2, ; iE_)

xG (p', r2, r7; iF_) (2-4)
iw--w+id /
The one-dimensional part of Eq. (2-4) is given by

6ID lim Im 2 R (q) X (q, ) (2-5)
w-o0 m2 [ 2 27J

where R (q) is the one dimensional rung in the particle-particle channel and

X (q, w) is the part containing the vertices:

R (q) = (r ,,2)2 GID(p1; ipi+) GID(q-pi; iF-), (2-6)

X(q, ) J J 2d pp (q p) G1D (p; i+) G1D (q p; i+)

x G1D(q p; iF_) G1D (p; i_). (2-7)

We use the linear spectrum approximation, f d (...) = J d1 (...) where
v = 1/27rv = vi/2 and vl is the 1D density of states, and assume small (total)

momentum, q < pF to obtain for the rung (w > 0):




(1 + wr)2+ q
R(q) (nuo2)22-rvlT- (1 +J j)q +2q2 (2 8)

For the vertex, linearizing and using q < pF, (we cannot set q = 0 in the vertex a

priori because our cooperon will acquire a mass) followed by the pole integration in
(, and the E-integration, we obtain for w > 0,

X (q, ) -2 1PFW (2-9)
(1 + wr) ((1 + wT)2 + q2f2)









and finally for the 1D part of the conductivity


61D -C(2rlH2)2. (2-10)
47r

Note that 6ald ~ OD, so perturbation theory breaks down in the UQL. The

transverse part of the conductivity is,


L J = dr'J dril dr2 Gi (ri,ri) GL (ri, r2) G (r2, r)

xGi (r ri) Gi (rl ,r2) G (r2,r), (2-11)

where r = (p, 0) and G_ is defined in Eq.(2-2). After performing the azimuthal

integration, we obtain
o lm 14rn r 2
Sn () dpp'R2 (/) dPm (P1) Rn (P1) 1 (P1) 1)
,mOn0 n=(20)3
oo l+rn T2(
>n o 2m 3 [Aimn ]2. (2-12)
i,m=0 n=0 (2)

where n' = 1 + m n. Notice that the second order fan diagram has two radial

integration, likewise third order fan diagrams will have three radial integration,

and so on. Using the integral representation of the Gamma function, Aimn,, is

1 1 1 12
A*Tnn' t 2t r2 (n + )! (2-13)
212 2m+' m!nln'!

and the transverse part of the conductivity becomes:
2
1 0o l+m rn 2m 1
4 (27rl/H2) 3 H 2m22(T+1)m!
H11n 1 (,r1 0 n =0(214)

x .! t* .[(' + 1)!]2 (2-14)
mWnI! (I + m n)!









The sum over n is done using the binomial property and the sum over I is a

tabulated sum [69],


S = 1 2H (2 2- 2(m+1), (2-15)
4(27lH2)3 0 2H 2mm!

1 (2-16)
2 (2 lH2)3

The coefficient of the transverse part of the second order fan diagram for the

conductivity is c2 = 1/2. Combining Eq. (2-10) and (2-16), the QCC from the fan

diagram at second order is 6Jao = -e2DvH/8.

Similarly, the higher order diagrams can be evaluated. The third order

fan diagram has the same vertex as the second order one but has one extra

factor of the rung R (q). This gives for the one dimensional part: 6 lD

-(3e2k/167) (27rl)3. The transverse part now has three radial integration (3

factors of A's) is given as:

To m+q 2
mL = Rt, 4 () [Amnqs] [As,'] [A m,] (2 17)
q,m=0n,n'=0 (

where s' m + q n and s = m + q n'. The radial integration can be performed

as before to obtain the A's. Performing the sums, we find
S o R2 m+q m+q
L (2x7)4 (21H 2)3 m!q!23(-+q) (m + q )!
Sm,q=0 n=0 n'=0
S) (2-18)
4 (2lH2)4

Thus for the third order fan diagram, c3 = 1/4, and the QCC is az

-3e2DvH/64. The nth-order fan diagram has n radial integration, each of which

gives a factor of 1/2 so that one has a coefficient (1/2)". The summation over

angular momentum indices gives a factor 2 regardless of the diagram's order. So,

the overall coefficient in the nth-order fan diagram due to the transverse integration

is c = 1/2-1.









n-1>
S 1 + 1/2 + 1/4 +.. (1/2) +*


Figure 2-5. Cooperon sequence for 3D electrons in the UQL. Unlike in 1D, each
term in the series comes with a different coefficient c,.

We construct the Cooperon sequence in UQL as shown in Fig. (2-5), with

the prefactors indicating c,, the numbers obtained after transverse integration at

each order. These numbers are responsible for the mass of the Cooperon. The DOS

factor at nth order is (1/27lH2) +1. The dashed line in the figure denotes g(gi), the

correlator in 3D (ID), where g = 1/27uvHT niuo2 = 2rlH2/(27vl') g127lH2. R

is the one dimensional rung in the particle-particle channel (small total momentum)

evaluated in the diffusive limit.

R = G D(P; i +)GID(q p; ) -1 (1 Drq2 +...), (2-19)

For 3D electrons in the UQL, the cooperon sequence gives:

g g2 R g3 R2
C(q; iw) + x-+ x-+...,
(2H1 2)2 (2 lH2)3 2 (21 1H2)4 4
S gR g12R2 (gR) (220)-1
7T'2 (t + + 4 + ...+ + ...) (220)
2lH 2 4 2"-1

and using Eq. (2-19), this becomes,

91/ (2xlH2)
C(q; iw) g (2 ) (221)
1/2 + D q2/2 |11|/2

In the limit q, u 0, C becomes a constant. There are no infrared divergence,

because we have a massive Cooperon. The mass in units of the scattering rate is

a pure number (1/2). It can be interpreted as 7-/-rH so that 7-TH is of the order

of the impurity scattering time 7r. This indicates that localization in a strong field

proceeds as if a strong phase-breaking process is operating simultaneously with

impurity scattering. This is dephasing by the field and it persists even at T -+ 0.










We now contrast this situation which arises in the UQL with that of any other

dimensions (1D,2D,3D) without the magnetic field. In the absence of the field the

c,'s are all one (for all dimensions) and the cooperon sequence is singular (there is

a diffusion like pole for real frequencies, for q, w ~ 0).


CHO (q;W) (I + gR + g2R2 +9 9 (2 22)
1- gR Drq2 Iw1

This gives the weak-localization correction to the conductivity (WL QCC discussed

in chapter 1) in 2D and 3D [12]. In 1D although the cooperon diagram has a pole,

all non-cooperon diagrams are also of the same order, and one needs to sum over

all the diagrams to get strong localization [15].

In the ultra quantum limit the transverse numbers for the particle-hole

diffusion propagator (the diffuson) are all equal to unity (c, = 1). Therefore

diffuson remains massless in a strong field and normal quasi-classical diffusion

occurs in the particle-hole channel. We will now evaluate the transverse part

k k

r' p-k r P k
P k


k
rl P,

Figure 2-6. First and second order diffuson


of the first and second order diffuson correction of Fig. (2-6), assuming a long

ranged impurity potential such that cld / 0. We do not attempt to calculate the

longitudinal part of the conductivity, (ald) as this will be more complicated due

to the long range disorder potential. We just assume that the longitudinal part is

finite. In the short ranged impurity case the diffuson correction to conductivity

is zero (because the ad = 0). The transverse part for the first order Diffuson









correction is,


aj= Jdri Jdr'G (r ,rl )G (r ,r') GI (r',rl )G (r, ,r1), (2-23)


where GI is defined in Eq. (2-2). Performing the azimuthal integration,


L (27)2 [Rm (P)]2 J dp1 [Rm (p )]2 [R (p1)]2 dt'' [ ( 2. (2-24)
(2'r)2" m,n=o o

Using orthonormality and completeness, we get or1 1/ (27lH2)2 X 1, so that

cl 1. For the second order diffuson we perform the azimuthal integration and

obtain,
OO
l (27 -)3 n [0' (P/)]2 dpp l[ k (p)]2l1 [R (p)]2 X
l,k,n 0

Jd2P2 [Rk (p2)]2 [Rn (P2)]2 J dp'p' [R (p')]2 (2-25)

and using orthonormality and completeness, we obtain aor 1/ (27rlH2)3 x 1, and

c2 = 1. Any nth-order diagram can be calculated in the same way, giving c, 1.

Therefore the longitudinal diffusion is free and the diffuson remains massless.

C(q;w)


P q--P


Figure 2-7. Interference correction to conductivity


Next we calculate the quantum interference correction to the conductivity in

the ultra quantum limit (see Fig. (2-7)):

1 e e2 ( dq
6cp = -lim C (q;u ) X (q; ) ), (2-26)
W->0 a) m 2 27ir+i









where the transverse integration have been performed. The vertex part of this

diagram has already been evaluated in Eq. (2-9). Using Eq. (2-21),


C (; )X(q;







The localization correct


coop -
bcoop =


[dq gl
J 27 (27lH2) (1/2 + wr/2 + Dq2r/2)

--2viT3pF2w ]
S(1 + wr) ((1t + r)2 + 2f2)
-2vr3ppF22g1 7T3 (v/ + 1)
(27w/H2) (1 + w;T) 27 (DTr2) (1 + wT)2 W

*tion (in the diffusive limit wur < 1,q < 1) is

lim -Im 2 T)2 ( -)
L-oa m2 [27f (27iH2) (+ ;T)3 i i6
- (1 ) D (2
27 27lH2 2


The above result indicates that perturbation theory fails in the UQL (6acoop/a =

-1/2) in the same manner as it does in a one dimensional system. However,

contrary to what happens in 1D, there is no strong localization in the UQL. The

crossed diffuson diagram (next order in l/kF, see Fig. 2-8), is also non singular

and massive. The transverse coefficient for the lowest order crossed diffuson

diagram is c = 1/3/2. It is different from the transverse coefficients of the 2nd and

3rd order fan and diffuson diagrams. Therefore one cannot sum the perturbation

series in the crossed diffuson diagram and obtain a simple geometric series. We

propose that intermediate localization (as opposed to weak or strong) occurs in a

3D metal in the UQL, by which we mean that although all interference corrections

are of the order of the Drude conductivity itself, the zero temperature value

remains finite (finite suppression of the Drude conductivity):

H 1
JT- a D < a < 1. (2-28)
2-


-27)









The lower bound for a is based on the fact that 6acoop + acD x aJH. The

non-cooperon type diagrams, at least in the lowest order, have the opposite sign as

that of the cooperon. They are also of the same order as the cooperon, so it is not

clear what they will add up to. It may happen that all the non-Cooperon diagrams

modify our prediction for a and may make a anywhere from 0 -i 1. To obtain a

better estimate for a one needs to generalize Berezinskii's [15] diagram technique

(developed for the 1D localization problem) to 3D UQL. Our results also agree with

those obtained by the authors of Ref. [66, 67]. These authors considered long range

disorder, < 1H, and obtained all (-)2cD. If one uses the author's formula

for the short range impurity case, where ~ 1H then one obtains all ~ JD, which

means that the number a ~ 1.

In the next section we use the finite temperature diagrammatic technique to

calculate the corrections to the conductivity due to electron-electron interactions

(interaction QCC). We will show that these corrections are logarithmic in

temperature and thus they confirm that the system behavior is one-dimensional.






Figure 2-8. Crossed diffuson diagrams. Left, a double-diffuson diagram, which also
acquires a mass. Right, a third-order non-cooperon diagram which,
up to a number, gives the same contribution as the third order fan
diagram.

2.2 Conductivity of Interacting Electrons in the Ultra-Quantum Limit:
Diagrammatic Approach

In this section we study the electron interaction corrections (Altshuler-Aronov

corrections) to the conductivity in UQL in the ballistic limit. We work in the

Landau gauge and use the basis,

ei(pz+pzz)
', ,,n(x, y, ) L= L (y + pl ) (2-29)
VL\LZ H









for the single-electron wave function, where

1 2 12
(u) 1 -_U/21H (U/1H) (2-30)
(2"n !i /1H 1) /2

Here 1H = 1/eH is the magnetic length and H, is a Hermite polynomial. The

Green's function in Matsubara representation can be written as


G(E, pz,p, y, y') H H (2-3 1)
i WnPz)

where the sum is over all Landau levels, ,(pz) = (p2 p2)/2m + nw,, and

c = eH/m is the electron cyclotron frequency. We will need only excitations near

the Fermi level for our calculation, so in the UQL (EF < c;,) contributions to the

Green's function coming from n > 0 terms in the sum in Eq. (2-31) are negligible

due to the large mass term (of order w,) in the denominator. Neglecting these

terms, the total Green's function is written as the product of a longitudinal and a

perpendicular part


G(, z,p,p, y, y') = G1D( pz)Gi(p,, (t') (2-32)

with Gi(p, ,'/1') =o(Y +Pxl),) o(y' + pl). As shown in the previous section the

disorder-averaged longitudinal Green's function corresponds to G1D(,Pz) = 1/(i -

jpz + isgn(E)/2r) where 1/7 = -2ImE. Calculating the conductivity using this

Green's function gives the Drude formula, with density of states vH = VlD/27rFl.

The (dynamically) screened Coulomb potential in the ultra-quantum limit is

given by [70]
47re2
VR (, q) q2 q 2q2 l R(q /D' (2-33)
q + q K2-q I R (H qz) V1D
where the screening wavevector is related to the density of states via the usual

relation K2 = 47e2H and HR (w, q,) is the polarization bubble of 1D electrons.

In what follows, we will need only some limiting forms of the potential. For









1/T < L < p E and l/f < q < kF,


n ((w, )= D (2-34)
g (a, + iO)

independent of the temperature (up to (T/Ep)2-terms). In the static limit, when

the transverse moment are small (ql2 2 < 1), the potential reduces to an isotropic

form
47re2
VR (0, q) = 2, (2-35)
2 2+2(2 35)

which differs from a corresponding quantity in the zero magnetic field only in that

K scales with H as K ~ 1H H. As it will be shown below, the leading correction

to the conductivity (as well as to the tunneling density of states [5, 6]) comes with

processes with q_ ~ K < 1H1. Therefore, the Gaussian factor in the denominator of

Eq. (2-33) can be replaced by unity for all cases of interest.

The polarization bubble exhibits a 1D Kohn anomaly for q, near 2kF. Such

large momentum transfers are important only in Hartree diagrams, where the

potential is to be taken at w = 0 in the ballistic limit. Near the Kohn anomaly, the

static polarization bubble can be written as

1 EF
II2k (0, q) IlD In EF, (2-36)
2 maxIIqz 2pF\, T/VFI

to logarithmic accuracy.

Finally, the pole of the potential in Eq. (2-33) corresponds to a collective

mode -magnetoplasmon. For w, qvF < EF and qlH < 1, the dispersion relation

of the magnetoplasmon mode is given by

2
p2 cO2 q, 22 2 22
U 2+vpO s+ Vz (2-37)


where upo = /4rne2/m is the plasmon frequency of a 3D metal in zero magnetic

field and s vpFV1 + 2/q2 is the magnetoplasmon velocity. In all situations of

interest for this problem, typical longitudinal moment is much smaller than the









transverse ones, |qz, < q so that one can write

I 2
S VF 1+ N. (2-38)










(a) (b)
Figure 2-9. First order interaction corrections to the conductivity where effects of
impurities appear only in the disorder-averaged Green's functions.

We now proceed to compute the first order interaction correction to the

conductivity in the ballistic limit (TTr > 1). This includes contributions from

diagrams shown in Fig. 2-9, where effects of impurities appear only in the

disorder-averaged Green's functions. It also includes diagrams with one interaction

line and one extra impurity line. These can be separated further into exchange

(Fig. 2-10) and Hartree (Fig. 2-11) diagrams. In this section we show that

diagrams 2-10(a), 2-10(b), 2-11(a) and 2-11(b) give a leading In (T/Ep)

-correction to the conductivity, whereas all other diagrams give sub-leading

contributions.

2.2.1 Self-Energy Diagrams

Diagrams Fig. 2-9(a), 2-10(a), 2-10(c), and 2-11(a) involve corrections to the

self-energy due to electron-electron interaction. Diagram 2-9(a) describes inelastic

scattering of an electron on a collective mode (plasmon), which would have existed

even for a system without disorder. As the electron-electron interaction cannot

lead to a finite conductivity in the translationally invariant case, this diagram

is canceled by the counter-correction of the vertex type [Fig. 2-9(b)]. Diagrams

Fig. 2-10(a), 2-10(c), and 2-11(a) describe correction to the self-energy due to















(a) (b)





-__




(c) (d) (e)



Figure 2-10. Exchange diagrams that are first order in the interaction and
with a single extra impurity line. The Green's functions are
disorder-averaged. Diagrams (a) and (b) give InT correction to the
conductivity and exchange diagrams (c), (d) and (e) give sub-leading
corrections to the conductivity.

interference between electron-electron and electron-impurity scattering. A general

form of the correction to the conductivity for all diagrams of the self-energy type

can be written as

y 2 lim -1 TpzT GI (En, P) GiD (En ,, Pz) 6ID (En, P,)

(2-39)

where 6E1D (,,pz,) is the correction to the (1 ii -ubara) self-energy of the effective

1D problem, to which the original problem is reduced upon integrating out

transverse coordinates. This is possible due to the fact that the Green's functions

are factorized into a 1D and a transverse part, as shown in Eq. (2-32), and the

integration over transverse variables can be carried out and simply give the

















(a) (b)



Figure 2-11. Hartree diagrams that are first order in the interaction and
with a single extra impurity line. The Green's functions are
disorder-averaged. Both diagrams give InT correction to the
conductivity.

degeneracy factor 1/27l In this effective 1D problem, electrons interact via an

effective potential

V (D z, q,) (22 V (, q) e-I (2-40)
S(24)
whereas each impurity line carries a factor nu^r /2rl = VF/2r, where ni is the

concentration of impurities and uo is the impurity potential. The overall factor of

2 in Eq.(2-39) is the combinatorial coefficient associated with each diagram of the

self-energy type.

Substituting (2-33) into (2-40) and using the condition KIH < 1, we obtain

1
V1D (w, q,) e2 In 1 2 (. (2-41)
1H [qz K2n (, q,) Iz D\

Performing the analytic continuation in Eq.(2-39), we obtain

C VF= i O 0 dp1 GR C 2 [-ImGR ImJZ + ReG DReJ6El]

(2-42)

where GD = 1/(E--p+i/2-) and GEZD is the interaction correction to the retarded

self-energy of the non-interacting electrons which is Yo = -i/2r. (For brevity, we

suppressed the arguments of GR1 and E1ZD, which are E,p).









2.2.1.1 Diagram Fig. 2-10(a)

We start with the exchange diagram Fig. 2-10(a) which corresponds to a

correction in the self-energy as shown in Figure 2-12. As u and q, are expected

to be large compared to 1/7 and 1/f, respectively, it suffices to replace the

Green's functions in the self-energy by those in the absence of disorder. In

the rest of the diagram for the conductivity, the Green's functions are taken

in the presence of disorder. In 1D, it is convenient to separate the electrons

into left- and right-movers described by the Green's functions G (,,p)

1/(in T vpp + isgnFn/2-), where p = pz T PF. Accordingly, there are also

two self-energies E, for left- and right- moving electrons. The contribution for

E+ is shown in Fig. 2-12. The Green's functions of right/left electrons are labeled

by in the diagram. Processes in which an electron is forward-scattered twice

at the same impurity do not contribute to the conductivity and are therefore

not considered in this calculation. The diagram with backscattering both at an

impurity and other electrons involves states far away from the Fermi surface and

is thus neglected. The only important diagram is the one shown in Fig. 2-12

where the electron is backscattered by an impurity and forward scattered by other

electrons.



+ + +
P \p's p'-q / p-q pe



Figure 2-12. The self-energy correction contained in diagram 2-10(a), denoted in
y(2--12)
the text as E 12


At first, we neglect the frequency dependence of the potential. The momentum

carried by the interaction line is small, q, E/vF T/vF, and at low temperatures,

such that T/vF < K, one can neglect q, compared to K in V1D and replace VID by a









constant, VID 2./,,* where


go= (e2vF) in 1 (2-43)
KiH

is a dimensionless coupling constant. The perturbation theory is valid for go < 1.

Having in mind that the retarded self-energy is obtained from the Matsubara

one by the analytic continuation ie -iE + iO for En > 0, we choose En to be

positive. Performing an elementary integration over p', we arrive at

S(2--12) 1 dq, 1
ST Wm>En 2r [L + I ][* ( o;) F (p q,)]'

where 1/T = niu~v1D/l1 Although the integral over q, is convergent at the upper

limit, it is instructive to calculate it with an ultraviolet cut-off qmax ~ PF. Doing so,

we obtain


y 2--12) ) = 290T 1x
7 i (2u; ,E) + VF

-1 a m -1 a;"m En
1 tan1 --- + tan- n (2-44)
7T VFqmax VFqmax

Now we see that to logarithmic accuracy it is safe to cut the sum at WUM -

VFqmax ~ EF and omit the factor in the curly brackets in Eq. (2-44):

12--12) ,) g 2T (1 (2-45)
+ p) 2T W i (2-; E, ) + UFP
rm >En
go InEF _(1 +n VFP
2T i[ 2-T 2 4+ T

Performing analytic continuation and separating real and imaginary parts, we

obtain


Re (2--12 (,) go tanh + V (246)
"4+ 4rT

Im (2--12) ( i) r In- t + Re i + VF
27T 27T (2 47T
go Ei
Sgo n EF (2-47)
27,T max {|lE +VUF, T}









To obtain the real part in a form given in Eq. (2-46) we used an identity

S( ix) 2 = r/cosh 7x, whereas the last line in Eq. (2-47) is valid to

logarithmic accuracy. The self-energy of left-moving electrons is obtained from

Eqs. (2-46), (2-47) by making a replacement E + vFp E vFp.

We pause here to discuss the physical meaning of the results contained in

Eqs. (2-46) and (2-47). Eq. (2-46) shows that the correction to the effective mass

is T-dependent: for IE + vFpp < T, 6m oc T-1. In principle, such a correction

might result in an additional T-dependence of the conductivity. However, this

T-dependence occurs only in the next-to-leading order in the parameter (T-r)- <1

1 of the ballistic approximation. The leading correction to the conductivity comes

from the imaginary part of the self-energy, Eq. (2-47). This correction exhibits a

characteristic 1D logarithmic singularity, which signals the breakdown of the Fermi

liquid (to the lowest order in the interaction).

The main contribution to the conductivity comes from the correction to the

imaginary part of the self-energy [Eq. (2-47)]. Substituting Eq. (2-47) into Eq.

(2-42) and adding a similar contribution from the left-moving electrons, we obtain

6,(2--10a) go EF e2 1 EF
--2In -I ln I n (2-48)
a 7r T TTVF IH T

We note that the above result was obtained using the static form of the

interaction potential. We now return to the full dynamic potential and show that

the frequency dependence of the potential does not change the results given by

Eqs. (2-46) and (2-47), to logarithmic accuracy. For a dynamic potential it is more

convenient to perform the integration over qj at the very end so that the potential

entering the calculation is of the 3D form

4xre2
V( W q) 472 (2-49)


g 2 + K2 qv z + 22
q,2+q +q2
42e 2vq2 + w2
q {_+ K2 2 q2 2 2 (2 50)
I F z T n d.









where we used that q, < q_ and introduced a2 (q2) = q/ (qj + 02). The integral

over p' gives the same result as for the static potential. Integrating over q,, we

obtain for the effective ID self-energy instead of (2-45),

(2--12) e2 d2q1 e-qlzH 1 EF (1 n iFp
+ ( ) J (27[)2 ( + 2 a (q2) 27T 2 27T [1 + a (q)]

To log-accuracy, the integral over q_ is solved by taking the limits q_/K -- oo and

qlH -- 0 in the integrand and cutting the integral at q_ = r and q_ = 1H as the
lower and upper limits, respectively. In this approximation, a (q ) is replaced by

1, and the result for (2-12b) coincides with that obtained for the static potential,

Eqs. (2-46) and (2-47). Coming back to Eqs. (2-49) and (2-50), we can interpret

this result in the following way. The difference between the dynamic potential and

the static one is in the presence of the dynamic polarization bubble multiplying

K2 in the denominator of Eq. (2-49). If the potential is taken in the static form,

typical frequencies are related to typical moment as u ~ vFpq, which means that

this factor is of order of unity and K must be replaced by K* ~ K. But because the

final result for E depends on K only via a (large) logarithmic term, log( ln KfB ),

such a renormalization of K is beyond the logarithmic accuracy of the calculation.

2.2.1.2 Diagram Fig. 2-11(a).

+ k+p-p'



k e'

+ I +



Figure 2-13. The self-energy correction contained in diagram 2-11(a), denoted in
the text a 2--13)
the text as +









Diagram Fig. 2-11(a) is a Hartree counter-part of the exchange diagram of

Fig. 2-10(a). Separating the contributions of left- and right movers, the diagram

corresponding to backscattering at the static impurity potential is shown in

Fig. 2-13. Again, diagrams corresponding to forward scattering at the impurity

potential do not contribute to the conductivity and do not need to be considered

here. The diagram of Fig. 2-13 also includes backscattering at a Friedel oscillation.

Although this diagram contains a particle-hole bubble, it is more convenient to

label the moment as shown in Fig. 2-13, integrate over p' first, and then over k.

For ', 1l i:-I I. lii,- the 1D potential of Eq. (2-41) becomes

V2 (m, q') V= (V q'+ 2kg)
= 2 In 1 (2kF)2 +K2 ln 2kF / ,T/ 12-51)
H .max {\q', UWm /UF,TI/VF})

where the last line is valid to logarithmic accuracy. As a first approximation, we

neglect the q-dependence of the interaction potential, replacing VT2 in Eq. (2-51)

by a constant V --2 2g2kpVF. This results in
EF
R-1) ( ) -2 92k T V (2-52)
7 2 (2 m Fn) + VFp

-2I [n E i- iF + (2-53)
27r7- 27T 2 47T )

which, up to a sign and overall factor of the coupling constant, is the same as the
R}{(2--13) in Eq. (245).
exchange contribution 213) in Eq. (2 45).

When the dependence of V'F on q' is restored, the result in Eq. (2-53)

changes only in that the coupling constant acquires a weak T- dependence


g2kF g92k (T) e2/2v1) ln 1 [(2kp)2 + 2Inn E/T] I.


(2-54)









Calculating the contribution of Eq. (2-53) with Eq. (2-54) to the conductivity,

we find the correction to the conductivity from diagram Fig. 2-11(a) to be:

6c(2--lla) e2 24k+ 2 In E/Tr EF
a 27VF Kk2 T

Notice that in the limit of very low T and/or very strong fields, the screening

wavevector drops out of the result and the net T-dependence of the conductivity

becomes In x In (In x) where x Ep/T.

2.2.2 Vertex Corrections

Two other important diagrams are the ones in Fig. 2-10(b) and Fig. 2-11(b).

These are the vertex corrections counterparts of the self-energy diagrams in Fig.

2-10(a) and Fig.2-ll(b), correspondingly.

2.2.2.1 Diagram 2-10(b)

Diagram 2-10(b) can be shown to give the same contribution as 2-10(a). In

this Section we show this by reducing diagram 2-10(b) to 2-10(a) without doing

explicit integration over q, and Matsubara summations.

Decomposing diagram 2-10(b) into contributions from left and right fermions,

we obtain

6 (2--0b) 2 lm T 2 M M_ V.D (, q,) im.Q+i
r 2w/ Q-O i72 2wQ 1q
wl En
(2-56)

where M are the vector vertices


M J G(En, p)G(E Qm,p)G(E 1, p q,).
27T

It is obvious that M_ = -M. When evaluating M1, one needs to consider all

choices of signs for Matsubara frequencies. For the cases


En > 0, n F- < 0, and F l < 0; (2-57)

En > 0, En m, < 0, and Fn ,l > 0, (2-58)









we have


M+1 / (2-59)
1
VF(Q + 1/T)[1*L T I + 1/T] ,

M( +/)[( ) /' (2-60)

respectively. For all other cases, the results can be shown either to vanish because

of the locations of the poles or to cancel each other. In the ballistic limit, the

product M+M_ can be simplified in both cases to

1
M+ M_ = (2-61)
M ( + 1/')2[w + 2, (2

The subsequent integration of this expression gives a Icu l--singularity and it is this

singularity which gives the In T-dependence of the correction to the conductivity.

Now we go back to diagram 2-10(a). In Sec. 2.2.1.1, we found the contribution

of this diagram by evaluating the self-energy first and then substituting the result

into the Kubo formula. To prove the equivalence of diagrams 2-10(a) and (b) it is

convenient to consider the full diagram 2-10(a) without singling out the self-energy

part. Summing up the contribution of left and right fermions, we obtain

(2__10a) 2v 1 (e22v2 2
i(2-"-10a) _- T t)2 PVD C1F ) im-(O,+i,,
-T 27 H U L 27- 7
(2-62)

where


P = G (,, p)G (E Q, p)G (E W,,p q,) x
27T

S G (En, p') GT (E z)

A non-zero result for Eq. (2-62) is obtained only for the case given in Eq. (2-57),

when













P++- P_ = + 2+ Q, (2-63)
vF (m + 1/L)2 [ + I i *+ i/ T[iu I i *i + 1Tr]

with


Q = (q -q,) *
V 2(i i, + i/r)2 (i( n) i )(i + i, + i/T) (q

Neglecting the q,-dependence of VID, we see that f 1. 0 0. An expansion in q

results in non-divergent terms which do not bring any non-trivial T-dependence.

Making a ballistic approximation in the rest of Eq. (2-63), we see that it coincides

with Eq. (2-61). The Matsubara summation goes over a twice smaller interval of

frequencies compared to that in Eq. (2-56). We see that Eqs. (2-62) and (2-56)

give the same result and thus


bo(2--10b) 6(2--10a). (264)


2.2.2.2 Diagram Fig. 2-11(b)

p-q
E-0o


C-0

E p-q p'q P P I~ P C



P e- --- p' e-n P C---- p' E-Q

(a) (b)

Figure 2-14. Diagram 2-10(b) vs diagram 2-11(b).


The diagram in Fig. 2-11(b) is a vertex correction counterpart of the Hartree

self-energy diagram Fig. 2-11(a), and it gives the same contribution as Fig.

2-11(a). To see this, we compare the diagrams in Figs. 2-10 (b) and 2-11(b)










labeling them as shown in Fig. 2-14. For a q- and cu-independent interaction,

diagram Fig. 2-14(b) is of the same magnitude but opposite sign as diagram Fig.

2-14(a). For a q-dependent interaction, the T-dependent parts of these diagrams

differ also in the overall factor of the coupling constant: diagram 2-14(a) contains

go whereas diagram 2-14(b) contains g2k,. Electron-electron backscattering in

diagram (a) and electron-electron forward scattering in diagram (b) give either

sub-leading or T-independent contributions. Thus


ba(2--11b) g 2F2kF (2--10b)
go


g2kF j(2--10a) 6(2--11)
go


2.2.3 Sub-Leading Diagrams

All other diagrams give sub-leading contributions. Diagram Fig. 2-10(c) gives

a self-energy type contribution to the conductivity so we use Eq. (2-42). If the

interaction potential is taken to be static, the contribution from this diagram is

zero. Using the dynamical potential, the leading contribution from this diagram is

a In T-correction to the conductivity

ba(2--1c) 1 C2 E.
IjC 2-6


-- .1T"
a 27 vF T

This contribution is smaller than that from diagrams Fig. 2-10(a) [Eq.(2-48)]

and Fig. 2-11(a) [Eq.(2-55)] (and diagrams Fig. 2-10(b) and 2-11(b)) by a

T-independent log-factor.

Diagrams 2-10(d) and 2-10(e) give mutually canceling contributions of the


)uu


form:


r(2--10d) e2 1
----- (2-67)
a 24vF TTr
6b(2--10c) e2 1
2-4 go-" (2-68)
a 24 1T

Each of these contributions is small since we are in the ballistic limit (Tr > 1).


(2-65)


*\









All the calculations shown here are done considering the dynamic interaction

potential at small frequencies. At high frequencies, i.e., at frequencies close to the

magnetoplasmon frequency, the contributions from all the conductivity diagrams

cancel out. That this has to be the case was pointed out recently in Ref. [19].

This is a very useful result because each individual diagram, taken separately,

may give singular corrections. In our case we have also explicitly checked that

this cancellation indeed occurs. Contributions from diagrams (2-9a), (2-10a) and

(2-10c) cancel each other. Contribution from (2-9b) cancels that of (210b), and

finally (2-10d) and (2-10e) cancel each other.

2.2.4 Correction to the Conductivity

Adding up the results from Eqs.(2-48), (2-55), (2-64), and (2-65), we find the

leading correction to the conductivity

c6 6,(2--10a) 7(2--11a) a(2--10b) a(2--11b)
+ + +
2 4p +2 In EF/T EF
In In (2 69)
TWV, K2 T

Eq. (2 -69) is the main result of this Section.

2.2.5 Effective Impurity Potential

The fact that only four diagrams, Figs. 2-10(a,b) and Figs. 2-11(a,b),

determine the leading correction to the conductivity -i.-.-. -I- that there must be

some simple reason for these diagrams to be the dominant ones. Indeed, only these

diagrams arise if one considers scattering of electrons by "effectil impurities that

consist of a combination of bare impurities and the Coulomb fields of electrons

surrounding the bare impurities. For weak delta-function bare impurities, the

effective impurity potential corresponds to "d' --:i, the impurity with the mean

field of Hartree and exchange interactions (see Fig. 3-1).


Vo(, p, p') = Vo + VH(p p') + V,(, p,p').









The first term in this equation is the strength of a bare impurity, the second one is

the Coulomb potential of electrons whose density is modulated due to the presence

of the bare impurity, and the third one is an exchange potential for electrons

interacting and scattering through a weak impurity.




X X x +



Figure 2-15. Effective impurity potential


Due to the exchange contribution, the effective impurity potential is non-local,

and it may depend on the energy, if the interaction is dynamical. Performing the

impurity averaging, we obtain the correlation function of the effective impurity

potential


C = nVo(, p,p') 2 = nV2 + 2nVo[VH(p p') + V(E, p,p')] + O(g2), (2-70)


where g = e2/vF is the interaction strength. Diagrammatically, C corresponds

to a dashed line of the cross-technique [18]. The first term (bare impurities) is

taken into account in the leading order in 1/EFr < 1 by summing infinite series

for the single-particle Green's function and then using the Kubo formula for

the conductivity. Because the bare impurities are short-range, there is only one

diagram for the conductivity-the usual !i iil!." diagram; the vertex correction

to this diagram vanishes. Corrections to the conductivity result from the Hartree

and exchange terms in Eq. (2-70). To first order there are two diagrams, shown

in Fig. 2-16. Although the bare impurity is point-like, the Hartree and exchange

potentials it generates have slowly decaying tails and also oscillate in space. Thus

the vertex correction, Fig. 2-16(b), is not zero. The self-energy diagram, Fig.









2-16(a), corresponds to two diagrams: Fig. 2-10(a) and Fig. 2-11(a). Diagram

Fig. 2-16(b) corresponds to the diagrams in Fig. 2-10(b) and Fig. 2-11(b). For

an arbitrary impurity potential, it can be shown that contributions of 2-16(a) and

2-16(b) coming from forward scattering cancels each other. For backscattering, the

contribution from 2-16(a) and 2-16(b) are the same.









(a) (b)



Figure 2-16. The handle diagram corresponds to diagrams 2-10(a) and 2-11(a) and
the crossing diagram corresponds to 2-10(b) and 2-11(b).


2.3 Impurity Scattering Cross-Section for Interacting Electrons

In this Section we apply a different approach to the conductivity of interacting

electrons in the UQL. In Section 2.2.5 we demonstrated that, to first order in

the interaction, the only diagrams which are important for 6a correspond to

scattering at an effective impurity potential. This -i-i.:- -I that the result for

6a can be obtained by calculating the interaction correction to the impurity

scattering cross-section and then substituting the corrected cross-section into the

Drude formula. In this section we show that to first order this procedure gives

a result identical to that of the diagrammatic approach of Sec. 2.2. Unlike the

diagrammatic series in the interaction for the conductivity, the perturbation

theory for the scattering cross-section can be summed up to all orders via

a renormalization group procedure. This will lead to a Luttinger-liquid-like

power-law scaling of the conductivity, discussed at the end of this section.









2.3.1 Non-Interacting Case

For electron scattering off an impurity potential Vimp(r) that is axially

symmetric about the direction of the magnetic field, the component of the

electron's angular momentum is conserved. In particular, any spherically symmetric

impurity satisfies this condition. For this reason, it is convenient to work in the

symmetric gauge, where the basis of single-electron states is labeled by pz, the

momentum in the direction of the magnetic field, and mz, the projection of the

angular momentum in the magnetic field direction:

CipzZ
'Pp ,m (r) Xm (C),
VILi

where = (x + iy)/1H and


Xm.() 1= 1m ('z exp(- (2/4) (2-71)
1HV 2m z+1xmz

with m = 0, 1, 2....

Electrons are restricted to the lowest Landau band and therefore there are only

two types of scattering events: forward and backward. Only backscattering events

contribute to the scattering cross-section, which can be written as A where

N is the number of electrons backscattered per unit time and J is the total flux

of incoming electrons. Using a Landauer-type scheme, the scattering cross-section

in each channel of conserved m, can be related to a reflection coefficient in this

channel via A, = 271i l r, 2 The total cross-section is obtained from the sum of

the cross-sections in each channel [71]:
00
A 2z |rFm2 (2-72)
m,=0









The coefficients rF are the reflection amplitudes of ID scattering problems, given

by a set of 1D Schrodinger equations

t 82
m 2 + vmz(z) () (+E ( /2) (z)

with effective 1D impurity potentials Vm,(z) = (ml Vip(r) m,) obtained by

projecting the impurity potential on the angular momentum channel mz. The

kinetic energy of the electron is denoted by p/2m. The cross-section A is

related to the backscattering time via the usual relation, 1/RH = ivFA, where ni

is the density of impurity scattering centers. When the electric field is along the

magnetic field and for T = 0, the corresponding component of the conductivity is

related to TH via

az = e21DVFTH/2. (2-73)

An impurity of radius a < 1H can be modeled by a delta-function: Vmp(r)

Vob(r). For a delta-function potential, only the m, = 0 component of Vmz (z) is
non-zero, Vm (z) = (Vo/2712l)6m ,o0(z). In this case, the scattering cross-section for

non-interacting electrons is simply
A= (V/~12 )2
A Am 0 2/02 1 2 122 (Vo/2711)2
A=A,0-o=2lro -27 1 2) 2 + (2-74)
H (Vo/21)2 + vz

where v, = pz/m. Consequently, at T = 0 the conductivity is given by

n e2 1 2i NiVF
2ol+'. (2-75)
z ni mv2 27lf Vo1

In the Born limit (when Vo < 27v pF) we recover the result for the conductivity as

found by using the Kubo formula for weak, delta-correlated disorder [Eq. (2-73)].

In the opposite (unitary) limit A = 27ln and


z, = e2/4 2/ 4 / .


(2-76)









2.3.2 Interacting Case

Now we turn to the calculation of interaction correction to the scattering

cross-section A. For an effective delta-function impurity potential Vmp(r) Vo6(r),

only the m, = 0 component is scattered. The free-electron wave function in this

channel is given by:
Pm(z)' 0Xm, O()
VzL
where far away from the impurity site, the .,i-i:,'l 1 ic form of the z-component of

the unperturbed wave-function is:


Sto ep z > 0
eiPz + r0 e-ipz Z < 0
(2-77)
e-ipzz + ro eip z > 0
to e-iz z < 0

By calculating the electron-electron interaction correction to the wave function,

one obtains the correction to amplitudes to and ro, and therefore to the scattering

cross-section via Eq. (2-72). Since now the problem has been mapped onto a 1D

scattering problem [21, 24], one can anticipate that this interaction correction has

an infrared logarithmic singularity, as it does in the pure 1D case.

The 1D nature of the system in the UQL is also clearly manifested by the

behavior of the Friedel oscillations around the impurity. The profile of the electron

density around the impurity site is given by Sn(r) = f dr'I(r, r')Vtp(r'), where

II(r, r') is the polarization operator. For a weak delta impurity potential, we obtain


n(r) no 2 ) sin( z exp(-r2/21 ), (278)

which shows only a slow, 1/z decay (see Fig. 2-17), characteristic of one-dimensional

systems (in contrast to the 1/r3 decay in 3D systems). Correspondingly, the

Hartree VH(r) and exchange Ve(r, r') potentials, that an incoming electrons feels









when being scattered from an impurity, also exhibit 2pF-oscillations and decay

as 1/z away from the impurity and along the magnetic field direction. In the

transverse plane, the density, and thus the potentials, have Gaussian envelopes

which fall off on the scale of the magnetic length (see Fig. 2-17).



















Figure 2-17. Profile of the Friedel oscillations around a point impurity in a 3D
metal in the UQL. The oscillations decay as 1/z along the magnetic
field direction and have a Gaussian envelope in the transverse
direction.

The interaction correction to the wave function due to the Hartree and

exchange potentials is


6p =o(r)= dr'G(r, r'; E) dr" [VH(r"(r' r") + Ve(r', r")] ,, o(r")
(2-79)

As discussed in the previous section, for the UQL the Green's function is the

product of a longitudinal (1D) and a transverse part, G(r, r'; E) = GlD(, z'p)Gi(rr, r'),

where the .*i-mptotic form of the longitudinal part as z oo is

S'' <
GD(Z, Z')= -+- <0 (2-80)
1pz ipz(z-z') + (z+z') z' > 0








and, in the symmetric gauge, the transverse part is

1 e (1(12 + (' 2(*(')4 (281)
Gi(ri, r) Z- Y X (r )x (r) 21 exp (2-81)
mr
For z > 0, Eq. (2-79) directly gives the correction for the transmission
amplitude t. We first consider the exchange potential,

V(r, r') -V(r- r') > j p [z,4 (r')]* z,(r) (2-82)

which can be factored as

V(r, r') -V(r r')f(z, z')Gi(ri, r) (2-83)

where

f zz') j P { [d ()] 0 (z') + [ O_ (z)]* v' (z')} (2 84)

From the form of f(z, z') one can see that the exchange potential also has terms
with 1/(z z') and 1/(z + z') decay. For example, for z, z' > 0,

sin pp( z') (eiPF(-) 1) (e- ipp(-z) 1)
f (z,z') + ro 2_- r*o (2-85)
7(z z') 27i(z + z') 2i(z + z')

The 1/(z + z') decay leads to a log-divergent correction to Itl. Decomposing the
screened Coulomb potential V(r r') into Fourier components, all the dependence
of Eq. (2-79) on the transverse coordinates rI can be collected into the factor

Tm o(ri) dr' i drGi(rir')Gi(r',r")e--iq .(r r)x(rrr') (2 -86)

Performing the integrals which appear in Eq. (2-86) for the exchange contribution,
we find that the part containing perpendicular coordinates simply enters the
interaction correction as a form factor:

Tm o(ri) Xm o(ri) exp (-q2H (2-87)









Therefore, the transverse part of the free wave function Xm,=o(r) simply remains

unchanged in the rhs of Eq. (2-79). The remaining exponential term appears in the

definition of the effective 1D potential, as in Eq. (2-40)

VlD (q) Jd V(q', q1) exp q .2H (2-88)

The same result is obtained for the transverse part of the Hartree contribution in

Eq. (2-79). Once the transverse part is solved and the effective 1D potential is

defined, the rest of the calculation is exactly equivalent to the calculation of Yue et

al. [21] for tunneling of weakly-interacting 1D electrons through a single barrier.

The interaction correction to the transmission amplitude t is directly obtained from

the correction to the wave function, Eq. (2-79). Just as in 1D, a logarithmically

divergent correction for t is obtained from the longitudinal part of this equation,

after integrating over z and z'.

It is straightforward to see why there is a log-divergent term. The Hartree

term of Eq. (2-79), after integration of the transverse coordinates, is


Z', (z) = dz'GID(z,z')VH(z', (z') (2-89)

where


VH(z) = d VWD(q )e-iqz(z-Z' ) (2-90)
Jo J-oo 27

Let's consider for simplicity |ro0 1 and Ito
gives GlD(z, z') = (2m/pz) exp(ipzz) sinpzz'. The Hartree potential behaves as

VH(z) V1D(2PF) sin(2pFZ)/z so that Eq. (2-89) gives

t to [0 sin(2ppz') / (2 91)
-o sin(pzz')V1D(2pF) s (PF) z (2-91)
to o Z/

The 1/z term gives a logarithmic singularity only in the limit pz pF, so

that Jt/to oc VD(2pF) ln[1/(p PF)]. The Hartree contribution corresponds










to enhancement of to. The exchange contribution has opposite sign and is

proportional to V1D(O). The general answer, can be written as [72]

S-a |r0o2 In (2-92)
to 1HIPz PFI

where a = (go g2k)/2r, and go and g2kF are defined in Eqs. (2-43) and (2-54),

respectively.

0


3 T

0 \0 /

(CO \ /P
a T


0 0 )
0"^- -- _____- ^
1/T W
temperature T


Figure 2-18.


Renormalized conductivities parallel (a,,) and perpendicular (axx)
to the direction of the applied magnetic field. Power-law behavior is
expected in the temperature region 1/7 < T < W.


The second-order contribution to the transmission amplitude was calculated

explicitly in Ref. [21]. The higher-order contributions can be summed up by using

a renormalization group (RG) procedure. Without repeating all the steps of Ref.

[21], we simply state here that in our case the transmission amplitude satisfies the

same RG equation, as in the purely 1D case. i.e.,

dt
at (1- t12), (2-93)









where In (1/ Ip pI 1H) and t( 0) to. The solution of Eq. (2-93) is

tto ) (p pF)H a"
;(0)2 + t2 (p pF)H 2a

The renormalized cross-section is given by Eq. (2-72), but now written in terms

of the renormalized reflection coefficient r|12 1 t12 The final result for the

conductivity can be cast in a convenient form by expressing the bare reflection and

transmission coefficients via bare conductivities in the Born and unitary limits, aoz

and a zu, given by Eqs. (2-75) and (2-76), respectively:

0T2
u0 + (0 u0 ) (2-94)


where W is an ultraviolet cut-off of the problem and a = (go g2kF)/2r, and go

and g2kF are defined in Eqs. (2-43) and (2-54), respectively which shows that a

scales with the magnetic field as a ~ H In H. We are interested in temperature

dependence of the conductivities due to electron-electron interaction corrections

and we assume here that the bare conductivities ao- and ao, have only weak

T-dependence which can be neglected.

Eq. (2-94) is the main result of this Section and is shown in Fig. 2-18. It has

a simple physical meaning: At T = W, the conductivity is equal to its value for

non-interacting electrons. At temperatures T < W, the conductivity approaches

its unitary-value limit, which means any weak impurity is eventually renormalized

by the interaction to the strong-coupling regime. However, if the impurity is

already at the unitary limit at T = W, it is not renormalized further by the

interactions. We emphasize that Eq. (2-94) is applicable only for high-enough

temperatures, i. e., T > max [1/7-, A]. The first conditions is necessary to

remain in the ballistic (single-impurity) regime, the second one allows one to

consider only the renormalization of the impurity's scattering cross-sections by the

interaction without renormalizing the interaction vertex. The latter process leads









eventually for a charge-density-wave instability at a temperature T ~ A, where

A is the charge-density-wave gap [1-3]. For the power-law behavior [Eq. (2-94)]

to have a region of validity, there should be an interval of intermediate energies

in which the renormalization of the interaction coupling constants due to CDW

fluctuations is not yet important but the corrections to the cross-section leading

to the formation of power-law is already significant. Such an interval exists for a

long-range Coulomb interaction (I KIH < 1) both for the conductivity and the

renormalization of the tunneling density of states [6].

The dissipative conductivity in a geometry when the current is parallel to

the electric field but both are perpendicular to the magnetic field, axx, occurs

via jumps between .,-li i,:ent cyclotron trajectories. In the absence of impurities,

electrons are localized by the magnetic field and a,, = 0. In the presence of

impurities, a,, is dir. /hl; rather than inversely, proportional to the scattering rate.

In particular, for short-range impurities, Ua, oc 1/T oc U- 1 and the temperature

dependence of Ua, is opposite to that of azz In the scaling regime, az oc T2" and

a,, oc T-2,. This situation is illustrated in Fig. 2-18, where a = (go g2k)/27,

and go and g2kf are defined in Eqs. (2-43) and (2-54), respectively which shows

that a(H) ~ H In H. In the next section we discuss possible experimental studies

for observing the localization and correlation effects mentioned in the first three

sections.

2.4 Experiments

For experimental observation of the effects described here, the right choice of

material is crucial. Firstly, a low-density material is needed so that the UQL may

be achieved at feasible magnetic fields. For a good metal, the quantizing field is too

high (of the order of 104 Tesla). Semi-metals, such as bismuth and graphite, and

doped semiconductors have low carrier density and quantizing fields of the order

of 1 10 Tesla. Another important condition for observing the Luttinger-liquid









like behavior is that the systems be relatively clean, so that there is a sizable range

of temperatures in which the system is in the ballistic regime (1/7 < T < EF).

This rules out doped semiconductors [73-75] since the charge carriers come from

dopants which act as impurity centers in the system. An additional condition for

occurrence of the power-law scaling behavior and formation of charge-density-wave

or Wigner crystal, is that the electron-electron interaction is strong enough.

Bismuth i --i I- can be made extremely pure; however, the charge carriers in

bismuth are extremely weakly interacting due to a large dielectric constant (~ 100)

of the ionic background. Therefore, the log-corrections calculated here can be

estimated to be very small and would be difficult to be observed experimentally.

C'i ge-density wave instability have been observed in graphite [4] -1 --. i ii-; that

interaction of charge carriers in this system is important in strong magnetic fields

and at very low temperatures. Thus graphite would be an ideal material to observe

the correlation and localization effects mentioned here. Below we present some

recent experimental results of transport measurements in graphite first in weak

magnetic fields [76] and then in ultra quantum regime and try to interpret them in

view of our findings.

Graphite has a low carrier density, high purity, relatively low Fermi-energy

(~ 220K), small effective mass (along the c-axis) and an equal number of electrons

and holes (compensated semi-metal). The metallic T dependence of the in-plane

resistivity in zero field turns into an insulating like one when a magnetic field of the

order of 10 mT is applied normal to the basal (ab) plane. Using magnetotransport

and Hall measurements, the details of this unusual behavior were shown [76], to

be captured within a conventional multiband model. The unusual temperature

dependence di-,1 i.' 1 in (Fig. 2-19) can be understood for a simple two-band case











20E-6
E 15


1E-6 o "
E 0 0 o--
T (K)
C Temeraur deedec of the ai

P 1 E-7 P 0 mT
20 mT
.P 2)2H40 mT
v 60 mT
o 80 mT
A 100 mT
1E-80 v 200 mT
0 70 140

T (K)

Figure 2-19. Temperature dependence of the ab-plane resistivity p,, for a graphite
crystal at the c-axis magnetic fields indicated in the legend


where p,, is given by [77],

SPP2(P + 2) + (P22 + P212)H2
p (p1 + p2 + ^) 2 + 92H2

with pi, and Ri = 1/qin (i = 1, 2) being the resistivity and Hall coefficient of the

two ii P ii fly electron and hole bands, respectively. At not too low temperatures

(where the measurements were performed) electron-phonon scattering is the

main mechanism for the resisitivity in the band. Assuming that p1,2 oc T" with

a > 0, we find that for perfect compensation, (R = -R2 = IRI), Eq. 2-95

can be decomposed into two contributions: a field-independent term oc T' and

a field-dependent term oc R2(T)H2/T. At high T, the first term dominates and

metallic behavior ensues. At low T, R(T) oc 1/n(T) saturates and the second

term dominates, giving insulating behavior oc T-". Although this interpretation

explains the qualitative features of the field induced metal-insulator behavior shown

in Fig.2-19, the actual situation is somewhat more complicated due to the presence









of a third (minority) band, T dependence of the carrier concentration and imperfect

compensation between the i, Pi i ily bands. For more details see Ref. [76].

Let us now direct our attention on transport measurements in the ultra

quantum regime in which we expect to see the power law conductivity behavior

similar to what is shown in Fig. 2-18. Below we present some recent data on the

same graphite samples on which the weak-field measurements were performed.


4.8E-2
8T
10T
12 T
14T
16T
17.5 T
c 4.0
N
N




3.2
0 5 10
T (K)

Figure 2-20. Temperature dependence of the c-axis conductivity a,, for a graphite
crystal in a magnetic field parallel to the c axis. The magnetic field
values are indicated on the plot, with the field increasing downwards,
the lowest plot corresponds to the highest field


Within the experimentally studied temperature range (5K < T < 10K), the

c-axis conductivity exhibits an '-:, ,l.i.:, linear temperature dependence, a, oN T

as shown in Fig. 2-20, whereas the transverse resistivity exhibits a metallic power

law temperature dependence, px oc T1/3, as shown in Fig. 2-21 and Fig. 2-22.

Although the insulating sign of the temperature dependence (for za, see Fig.2-18)

is consistent with the model of a field-induced Luttinger liquid, the independence

of the exponent of the field is not. The slope (exponent) of both a,, and px, (for

various magnetic fields), are independent of the magnetic field whereas in the










25
20
15

N 10 T
D 4T
S6T
8T
.5 io. o
14T
16T
17.5T

1 10 100
T(K)

Figure 2-21. Temperature dependence (log-log scale) of the ab-plane resistivity
scaled with the field p,,/B2 for a graphite crystal at the c-axis
magnetic fields indicated in the legend


field-induces Luttinger liquid model the exponent of the power law should depend

on the field (see Eq. 2.94, a ~ H In H). Also, the exponents of the T scaling in a,,

(which is 1), and pxx (which is 1/3) are different (as seen in experiments) whereas

they were predicted to be the same in the Luttinger liquid model.

We are going to argue that the unusual temperature behavior of azz and pxx,

can be understood within a model which includes phonon-induced dephasing of

one-dimensional electrons (in the UQL) and the correlated motion in the transverse

direction due to the memory effect of scattering at long ranged disorder. Before

we get into the details of the model, let us keep in mind a few numbers for the

system we are about to describe. For our graphite samples the Fermi energy is

EF = 220K, the Bloch-Gruniesen temperature (which separates the region of T and

T5 contribution to the resistivity) is o = 2kFs ~ 10K and the Dingle temperature

(which gives the impurity scattering rate) is 3K. Also the transport relaxation

time is much longer (by a factor of 30), than the total scattering time (or life time)

Ttr > T, indicating the long range nature of the impurities.










1800
H=17.5T
1500 slope =0.312(1)

1200

S900


600

1 10 100
T (K)

Figure 2-22. Temperature dependence (on a log-log scale) of the ab-plane
resistivity p,,/B2 at the highest attained c-axis magnetic field of
17.5T for the same graphite ( i-- I 1

We first outline an argument by Murzin [73] which shows that the transverse

motion of the electron is correlated due to drift motion in a crossed magnetic and

electric field. The disorder model is assumed to be ionized impurity type and is

therefore long ranged. The transverse displacement (after a single scattering act)

is assumed to satisfy ri < 1H < rD (rD being the screening radius). Electrons

are assumed to diffuse in the z direction. An electron re-enters the region where

the impurity's electric field is -I i, i-. (rl < rD), many times as it moves in

the transverse direction. Thus electron's motion in the transverse direction is

correlated. Only after an electron has traveled a distance greater than rD in the

transverse direction, its motion becomes diffusive with the diffusion coefficient

D,, rD2/TD. We estimate TD (the time in which electron has moved a distance

rD I to H) by finding the transverse displacement AX(t) for AX(t) < rD, and
then obtain TD by setting AX(TD) rD. The probability to find an electron again