Role of Defects in the Supersolid Phenomena

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Role of Defects in the Supersolid Phenomena
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Goswami,Debajit
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Doctorate ( Ph.D.)
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Physics
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Dorsey, Alan T
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Lee, Yoonseok
Kumar, Pradeep P
Fry, James N
Phillpot, Simon R

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arnoldi -- biconjugate -- bose -- burger -- coulomb -- dipole -- edge -- elasticity -- exact -- ginzburg -- hartree -- jacobi -- lattice -- mean -- non -- perturbation -- phase -- screw -- semiclassical -- solid -- statistical -- strained -- superfluid -- supersolid -- topological -- variational -- weakly -- xy
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Abstract:
We investigate the role of topological defects, particularly dislocations, in the observed phenomena of supersolidity - a state of matter with simultaneous long range crystalline order and superfluid behavior. First, we determine bound state solutions of the 2D Schr\"odinger equation with a dipole potential originating from the elastic effects of a single edge dislocation. Along with a variational estimate of the ground state energy, we numerically solve the eigenvalue problem and calculate the energy spectrum using methods of exact diagonalization and basis expansion. A comparison of the behavior of the calculated energy spectrum and that obtained from semiclassical considerations is briefly discussed. The quantum mechanics of the inverse square screw dislocation potential is briefly discussed and the ground state energy values calculated for different cut-offs to the potential. We next propose an explanation for the heat capacity peak observed by Lin {\it et al.} Nature {\bf{449}}, 1025 (2007), by developing a lattice gas model of desorption (absorption) of $^3$He impurities from dislocations in solid $^4$He. The thermodynamics of the model is discussed and the associated heat capacity is calculated. We find that for various $^3$He concentrations and suitable dislocation densities the heat capacity shows quantitative agreement with the experiment, suggesting that the specific heat peak observed in the experiments may be due to a Schottky anomaly. Finally, within a phenomenological Landau theory, we study the effect of an edge dislocation in promoting superfluidity in a Bose crystal. We couple the elastic strain field of the dislocation to the superfluid density, and use a linear analysis to show that superfluidity nucleates on the dislocation before occurring in the bulk of the solid. Moving beyond the linear analysis, we develop a systematic perturbation theory in the weakly nonlinear regime, and use this method to integrate out transverse degrees of freedom and derive a one-dimensional Landau equation for the superfluid order parameter. We then extend our analysis to a network of dislocation lines, and derive an $XY$ model for the dislocation network by integrating over fluctuations in the order parameter. Our results show that the ordering temperature for the network has a sensitive dependence on the dislocation density, consistent with numerous experiments that find a clear connection between the sample quality and the supersolid response.
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by Debajit Goswami.
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Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Dorsey, Alan T.

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ROLEOFDEFECTSINTHESUPERSOLIDPHENOMENA By DEBAJITGOSWAMI ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c 2011DebajitGoswami 2

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Tomyfamilyandfriends 3

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ACKNOWLEDGMENTS Iowemydeepestgratitudetomyadvisor,ProfessorAlanT.Dorsey,forhisexcellent guidance,encouragementandunabatedpatiencehehadexhibitedthroughoutthis work.Ithasbeenapleasuretoworkunderhissupervisionandexploretheexoticand challengingeldofsupersolids. MuchofthisworkhasbeenacollaborativeeffortwithmycolleaguesChi-Deuk YooandKinjalDasbiswas.Iwouldliketothankthemforexchangeofideas,resultsand fosteringateamspiritinourgroup. IwouldliketothankmycommitteemembersProfessorJamesFry,Professor PradeepKumar,ProfessorYoonseokLeeandProfessorSimonPhillpotforvaluable discussions,timeandconsideration.IamspeciallythankfultoProfessorYoonseokLee forstimulatingdiscussionspertainingtoourresultsduringthecourseofthiswork. IwouldliketoacknowledgetheNationalScienceFoundationfortheresearch fundingforpartofthiswork.IamgratefultoProfessorH.B.Thackerforkindlyproviding uswithaSchr odingerequationsolverbasedonthebiconjugategradientmethod.I amthankfultotheUniversityofFloridaHigh-PerformanceComputingCenterforits computationalresourcesforthenumericalstudiesinthiswork. Finally,Ithankmyparents,ManosijGoswamiandRinaGoswami,fortheirconstant support,faithandenthusiasmthroughoutmyacademiccareer. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................7 LISTOFFIGURES.....................................8 ABSTRACT.........................................10 CHAPTER 1INTRODUCTIONTOSUPERSOLIDS.......................12 1.1EarlyTheoreticalIdeas............................12 1.2FirstObservationsofthePhenomena....................13 1.3OverviewofExperimentalandTheoreticalResults.............14 2BOUNDSTATESOFDISLOCATIONS.......................26 2.1EdgeDislocations...............................26 2.1.1ElasticFieldandPotential.......................27 2.1.2GroundStateEnergyEstimation:VariationalApproach.......29 2.1.3NumericalMethodstoDetermineCompleteSpectrum.......32 2.1.3.1Realspacediagonalization.................33 2.1.3.2Coulombbasismethod...................35 2.1.4SemiclassicalAnalysis.........................37 2.2ScrewDislocation...............................39 2.2.1ElasticFieldandPotential.......................39 2.2.2GroundStateEstimate.........................40 3HEATCAPACITYOFSOLID 4 He..........................54 3.1ExperimentalMeasurementoftheAnomalousHeatCapacityofSolid 4 He54 3.2LatticeGasModelfor 3 HeImpuritiesinSolid 4 He..............55 3.3StatisticalMechanicsoftheLatticeGasModel...............57 3.4DiscussionofResultsObtainedFromtheModel..............60 4DISLOCATIONINDUCEDSUPERFLUIDITYINAMODELSUPERSOLID...67 4.1Introduction...................................67 4.2Superuid-DislocationCouplinginLandauTheory.............68 4.3LinearStabilityAnalysis............................69 4.4WeaklyNonlinearAnalysisNearThreshold.................72 4.5DerivationoftheDislocationNetworkModel.................75 5

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5CONCLUSION....................................87 APPENDIX AELASTICFIELDOFANEDGEDISLOCATION..................90 BELASTICFIELDOFASCREWDISLOCATION..................94 CFERMITEMPERATUREOF 3 HePARTICLESINTHE1DLATTICEGAS....97 DHEATCAPACITYFOR3DGASOF 3 HePARTICLESIN 4 HeLATTICE.....98 EMETHODOFSTRAINEDCOORDINATES....................100 FANALYSISOFALANDAUMODELWITHACOULOMBPOTENTIAL......103 GANALYSISOFATIME-DEPENDENTMODEL..................107 REFERENCES.......................................109 BIOGRAPHICALSKETCH................................114 6

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LISTOFTABLES Table page 2-1Summaryofgroundstateenergyestimatesoftheedgedislocationpotential..44 2-2Comparisonofrstfewenergyeigenvaluesobtainedfromdifferentnumerical methods........................................44 2-3Comparisonofresultsforedgeandscrewdislocations..............45 7

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LISTOFFIGURES Figure page 1-1Resonantperiodchangewithtemperatureofsolid 4 HeinVycorglassfordifferentrimvelocities..................................21 1-2Resonantperiodchangewithtemperatureofsolid 4 HeinVycorglassfordifferent 3 Heconcentrations..............................22 1-3Resonantperiodandamplitudeofoscillationofunblockedandblockedannularcelllledwithbulksolid 4 He...........................23 1-4Annealingeffectintheresonantperiod.......................24 1-5Shearmodulusanomalyinsolid 4 He........................25 2-1Schematicdiagramforanedgedislocation....................46 2-2Plotofgroundstateenergyversuscut-offparameter...............46 2-3Comparisonofeigenvaluesobtainedfromdifferentmethods...........47 2-4FitfortheeigenvaluespectrumobtainedfromJADAMILU............48 2-5Eigenfunctionforthegroundstate.........................49 2-6Eigenfunctionforthe1stexcitedstate.......................49 2-7Eigenfunctionforthe2ndexcitedstate.......................50 2-8Eigenfunctionforthe3rdexcitedstate.......................50 2-9Eigenfunctionforthe4thexcitedstate.......................51 2-10Eigenfunctionforthe10thexcitedstate......................51 2-11Eigenfunctionforthe23rdexcitedstate......................52 2-12Eigenfunctionforthe24thexcitedstate......................52 2-13Eigenfunctionforthe50thexcitedstate......................53 2-14Eigenfunctionforthe100thexcitedstate......................53 3-1Heatcapacityoffoursamplescontainingdifferentamountsof 3 Heimpurities.63 3-2Specicheatpeaksofthe1ppb,0.3ppmand10ppmsamples..........64 3-3Hysteresisandexcessheatcapacityofsolid 4 Hesampleswithvarious 3 He concentration.....................................65 8

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3-4Specicheatcapacityforvariousconcentrationsof 3 Heobtainedfromthe model.........................................66 4-1Schematicdiagramshowingthedislocationaxisandthetubularsuperuid regionthatdevelopsaroundit............................80 4-2Qualitativeplotof1Dcorrelationlengthfrommean-eldandHartree-Fockapproximation......................................81 4-3Orderparameter r versus r for =0 : 09 .....................82 4-4Orderparameter r versus r for =0 : 43 .....................83 4-5Orderparameteramplitude asafunctionof ................84 4-6Plotof = 2 1 = 2 asafunctionof .........................85 4-7Comparisonofthenumericalshootingmethod,naiveperturbationtheoryand strainedcoordinatesperturbationtheoryfor =0 : 43 ................86 A-1Figureshowinggeneraldislocationlineandforanedgedislocation,thevectors b and ....................................90 D-1Heatcapacityof 3 Heparticlesina 4 Helattice...................99 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ROLEOFDEFECTSINTHESUPERSOLIDPHENOMENA By DebajitGoswami August2011 Chair:AlanT.Dorsey Major:Physics Weinvestigatetheroleoftopologicaldefects,particularlydislocations,inthe observedphenomenaofsupersolidity-astateofmatterwithsimultaneouslongrange crystallineorderandsuperuidbehavior.First,wedetermineboundstatesolutionsof the2DSchr odingerequationwithadipolepotentialoriginatingfromtheelasticeffectsof asingleedgedislocation.Alongwithavariationalestimateofthegroundstateenergy, wenumericallysolvetheeigenvalueproblemandcalculatetheenergyspectrumusing methodsofexactdiagonalizationandbasisexpansion.Acomparisonofthebehavior ofthecalculatedenergyspectrumandthatobtainedfromsemiclassicalconsiderations isbrieydiscussed.Thequantummechanicsoftheinversesquarescrewdislocation potentialisbrieydiscussedandthegroundstateenergyvaluescalculatedfordifferent cut-offstothepotential. WenextproposeanexplanationfortheheatcapacitypeakobservedbyLin etal. [Nature 449 ,1025],bydevelopingalatticegasmodelofdesorptionabsorption of 3 Heimpuritiesfromdislocationsinsolid 4 He.Thethermodynamicsofthemodelis discussedandtheassociatedheatcapacityiscalculated.Wendthatforvarious 3 He concentrationsandsuitabledislocationdensitiestheheatcapacityshowsquantitative agreementwiththeexperiment,suggestingthatthespecicheatpeakobservedinthe experimentsmaybeduetoaSchottkyanomaly. 10

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Finally,withinaphenomenologicalLandautheory,westudytheeffectofanedge dislocationinpromotingsuperuidityinaBosecrystal.Wecoupletheelasticstrain eldofthedislocationtothesuperuiddensity,andusealinearanalysistoshowthat superuiditynucleatesonthedislocationbeforeoccurringinthebulkofthesolid. Movingbeyondthelinearanalysis,wedevelopasystematicperturbationtheoryin theweaklynonlinearregime,andusethismethodtointegrateouttransversedegrees offreedomandderiveaone-dimensionalLandauequationforthesuperuidorder parameter.Wethenextendouranalysistoanetworkofdislocationlines,andderive an XY modelforthedislocationnetworkbyintegratingoveructuationsintheorder parameter.Ourresultsshowthattheorderingtemperatureforthenetworkhasa sensitivedependenceonthedislocationdensity,consistentwithnumerousexperiments thatndaclearconnectionbetweenthesamplequalityandthesupersolidresponse. 11

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CHAPTER1 INTRODUCTIONTOSUPERSOLIDS Asolidmaybedenedasastateofmatterwithnon-zeroshearmodulus.In otherwords,asolidwillresistdeformationsthatchangeitsshapeatxedvolume.A crystallinesolidalsohaslongrangespatialorder.Asuperuidisastatewhichallows non-viscousmasstransportforinstance, 4 Hebelow2K.A supersolid isanexotic phaseinwhichthecrystallinesolidandsuperuidbehaviorcoexist. 1.1EarlyTheoreticalIdeas Thehistoryofsuperstatesofmatterdatesbacktoonehundredyearswhenin1911, KammerlinghOnnesrstdiscoveredthephenomenaofsuperconductivity[1]inmercury. ThiswasfollowedbythediscoveryofsuperuidityinliquidHeliumsimultaneouslyby Kapitza[2],andAllenandMisener[3]in1938. Speculationsofastateofmatterwithcoexistinglongrangeorderandsuperuidity weretheoreticallyinvestigatedasearlyas1956byPenroseandOnsager[4].Theyused thedensitymatrixformalisminwhichtheexistenceofacondensateappearsasoff diagonallongrangeorderODLROandshowedthatinacrystalODLROcouldnot exist.However,itwaslaterpointedoutbyChester[5]thattheconclusionofPenrose andOnsagerisincorrectastheyhaveusednon-symmetrizedwavefunctions.In1969, AndreevandLifshitz[6]proposedtheBosecondensationofzero-pointvacanciesin a 4 Hesolid.Withtheideathatthesezero-pointvacanciesinasolid 4 Hecrystala BosesolidarealsoBoseparticles,theyarguedthatthevacanciesshouldundergo acondensationbelowacertaincriticaltemperature.Thissuperuidityofvacancies wouldallowmass-transportthroughthelatticewithoutdissipationandwillallowthe coexistenceofcrystallineorderasinsolidsalongwithaBosecondensateofvacancies asinasuperuid.AyearlaterChester[5]revisitedtheworkofPenroseandOnsager suggestedusingsymmmetrizedwavefunctionsthatasystemofinteractingbosons couldexhibitbothcrystallineorderandBose-Einsteincondensation. 12

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TheideathattheNon-ClassicalRotationalInertiaNCRIpossessedbysuperuid liquid 4 He,thephenomenonthatthemomentofinertiadecreasesbelowacritical temperaturefromitsclassicalvalue,maybesharedbysomesolidswasintroducedby Leggett[7].Heproposeddirectexperimentsliketherotationofasolidintheformofan annulusbelowacertaintransitiontemperature.Heexpectedadecreaseintheapparent momentofinertiawithtemperatureascomparedtotheclassicalinertiaandestimated theupperlimitofthesupersolidfraction s = tobe 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(4 Motivatedbythesetheoreticalideas,anumberofexperimentson 4 Hefollowedto ndthisexoticphaseofmatter.Theseexperimentsincludedtorsionaloscillator,mass owandsoundspeedmeasurementstondapossiblesupersolidsignature.However, noneoftheexperimentsfoundanythingsignicant.Areviewoftheexperimentalresults uptotheyear1992canbefoundinRef.[8]. 1.2FirstObservationsofthePhenomena Amajorexperimentalbreakthroughinthesearchforthesupersolidphasewasrst reportedbyKimandChanin2004[9].Theyreportedatorsionaloscillatormeasurement onsolidHeliumconnedinaporousmediumVycorandfoundanabruptdropin therotationalinertiaoftheconnedsolidbelowacertaincriticaltemperature.They interpretedtheirobservationsasthepossiblesupersolidphase. TheapparatususedbyKimandChanisatorsioncellconsistingofacylinder suspendedbyaverticalhollowrod,bothmadeofBe-Cu,containingall-lineforHelium. Thedetectionelectrodeproducesanacvoltagewhenthecelloscillatesandthedrive electrodekeepstheoscillationinresonance.Theresonantperiod wasmeasuredwith highaccuracyastheyusedahighQuality-factoroscillator 10 6 .Theperiodgivenby =2 p I=G; where I isthemomentofinertiaofthetorsioncelland G thespringconstantofthe Be-Cutorsionrod,ismeasuredasafunctionoftemperature.Theresonantperiodfor 13

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varyingvaluesoftherimvelocityofthecellasshowninFig.1-1wasmeasuredand remarkablytheyfoundadecreaseoftheperiodbelow 200mK.Thisdecreasewas observedbelowacertainrimvelocityofafew m/s,thussuggestingtheexistenceof acriticalvelocitythemaximumvelocityofsuperowwithoutanydissipationasin superuids.ThiswasinterpretedasLeggett'sNCRI,theresonantperioddecreases duetothemassdecouplingthatoccursattheonsetofsuperuidityinthecrystaland thesuperuidfraction S = orNon-ClassicalRotationalInertiaFractionNCRIFwas estimatedtobeabout1%. Theresonantperiodwasalsomeasuredforavarietyofsolid 4 Hesampleswith different 3 Heimpurityconcentrations,ataxedrimvelocityseeFig.1-2.Adecrease intheperiodwasalsoseeninthiscasealongwiththeobservationthattheeffect disappearsfor 3 Heconcentrationbeyond0.1%.Acontrolexperimentwithsolid 3 Hea Fermionicsolidhadalsobeenperformed,whichinsharpcontrasttosolid 4 Hedoesnot showaperioddrop.Onthebasisofalltheseobservations,itwasconcludedbyKimand Chanthatthelongelusivesupersolidphasehadbeendiscoveredinsolid 4 He. Inafollowingwork,publishedlaterin2004[10],thesupersolidargumentwas furtherstrengthenedbyKimandChanwhentheyreproducedtheNCRIeffectinbulk solid 4 He.Acontrolexperimentwasalsoreportedinthisworkusingablockedannulus whichdidnotshowanyperioddrop,suggestingthatsuperowinsolid 4 Heisirrotational asinasuperuidseeFig.1-3. 1.3OverviewofExperimentalandTheoreticalResults TheexperimentsbyKimandChaninspiredfurtherexperimentsonsolid 4 He. Todateseveralexperimentalgroupshaveindependentlyreproducedthetorsional oscillatorNCRIresults[11,12,14,15,16].Inthissectionwegiveanoverviewofmajor experimentalndingsalongwiththeoreticalapproachestounderstandtheputative supersolidstate.References[18,19]provideacomprehensivereviewonthestateof currentresearchinthiseld. 14

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AlongwithrecoveryofNCRIdata,theobservationthattherewasscatterbetween differentsamples,promptedthestudyofannealingeffects.Annealingisamethodto improvethequalityofcrystalsandisperformedbywarmingupthesamplecloseto themeltingpointforsometimeandcoolingdownslowlyafterwards.Therstsuch experimentswereperformedbyRittnerandReppywhofoundthemagnitudeofNCRIF greatlydecreasedandeveneliminated[11]throughannealingasshowninFig.1-4. ThissuggestedtothemthattheNCRIobservedintheseexperimentswasduetocrystal imperfections. RittnerandReppyfurtherstudiedtheeffectsofsampledisorderbyconningthe solid 4 Hesampleinanarrowannularregion[12]andrapidlyfreezingtothesolidstate. TheyobtainanNCRIFaslargeas20%inthiscase.Theyalsostudiedthesurfaceto volumeratioofothertorsionaloscillatorexperimentsandfoundamonotonicincrease ofNCRIFwiththisratio.Theyconcludedthatasthesamplethicknesswasdecreased solidicationwouldproceedmorequicklyandproducemoredefectsdislocations/grain boundarieswhichenhancethesupersolidsignals[13].Thisseriesofexperimentsrst pointedouttheimportanceofdisorderresultingfromsamplepreparation. ApartfromtheNCRIeffecttherehadbeenotherattemptstoprobethesignatureof superuidityinsolid 4 He.TherstsuchexperimentafterthediscoveryofNCRIwas anattempttodetectapressure-induceddcmassowbyDay etal. [20],insolid 4 He connedinVycorglasspores,andlaterinanarrayofcapillaries[21].However,noneof theseexperimentsprovidedanysignatureofsuchow.Inarelatedsetup,Sasaki etal. [23]didhoweverobserveadcmassowbuttheowwasinterpretedtobeduetothe superuidityofgrain-boundariesattheliquid-solidinterface.Also,theeffectmeasured, extendedbeyond1K,thussuggestingthatitmaynothaveanycorrelationwiththe NCRIeffect.Animportantcontributiontothesesetofexperimentsisthedetectionof unusualmasstransportinhcpsolid 4 Helaterin2008,byRayandHallock[24],viaan injectionof 4 Heatomsfromthesuperuiddirectlyintothesolid 4 HeusingVycorglass 15

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rods.Theowwasreportedtobepressureandtemperaturedependent:forpressures exceeding27barandtemperaturesbeyond800mKnoowwasobserved.Atlower pressuresandtemperaturesbelow400mKtheowwasobservedtobelinearintime, asexpectedforasuperuidmovingatacriticalvelocity. Adouble-resonancetorsionaloscillatorwasusedbyAoki etal. [14]tostudythe NCRIeffect.Thisallowedthemtostudythephenomenaattwodifferentfrequencies forthesamesample.TheyfoundnofrequencydependenceofNCRIontheonset temperature.ThisconcludedthattheNCRIeffectwasindeedcharacterizedbyacritical velocity,andnotduetoadisplacementamplitudeoracceleration.Theyalsoreported ahysteresisthatdependsontherimvelocity:atatemperatureof19mKtheNCRIF increaseduponloweringtherimvelocity,butsaturatedatthemaximumvalueastherim velocitywasagainincreased. Anelasticanomalyintheshearmodulusofsolid 4 HewasreportedbyDayand Beamish[22].Theyfoundthatasthetemperaturedecreasesbelowabout100mK, theshearmodulusisfoundtotypicallyincreasebyabout10%asshowninFig.1-5. Theyalsoobserveannealingandhysteresiseffectsonthemeasuredshearmodulus. Anincreaseinthe 3 Heconcentrationincreasestheonsettemperaturebelowwhichthe shearmodulusvaries.Itwasproposedthattheanomalyisduetodislocationsinsolid 4 He,whichbecomeimmobileatlowtemperaturesduetothepinningon 3 Heimpurities thusstiffeningthesolidandincreasingtheshearmodulus.Theseobservationsshare thesamecharacteristicsastheNCRIeffectsinthetorsionaloscillatorexperiments, howeverthesimultaneousstiffeningofthesolidiscounterintuitivewithitsdissipationlessowatlowtemperaturesanditisunclearasofnowastowhetherthetwoeffects havethesamephysicalorigin. Atruesupersolidphasetransitionwouldbeestablishedbyanequilibriumthermodynamicmeasurement,suchasspecicheatandameasurementofthesamewas 16

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conductedbyLin etal. [25].Theobservationofaheatcapacitypeakalongwithrelated experimentsisdiscussedindetailinChapter3. NeutronandX-rayscatteringonsolid 4 Hehadalsobeenperformed,althoughthe interpretationofresultsremaininconclusive.Neutrondiffractionstudiesinbccsolid 4 He showstructuraluctuations[26],whilefortemperaturesbelow140mKtheDebye-Waller factorinhcpsolid 4 Hehasshownnoanomalies[27]. Wenowhighlightsomekeytheoreticaldevelopmentintheeldaftertherst observationofNCRI.Asthenatureofmicroscopicinteractionsinthisexoticphaseis unknown,mostlytheseapproacheshadbeenphenomenologicalornumerical. Assumingthenormaltosupersolidtransitiontobecontinuous,Dorsey etal. [28] withintheLandautheorycoupledthesuperuidorderparametertotheelasticityofthe crystallinelattice.Theyfoundthatthecouplingdoesnotaffecttheuniversalproperties ofthesuperuidtransitionandhencea -anomalyinthespecicheatasexpectedfora superuidtransitionshouldalsoappearinthesupersolidtransition.Theyalsopredicted anomaliesintheelasticmoduliofthesolidnearthetransitionandlocalvariationsinthe superuidtransitionowingtoinhomogeneouslatticestrains. Therehadbeenbeentheoreticalmodelstoprovideanon-supersolidinterpretation totheseriesofexperiments.Onesuchtheory,proposedbyNussinov etal. [29], attemptstoexplaintheNCRIeffectintermsofglassphysics.Withintheframeworkofa linearresponsetheory,theyproposethatasmallliquid-likecomponenttransformsinto aglassatlowtemperatures,possiblyduetothequenchingoftopologicaldefects.Using ttingparameters,theyprovidetstoboththeperiodshiftandthedissipationpeakdata andalsoclaimtoexplaincertainfeaturesofthespecicheatexperimentbyLin etal. [25]. Abulksuperuidorderinginsolid 4 Heduetoaquenchededge-dislocationnetwork, withinaLandautheory,wasproposedbyToner[30].Thispossibilityhadearlierbeen exploredinaBosecrystalbyShevchenko[31,32].Anorderingtemperatureinterms 17

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ofthedislocationdensityareestimatedintheseworks.Westudyadislocationnetwork modelindetailinChapter4,inwhichweextendonexistingresults. Aviscoelasticmodelforsolid 4 He,wasproposedbyYooandDorsey[33],takinga frequencydependentcomplexshearmodulus.However,theyfoundthatthoughthey couldobtainagoodttothedissipationpeak,onlyabout10%oftheperiodshiftcould beaccountedforusingtheirmodel. Anderson[34,35]suggestedavortexliquidpicturetounderstandtheobserved phenomena.Inhisview,adissipativeregimedominatedbyquantizedvorticesismanifestedinthecurrentexperiments,thoughtruesupersolidbehaviormaybeobservedat lowertemperatures.Thispicturehadbeendevelopedinanalogytohigh-temperature superconductors,inwhichthereemergesaCooper-pairgapandthusananomalous diamagneticsusceptibilityabovethetruesuperconductingtransition. Morerecently,S oyler etal. [36]proposedasuperclimbactionwithamotivationto understandtheunusualmasstransportobservedintheRayandHallockexperiments [24].Intheirpicture,anedgedislocationinsolid 4 Heshowntohavesuperuidcore insimulationsreportedintheirwork,couldclimbbecauseofmasstransportalongits superuidcore.Thiswassuggestedtobethereasonbehindananomalousisochoric compressibilityassociatedwithsuperowintheexperiments. Alongsidetheabovementionedmodelstherehasbeenextensivenumericalwork, mostlyusingtheMonteCarlotechnique,tounderstandthesupersolidphase.Firstly, in2006,ClarkandCeperley[37]andBoninsegni etal. [38]usingPathIntegralMonte CarloPIMCmethodsshowedthatadefect-freesolid 4 Hecrystaldoesnotpossess ODLROandhencenosupersolidity.Thelaterworkalsofoundthatthedisorderedsolid however,freezesintoasuperglassphaseandfeaturesODLRO. Followingthis,thesimulationshadtriedtoinvestigatetheroleofvacanciesand topologicaldefectssuchasdislocationsandgrainboundariesinsolid 4 He.Boninsegni etal. [39]throughtheirPIMCmethodsfoundthatvacanciesareunstableincrystalline 18

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4 Heandphase-separateintovacancy-richphaseandperfect,insulatingcrystal.Thisis contrarytotheAndreev-Lifshitzscenario[6]ofvacancy-inducedsuperowinaquantum crystal. Pollet etal. [40]simulatedgrainboundariesinsolid 4 Heandfoundthatthese aregenericallysuperuid,althoughforspecialorientationsofgrainstheseturnnonsuperuid.Thisresultsupportedthegrain-boundarysuperowinterpretationofthe Sasaki etal. [23]experiment.Thecoreofascrewdislocationwasshowntobeasuperuidbyrst-principleMCsimulations[41].Thesuperuidresponsewasinterpretedin termsofaLuttingerliquidtheory,wheretheLuttingerparametersisaone-dimensional superuiddensityandthecompressibility,whichcouldbedirectlyrelatedtotheparametersofthesimulations. Thesearchofpossiblesupersolidphasesarealsobeentheoreticallyexploredin othersystems,mostnotablyinultracoldatomsinopticallatticesseeforeg.Scarola andDasSarma[42]foroneoftheearliestreferencesintheeld.Thetunabilityof theparametersinanopticallatticemakethesesuitabletosimulatecondensedmatter systems.Thegenericapproachtostudythesekindofsystemsarethroughlattice modelssuchastheBose-Hubbardmodel,whichforvariousregimesofthemodel parameterexhibitsuperuid,Mott-insulator,densitywaveorsupersolidphases. Tosummarize,thequestionofwhetherapossiblesupersolidphaseofmatter exists,thoughtheoreticallyspeculatedbefore,hasgainedimpetusaftertheNCRI effectreportedbyKimandChan.Duetosensitivedependencesoftheexperimentally measuredquantitiesonthecrystalgrowth,annealingand 3 Heimpuritiesitseemsto begrowingconsensusthatthephaseisduetoextrinsicdefectsinsolid 4 Hesuchas dislocationsandgrainboundaries.Inordertoinvestigatethepossiblerolethatthese topologicaldefectsplayintheputativesupersolidstate,inthisworkwefocuson quantummechanicalboundstatesoflinedefectssuchasedgeandscrewdislocations; 19

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proposinganon-supersolidinterpretationtotheheatcapacityexperiment,interms ofabsorptionordesorptionof 3 Heimpuritiesfromdislocationinsolid 4 He; developingadefect-basedmodelforasupersolid,wherebulksuperuidorder maybeestablishedinasolidbyatanglednetworkofdislocationswhichpromote one-dimensionsionalsuperuidity. 20

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Figure1-1:Resonantperiodasfunctionoftemperatureofsolid 4 HeinVycorglassfor differentrimvelocitiesofthedisk.Adropintheperiodisseenbelow175mK.Theordinateshows P )]TJ/F24 11.9552 Tf 10.826 0 Td [(P ,thedifferenceoftheactualperiod P and P =971 ; 000 ns.Reprinted bypermissionfromMacmillanPublishersLtd:Nature[E.KimandM.H.W.Chan,Nature 427 ,225],copyright. 21

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Figure1-2:Resonantperiodchangewithtemperatureofsolid 4 HeinVycorglassfordifferent 3 Heconcentrations.Theordinateshows P )]TJ/F24 11.9552 Tf 9.299 0 Td [(P ,thedifferenceoftheactualperiod P and P =971 ; 000 ns.ReprintedbypermissionfromMacmillanPublishersLtd:Nature [E.KimandM.H.W.Chan,Nature 427 ,225],copyright. 22

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Figure1-3:Resonantperiodandamplitudeofoscillationofunblockedannularcell panelAandblockedannularcellpanelBlledwithbulksolid 4 He,asafunctionof temperature. istheresonantperiodatatemperatureof300mK.ReprintedwithpermissionfromAAAS,[E.KimandM.H.W.Chan,Science 305 ,1941],copyright AAAS. 23

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Figure1-4:Annealingeffectintheresonantperiod. P =5 ; 428053 ns.Reprinted Fig.3withpermissionfromA.S.C.RittnerandJ.D.Reppy,Phys.Rev. Lett. 97 ,165301,CopyrightbytheAmericanPhysicalSociety. http://link.aps.org/doi/10.1103/PhysRevLett.97.165301 24

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Figure1-5:Shearmodulusanomalyinsolid 4 Heforarangeof 3 Heimpurityconcentrations.Changesinshearmodulus andNCRIdatafromtorsionaloscillatormeasurementshavebeenscaledbythevaluesatthelowesttemperaturemKtocompare temperaturedependence.ReprintedbypermissionfromMacmillanPublishersLtd: Nature[J.DayandJ.Beamish,Nature 450 ,853],copyright. 25

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CHAPTER2 BOUNDSTATESOFDISLOCATIONS 2.1EdgeDislocations Iwouldliketoacknowledgeherethatpartofthecontentofthischapteristaken fromRef.[54],copyrightbytheAmericanPhysicalSociety. Fromthediscussionsinthepreviouschapter,itisclearthattheobservedexperimentsonsolid 4 Hearestronglydependentonlatticedefectssuchasdislocations.In amoregeneralcontext,thestraineldassociatedwithdislocationsbindsorscatters chargecarriersinmetalsandsemiconductorsorsoluteatomsinagenericsolid.As such,thepresenceofdislocationshasasignicanteffectonthetransportandmechanicalpropertiesofthesolid.Itisthusimportanttohaveanunderstandingofthespectrum ofboundstatesduetoadislocation.Inthissection,wefocusonsuchstatesforanedge dislocation. Westartourdiscussionwithabriefintroductiontotermswhichwewillfrequently encounter. Edgedislocation :Anedgedislocationisalinedefectinacrystal.Itcanbe thoughtofasanextrahalf-planeofatomsinsertedwithinthecrystal.Thedislocationcanbecharacterisedbyalinedirectionwhichrunsalongthebottomofthe extrahalf-planeandaBurger'svector. Burger'svector :TheBurger'svectorgivesthemagnitudeanddirectionofthe distortioninthedislocatedlattice.Ifweimagineacontour L surroundingtheline directionofanedgedislocationand u bethedisplacementeldthen I L d u = b ; where b istheBurger'svector.Foraperfectlatticetheaboveintegralwouldbe zero.SeeFig.2-1foraschematicdiagramofanedgedislocationshowingthe Burger'svector. 26

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Poisson'sratio :Itistheratioofthetransversestrainwhichisnormaltothe appliedstresstothelongitudinalstrainwhichisalongtheappliedstress.Fora stablematerialthePoisson'sratioliesbetween-1and0.5. 2.1.1ElasticFieldandPotential Letusnowdiscusstheelasticeldproducedduetoanedgedislocation.Thiswill alsoleadustondaneffectiveelasticpotential.Weconsiderastraightedge-dislocation inanisotropicsolid,withinlinearelasticitytheory.Letthedislocationbealongthe positive z -axis.If b istheBurger'svectorand isthePoisson'sratiothen,itisshownin theAppendixAthatthecomponentsofthedisplacementeldduetothedislocationare givenby u x = b 2 tan )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 y=x + xy 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( x 2 + y 2 # ; u y = )]TJ/F24 11.9552 Tf 14.467 8.088 Td [(b 2 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( ln p x 2 + y 2 =r 0 + 1 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( x 2 x 2 + y 2 # ; where r 0 isacut-off.Inpolarcoordinates r; thecomponentsare u r = b 2 cos )]TJ/F15 11.9552 Tf 17.703 8.088 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin ln r=r 0 # ; u = )]TJ/F24 11.9552 Tf 14.467 8.087 Td [(b 2 sin + cos + )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 cos ln r=r 0 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( # : If r isthedistancefromthedislocationaxisand theazimuthalangle,thecomponents ofthestresstensorexpressedinpolarcoordinates[43]are rr = = )]TJ/F24 11.9552 Tf 28.999 8.088 Td [(b 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin r ; r = b 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( cos r ; zz = rr + = )]TJ/F24 11.9552 Tf 22.803 8.088 Td [(b )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin r ; rz = z =0 ; 27

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where referstotheshearmodulusofthesolid.Fromtheabovewecancalculatethe meanhydrostaticpressure p generatedbyanedgedislocationinanisotropicmedium, p = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 3 ii = b 3 1+ 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin r : Now,letusconsiderasoluteatominthepresenceofastrainedduetothedislocationsolventlattice.Forinstance,inthecontextofsolid 4 He,thesoluteatomsare 3 He impuritiesandthesolventdislocatedlatticeisthatof 4 He.Ifthehydrostaticpressurein whichthesoluteatomisinsertedis p ,thentheworkdoneininsertingthesoluteatomin thesolventmatrixis E = pV; where V istheamountbywhichthevolumeofthesoluteatomexceedsthevolumeof thesolventatom[44].Thisworkmaybethoughtofastheeffectivepotentialwhichthe solventlatticecontaininganedgedislocationpresentstoasoluteatom. Forasmalldifferenceinatomicradiibetweensoluteatom r 1 andsolventatom r 2 V = 4 3 r 1 3 )]TJ/F24 11.9552 Tf 11.955 0 Td [(r 2 3 : If r 1 = r 2 + a a beingthesize-mistparameter,then V 4 a r 2 3 : Sotheeffectivepotentialtheedge-dislocationpresentsis U r; = U 0 sin r ; where U 0 = 4 3 + )]TJ/F25 7.9701 Tf 6.587 0 Td [( b a r 2 3 Asimplerotationofthecoordinateaxesinthe x )]TJ/F24 11.9552 Tf 12.334 0 Td [(y planecaststhispotentialinto themorestandardformofthedipolepotentialintwodimensionsD U r; = U 0 cos r ; 28

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where U 0 isthedipolemoment.Inanelectrostaticscontextthispotentialcanberealized asadipolebuiltbybringingtwoinnitelinechargesofoppositesignsclosetogether. For U 0 > 0,thispotentialisrepulsivefor )]TJ/F25 7.9701 Tf 10.494 4.707 Td [( 2 << 2 andattractivefor 2 << 3 2 .Since thedipolepotentialalwayshasanattractivepartweexpecttohavequantummechanical boundstates. 2.1.2GroundStateEnergyEstimation:VariationalApproach Inthissectionweaimatobtainingaquantummechanicalestimatefortheground stateenergyofaparticleinthedipolepotentialin2DasgivenbyEq.2.Hence, wefocusonthegroundstateeigenvalueoftheSchr odingerequationforthe2Ddipole potentialgivenby )]TJ/F15 11.9552 Tf 15.524 8.088 Td [( h 2 2 m r 2 + U 0 cos r = E: Wenoteherethatthedipolepotentialisnon-centralandtheEq.2isnonseparable,whichimpairstheapplicabilityoftheWKBapproximation.Anyexactanalyticalsolutionforthisproblemisyettoappear[45]. Theproblemofestimatingthegroundstateenergyofthequantumdipoleproblem hasalonghistory,startingfromtheworkbyLandauerin1954[46],whousedavariatonalapproach.Subsequentauthorsusedavarietyoftechniquesforthisestimate: semiclassical[47]orpurelyvariational[48,49]methods,acombinationofvariational andperturbativemethods[50]oranexpansionintermsofknownbasisfunctions [51,52].Someoftheseworks[49,50],havealsoestimatedthespectrumofthebound eigenstates.ThegroundstateenergiesestimatedintheseworksareshowninTable 2-1[54],togetherwithourvariationalandnumericalestimateforthegroundstatevalue obtainedintheongoingandfollowingsubsection. Wenowelaborateonthevariationalmethodusedbyustoobtainanupperbound forthegroundstateenergy.Avariationalmethodusesafunctionalwhichwhenextremizedgivestheequationofmotionforinstance,inclassicalmechanicstheaction S is 29

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extremizedtoobtaintheLagrange'sequationsofmotion.Similarly,wemaythinkofa functionalwhichwhenextremizedwillgiveustheSchr odingerequation. Givenanormalizedwavefunction r; anenergyfunctionalforEq.2is[54] F [ ; ]= Z h 2 2 m jr j 2 + U 0 cos r j j 2 # d 2 r : ThisfunctionalhasitsextremaatthesolutionsoftheSchr odingerequation.Wenote thatthelengthandenergyscalesthatemergefromEq.4orEq.2are h 2 = 2 mU 0 and 2 mU 2 0 = h 2 ,respectively.Now,weintroduceanormalizedtrialwavefunctionin dimensionlessvariablesinordertominimizetheenergyfunctionalinEq.42[54]: r; = 2 AB C p )]TJ/F24 11.9552 Tf 11.955 0 Td [(r=BC p )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 B +2 B 2 exp )]TJ/F24 11.9552 Tf 14.967 8.088 Td [(r C )]TJ 10.494 17.922 Td [(p 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(A 2 C 2 r 8 3 r cos exp )]TJ/F24 11.9552 Tf 14.967 8.088 Td [(r C ; where A B and C arevariationalparameters.Wechoosethetrialwavefunctionsoas toaccountfortheanisotropyofthepotential.Further,theasymptoticbehaviorofthe potentialiscapturedbytheexponentiallydecayingfactors.Theminimumexpectation valueoftheenergyoccurswhen A =0 : 803 ;B = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 774 and C =2 : 14 withaground stateenergyof )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 1199 .Thisvalueislowerby2.5%comparedtothepreviouslowest variationalestimate )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 1196 obtainedbyDubrovskii[50]. Amorephysicalapproachtotheproblemwouldbetointroduceacut-offtothe potentialforthedistance r fromthedislocationcore.Thecut-offtakescareofthe singularityofthepotentialas r 0 .Weintroduceasoftcut-offtothepotentialin 2,withacut-offparameter c .Thepotentialwithcut-offinpolarformis U c r; = U 0 r cos r 2 + c 2 : 30

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Weagaintakeavariationalapproachtoestimatethegroundstateenergyandproceed withthenormalizedtrialwavefunction r; = A a r 2 e )]TJ/F26 5.9776 Tf 7.97 3.259 Td [(r a )]TJ 13.151 17.922 Td [(p 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(A 2 a 2 r 8 3 re )]TJ/F26 5.9776 Tf 7.969 3.259 Td [(r a cos ; where a and A arevariationalparameters.Next,weminimizetheenergyfunctional 4withthepotentialreplacedbythecut-offpotential U c r; ,in2.Doingthis, weexpecttoobtainanupperboundtothegroundstateenergyasafunctionofthe cut-offparameter c .Theexpectationvalueoftheenergyusingthetrialwavefunctionin 2obtainedusingMapleis E A;a;c = 1 a 2 )]TJ/F15 11.9552 Tf 18.132 8.087 Td [(8 p 3 A p 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(A 2 a 3 Z 1 0 r 3 r 2 + c 2 e )]TJ/F23 5.9776 Tf 7.782 3.258 Td [(2 r a dr: Thisexpressionneedstobeminimizedwithrespecttothevariationalparameters A and a ,toobtainthevariationalground-stateenergy.Minimizationwithrespectto A gives A = 1 p 2 .However,theintegralintheexpressiongivestheCosineandSineintegral functions[55]andminimizationisnottrivialwithrespectto a .Toovercomethisproblem wechosediscretevaluesof c from0.5to5andgeneratedplotsfor E A = 1 p 2 ;a versus a foreach c usingMaple.Wedeterminetheminimaintheenergyandgenerateaplot ofthisminimizedenergy E versusthecut-offparameter c asshowninFig.2-2. Atthispoint,wemakeanestimateofthelengthandenergyscalesandhencethe bindingmagnitudeofgroundstateenergyof 3 Heimpuritiesinsolid 4 Helatticeusing physicalparametersfor 4 Heand 3 He.Wedetermine r 0 =1 : 99 Atheatomicradiiof 4 Heand a =0 : 077 thesize-mistparameterfor 4 Heand 3 Hefromthemolarvolume of 4 Heand 3 Hesolids[56].Fromthelongitudinalsoundspeedinsolid 4 He[57]we obtain =0 : 364 .Wetake b tobeoftheorderofalatticeconstantwhichis 2 r 0 and =1 : 5 10 7 Pa[22].ThemassappearingintheSchr odingerequationisthemassofa 3 Heatomwhichistakentobe m =3 m proton =0 : 5 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(23 gm.Thesevaluesgivethe strengthoftheedgedislocationpotential U 0 = 4 3 + )]TJ/F25 7.9701 Tf 6.587 0 Td [( b a r 0 3 .Thisgivesthelengthand 31

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energyscalesas h 2 2 mU 0 =2 : 44 A ; 2 mU 0 2 h 2 =1 : 33 K ; respectively.Theenergyscaleisgivenintemperatureunitsabove. Thisgivesthebindingenergyforthebarepotentialwithoutacut-offi.e, c =0 usingthetrialwavefunction2tobe0.856K.FromtheplotinFig.2-2,wecan inferthatthebindingenergyforthepotentialwithasoftcut-offhasarangeofvalues dependingonthecut-offparameter.Therangeofvaluesareroughlyfrom37mKfor c =5 ,i.e.12.2 Ato95mKfor c =0 : 5 ,i.e.1.22 A.Asummaryoftheseresultsappear inTable2-3. 2.1.3NumericalMethodstoDetermineCompleteSpectrum Iwouldliketoacknowledgeherethatthenumericalstudyabouttobediscussed hereisacollaborativeeffort,withK.Dasbiswasandmyselfworkingontherealspace diagonalizationandC.-D.YoocontributingwiththeCoulomb-basiscalculation. Adetailednumericalsolutionofthetwo-dimensionalSchr odingerequationwiththe dipolepotential,Eq.2,islikelytoprovidemoreaccurategroundstateeigenvalues inadditiontodeterminingtherestoftheboundstateeigenvaluesandcorresponding wavefunctions.Wedothisbothbyarealspacediagonalization,wheretheSchr odinger equationisdiscretizedonasquaregrid,andbyexpandinginthebasisoftheeigenfunctionsofthetwo-dimensionalCoulombpotentialproblem.Twospecialfeaturesofthis dipolepotentialmakeitanumericallydifcultproblem:thesingularityattheorigin,and thelongrangebehaviorofthepotential.ItisexpectedthattheCoulombwavefunctions wouldbebettersuitedtocapturingthislongrangebehavior,andconvergencewould consequentlybefaster.OurresultsshowthattheCoulombbasismethodismoreaccurateforthehigherboundstateswhichareexpectedtoextendmoreinspace,asthe 32

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realspacemethodsarelimitedbysizeissues.However,therealspacemethodworks betterforthegroundstate. 2.1.3.1Realspacediagonalization FornumericalpurposestheSchr odingerequationisconvertedtoadifference equationonasquaregridofspacing h ,withtheLaplacianapproximatedbyitsvepointnitedifferenceform[58],resultinginablocktridiagonalmatrixofsize N 2 N 2 wherethegridhasdimensionsof N N .Eachdiagonalelementcorrespondstoa gridpointandhasvaluesof 4 =h 2 + V x;y ,whereasthenonzerooffdiagonalelements allequal )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 =h 2 .Thematrixisthusverylargebutsparse.Weusethreedifferent numericalmethodstodiagonalizethismatrix:thebiconjugategradientmethod[59],the Jacobi-Davidsonalgorithm[60]andArnoldi-Lanczosalgorithm[61],withthelattertwo beingmoresuitedtolargesparsematriceswhoseextremeeigenvaluesarerequired. WeusefreelyavailableopensourcepackagesJADAMILU[62]andARPACK[63] writteninFORTRANforboth.AllthreeapproachesareprojectiveKrylovsubspace methods,whichrelyonrepeatedmatrix-vectormultiplicationswhilesearchingfor approximationstotherequiredeigenvectorinasubspaceofincreasingdimensions. Reference[64]providesaconciseintroductiontotheJacobi-Davidsonmethod,together withcomparisonstoothersimilarmethods.TheimplicitlyrestartedArnoldipackage ARPACKisdescribedingreatdetailinRef.[65].Somegeneralissuesaboutthereal spacediagonalizationaswellassomespecicfeaturesofthethreemethodsusedforit arediscussedbelow. Theaccuracyoftherealspacediagonalizationmethodsiscontrolledbytwomain parameters:thegridspacing h andthetotalsizeofthegrid,whichisgivenby Nh .The nitedifferenceapproximationtogetherwiththerapidvariationofthepotentialnearthe originimplythatthesolutionofthepartialdifferentialequationwouldbemoreaccurate forasmallergridspacing.Weworkwithopenboundaryconditions,whichmeansthat aboundstatewavefunctioncouldbecorrectlycapturedonlyifthetotalsizeofthegrid 33

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weretobegreaterthanthenaturaldecaylengthofthewavefunction.Inotherwords, theeigenstatehastobegivenenoughspacetorelax.Thislimitsthenumberofbound stateswecancalculateaccuratelybecausealargegridsizetogetherwithsmallgrid spacingscallsforalargenumberofgridpoints,thusquadraticallyincreasingthesizeof thematrixtobediagonalized.Computationalresourcesaswellasthelimitationsofthe algorithmsthemselvesplaceaneffectiveupperboundonthesizeofadiagonalizable matrix.Weexperimentedtondthata 10 6 10 6 sizesparsematrixwasaboutthe maximumthatcouldbediagonalizedwithourcomputationalresourcesavailableatthe High-PerformanceComputingCenter,UniversityofFlorida. Theoriginofthesquaregridissymmetricallyoffsetinboth x and y directionsto avoidthe 1 =r singularity.Wetestedrsttheaccuracyoftherealspacetechniquesfor thecaseofthetwodimensionalCoulombpotential,thespectrumofwhichiscompletely known[66].Weobservethatforvariouslatticesizesthebiconjugatemethodcapturesat mosttherstfourstateswhereastheJacobi-Davidsonmethodreturns20eigenstates. Theeigenvaluesobtainedfrombothmethodsareaccuratetowithin2%oftheexact values[66]. Wehaveappliedthebiconjugatemethodtotheedgedislocationpotentialfor variouslatticesizes,varyingfrom10 10to600 600.Thenumberofeigenstates capturedincreaseswiththesizeofthelattice,asexpected.Thegroundstateenergy isobservedtovaryfrom-0.134to-0.142.Wealsoobservethatforthenumberofgrid pointsexceeding N =2000 weencounteranumericalinstabilityduetotheaccumulation ofroundofferrors.Forthelargestrealspacegridsizeof600 600 N =1200 ;h =0 : 5 weobtainseveneigenstateswithagroundstateenergyof-0.1395. ThegroundstateenergyfromtheJacobi-Davidsonmethod,employedforthesame latticesizegives-0.1395,whichmatcheswellwithourexpectationsfromthevariational calculation.Weareabletoobtain 20 boundstateeigenvaluesinthismethodusing 34

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N =1000 ;h =0 : 5 .Itischeckedthatthelow-lyingeigenvaluesarenotverysensitiveto valuesof h inthisregime,soarelativelylargevalueof 0 : 5 servesourpurpose. TheArnoldi-Lanczosmethodyieldsverysimilareigenvalues.Ittakesmoretime andmemoryresourcestoconvergebutcancalculatemoreeigenvalues,withgreater accuracyforthehigherexcitedstates.Itprovides 30 boundstateeigenvaluesforthe samesetoflatticeparametersastheabove. 2.1.3.2Coulombbasismethod Wealsocalculatethespectrumnumericallybyusingthelinearvariationalmethod withthebasisofthe2Dhydrogenatomwavefunctions[66].Therearetwoadvantages ofthismethodovertherealspacediagonalizationmethods.First,thelinearvariational methodiscapableofcapturingmoreexcitedstatesbecausethenumberofcalculated boundstatesisnotlimitedbythesizeoftherealspacegridbutbythenumberoflongrangebasisfunctions.Second,thesingularityattheoriginoftheedgedislocation potentialdoesnotposeaproblemanymorebecauseelementsoftheHamiltonianmatrix becomeintegrable. NowwecalculatetheelementsoftheHamiltonianmatrixwitha2Dedgedislocation potential.TheSchr odingerequationwiththe2DCoulombpotentialisanalytically workedoutinRef.[66].Thenormalizedwavefunctionsofa2Dhydrogenatomare givenby H n;l r; = r 1 R n;l r 8 > > > > > > < > > > > > > : cos l for 1 l n; 1 p 2 for l =0 ; sin l for )]TJ/F24 11.9552 Tf 9.299 0 Td [(n l )]TJ/F15 11.9552 Tf 21.918 0 Td [(1 ; where R n;l r = n j l j s n + j l j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1! n )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 n )-222(j l j)]TJ/F15 11.9552 Tf 17.933 0 Td [(1! n r j l j exp )]TJ/F24 11.9552 Tf 10.494 8.088 Td [( n r 2 1 F 1 )]TJ/F24 11.9552 Tf 9.299 0 Td [(n + j l j +1 ; 2 j l j +1 ; n r ; 35

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with n =2 = n )]TJ/F15 11.9552 Tf 12.003 0 Td [(1 and 1 F 1 beingtheconuenthypergeometricfunction.Theelements oftheHamiltonianwiththe2Ddipolepotentialare H n 1 ;l 1 j)-222(r 2 j H n 2 ;l 2 = l 1 ;l 2 Z 1 0 dr 1 )]TJ/F24 11.9552 Tf 13.15 8.725 Td [( 2 n 2 4 r R n 1 ;l 1 r R n 2 ;l 2 r ; H n 1 ;l 1 cos r H n 2 ;l 2 = ~ V Z 1 0 drR n 1 ;l 1 r R n 2 ;l 2 r ; where ~ V = l 1 ;l 2 1 = 2 ifboth l 1 and l 2 arelessorgreaterthan0,or ~ V =1 = p 2 if l 1 is 0and l 2 positiveorviceversa.Thespectraareobtainedforseveraltotalnumbersof basisfunctions N basis .Duetothenumericalprecisionincalculatingelementsofthe Hamiltonianmatrix N basis cannotbeincreasedtomorethan400.For N basis =400 we obtainabout149boundstatesandthegroundstateenergyof-0.0969.Inorderto improvethegroundstateenergy,weintroduceanadditionaldecayingparameterinthe basisfunctions,andoptimizetheenergylevelsforacertainvalueofthisparameter. Withthedecayingparameterweobtainthebestvariationalestimateforthegroundstate energyof-0.1257for N basis =400 ItmightbeworthwhileinthiscontexttomentionofarecentworkbyAmore[67], whoimprovesuponthegroundstateenergyobtainedusingtheCoulomb-basismethod. UsingthedecayingparameterinthebasisfunctionsinaselectedportionoftheHilbert space,alongwiththeimplementationofthe`principleofminimalsensitivity'andShanks transforms,theauthorobtainsagroundstateenergyof-0.1314andalsoestimates someoftheexcitedstates. WeshowthersttwentyeigenvaluesobtainedfromdifferentmethodsinFig.2-3 andtherstverepresentativeeigenvaluesinTable2-2.Asmentionedearlier,thereal spacediagonalizationmethodsprovidesabestestimateofthegroundstateenergy whereastheCoulombbasismethodismoresuitableforhigherexcitedstates.The eigenvaluesofboththeCoulombbasismethodandtherealspacediagonalization 36

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methodsarefoundtomatcheachotherforexcitedstates,andthentheybeginto deviateagainseeFig.2-3.Thiscanbeunderstoodbythefactthattheextentofwave functionsofthe2Dedgedislocationpotentialdoesnotalwaysincreaseasonegoesto higherexcitedstatesthewavefunctionsofsomeexcitedstatesextendlessthanthose oflowerenergy.Therefore,thereareintermediateboundstatesthataremissedinthe realspacecalculationbecausethegridsizeusedinthecalculationisnotlargeenough tocapturethem.Forexample,wendfourmoreboundstateswiththeCoulombbasis calculationbetweenthe18 th and19 th excitedstatesascalculatedfromtherealspace diagonalizationmethod.Thisfeaturealsoexplainstheabruptincreaseoftheeigenvalue ofthe19 th statecalculatedbyusingtheArnoldi-LanczosmethodARPACKroutinein Fig.2-3. 2.1.4SemiclassicalAnalysis Itisusuallyinsightfultoconsiderthesemiclassicalsolutionofaquantummechanics problem,sincethehigherenergyeigenstatestendtoapproachclassicalbehavior.A semiclassicalestimateoftheenergyspectrumhasbeenprovidedinRef.[47].Here thetotalnumberofeigenstatesuptoavalueofenergy E isproportionaltothevolume occupiedbythesystemintheclassicalphasespace.ThisisexpressedbyWeyl's theorem[68]: n E = A 4 2 m j E j h 2 + O 0 @ s h 2 2 mU 0 2 j E j 1 A ; where A istheclassicallyaccessibleareainrealspaceand j E j theabsolutevalueof energyofthestate.Thehigherordercorrectionscanbeshowntobelessimportantfor higherexcitedstates,whichiswherethesemiclassicalpictureapplies.Tond A ,we needtheclassicalturningpointsforthispotentialdeterminedbysetting E = V r; Thentheaccessibleareaistheinteriorofacirclegivenby x )]TJ/F25 7.9701 Tf 13.936 4.813 Td [(U 0 2 E 2 + y 2 = U 0 2 E 2 ,with area A = U 0 = 2 E 2 .Therefore,weobtainwritingthenondimensionalizedenergyinour 37

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systemofunitsas : n = )]TJ/F15 11.9552 Tf 15.783 8.088 Td [(1 16 ; where n isthequantumnumberoftheeigenstate,and thecorrespondingenergy.Note thatthedensityofstates dn=d scalesas 1 = 2 Tocheckthisresultwetthenumericalspectrumwiththefollowingfunctionalform: n = a n )]TJ/F24 11.9552 Tf 11.955 0 Td [(b c ; where n takesintegralvaluesstartingfrom1andthettingparametershasvalues a = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 06 ;b =0 : 5 ;c = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 98 ,eachcorrecttowithin 5% .Wenotethatthetgives = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 16 n )]TJ/F15 11.9552 Tf 12.039 0 Td [(1 = 2 ; incomparisonto = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 = 16 n asinEq.2sincewearedealing withboundstateshere,alltheenergyeigenvaluesarenegative,andthehigherexcited stateshavelowerabsoluteeigenvalues.Weshowthettothespectrumobtained fromJADAMILUroutineinFig.2-4.Thesemiclassicallyderiveddependenceisfoundto closelymatchwiththetfornumericallycalculatedenergyeigenstates,exceptforthe b =0 : 5 factor.Inthelimitoflarge n valuesi.e,higherexcitedstates,thetrelationtends tothesemiclassicalresultasexpected. Theclassicaltrajectoriesforthispotentialbearthesignatureofchaoticdynamics showingspace-llingnatureandstrongdependenceoninitialconditions.However,for reasonsnotyetcleartous,theyarenotergodic,llinguponlyawedge-shapedregion inrealspaceinsteadofthefullclassicallyallowedcircle.Thequantummechanical probabilitydensityascalculatedfromtheeigenfunctionsalsoexhibitssuchwedgeshapedregions.Someofthewavefunctionsobtainedfromournumericalcalculationare presentedinFigs.2-52-14.Theparityofthepotentialshowsupinthewavefunctions beingeithersymmetricorantisymmetricaboutthe x -axis,althoughstatesofsuchodd andevenparitydonotalwaysalternate.Forexample,the2ndand4thexcitedstates areodd,andthe3rdexcitedstatecorrespondinglyeven,butthegroundstateandrst excitedstatesarebotheven.Similarly,the10th,50thand100thexcitedstatesareall 38

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oddwhilethe23rdand24thareodd.Also,thespatialextentofthewavefunctionsdoes notscalemonotonicallywithquantumnumber.Somecasesarefoundwhereahigher excitedstatehaslessspatialextentthanalowerone.ForexampleinFig.2-12,the24th excitedstateislessextendedinthe x directioncomparedtothe23rdexcitedstatein Fig.2-11.Wedonothaveanysatisfactoryexplanationyetfortheseirregularfeatures. 2.2ScrewDislocation Ascrewdislocationisanotherkindofalinedefectinacrystal.Inthecaseof ascrewdislocationthedislocationlineandtheBurger'svectorarealongthesame direction. 2.2.1ElasticFieldandPotential Theelasticeldforastraightscrewdislocation,withitsdislocationaxisalongthe z -axisisderivedinAppendixB.Theonlycomponentofthedisplacementeldinthis caseisalongthe z -axisandisgivenby, u z x;y = b 2 tan )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 y x : Thus,wehave r u =0 andthereisnovolumedeformationduetoascrewdislocation ifweconsideronlylinearstrains.Avolumedeformationandhenceaneffectiveelastic potentialforascrewdislocationishoweverpossibleifweconsidernon-linearstrains. Consideringnon-linearstrains,asshowninAppendixB,wegetthehydrostaticpressure duetoastraightscrewdislocationwithinlinear,isotropicelasticitytheoryas p = )]TJ/F24 11.9552 Tf 10.494 8.088 Td [( 3 1+ 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 b 2 2 1 r 2 ; where istheshearmodulus, isthePoisson'sratio, b theBurger'svectorand r the radialcoordinateofthecylindricalpolarcoordinatesystem. Sotheeffectivepotentialduetoascrewdislocationmaybewrittenas V r = )]TJ/F24 11.9552 Tf 9.298 0 Td [(V 0 b r 2 ; 39

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where V 0 isaconstantwithdimensionsofenergyanddependsontheshearmodulus, Poisson'sratioandthevolumedifferencebetweenhostandsoluteatoms. Hence,thepotentialduetoascrewdislocationincontrasttoanedgedislocationis centralandalwaysattractive.Inthefollowingsectionweinvestigatetheboundstatesfor thepotentialin2. 2.2.2GroundStateEstimate Theboundstatesofaparticleforthepotentialin2isaproblemwhichhas beenaddressedasearlyas1931byShortley[69]andhasbeendiscussedindetail inaseriesoflaterworks[70,71,72].Thepotentialispathologicalanddoesnotallow theexistenceofdiscreteboundstates.Thiscanbeseenfromthefactthatthepotential 1 =r 2 ,hasthesamescalingasthekineticterm r 2 thusrenderingthelengthandenergy scale-invariant. Whenregularized,thepotentialdoesgiverisetoboundstates.Weregularizethe potentialbyintroducingtwocut-offschemes,thehardandsoftcut-offschemes,and ndthegroundstateenergybyvariationalandnumericalmethods.Inwhatfollowswe investigatethelengthandenergyscalesofthegroundstateforthepotentialwithahard cut-off,withinavariationalframework. Thepotentialin2isreplacedbythefollowing V r = 8 > > < > > : )]TJ/F24 11.9552 Tf 9.299 0 Td [(V 0 b r 2 ; for r>r c )]TJ/F24 11.9552 Tf 9.299 0 Td [(V 0 b r c 2 ; for r
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functional F [ ; ]= Z h 2 2 m jr j 2 + V r j j 2 # d 2 r : Thegroundstateisexpectedtobea s -waveasthepotentialiscentrallysymmetric.The simplestnormalizedvariationalwavefunctionisanexponentialgivenby f r = r 2 1 a e )]TJ/F25 7.9701 Tf 6.586 0 Td [(r=a ; where a isavariationalparameter. Theexpectationvaluesforthekineticandpotentialenergiesforthewavefunctionin 2isgivenby h T i = h 2 8 mr 2 c x 2 and h V i = )]TJ/F24 11.9552 Tf 9.299 0 Td [(V 0 b r c 2 x 2 Ei x +1 )]TJ/F24 11.9552 Tf 11.956 0 Td [(e )]TJ/F25 7.9701 Tf 6.586 0 Td [(x )]TJ/F24 11.9552 Tf 11.955 0 Td [(xe )]TJ/F25 7.9701 Tf 6.586 0 Td [(x respectively.Here x = 2 r c a ,isadimensionlessparameterandEi x istheexponential integralgivenby Ei x = Z 1 x e )]TJ/F25 7.9701 Tf 6.587 0 Td [(t t dt: Theexpectationvalueforthetotalenergyisgivenbyaddingtheexpectationvaluesof thekineticandpotentialenergiesasgivenby2and2toget h E i = h 2 8 mr c 2 F x ; where F x = x 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(gx 2 Ei x )]TJ/F24 11.9552 Tf 11.956 0 Td [(g )]TJ/F24 11.9552 Tf 11.955 0 Td [(e )]TJ/F25 7.9701 Tf 6.587 0 Td [(x )]TJ/F24 11.9552 Tf 11.955 0 Td [(xe )]TJ/F25 7.9701 Tf 6.587 0 Td [(x and g = V 0 h 2 = 8 mb 2 givesameasureoftherelativestrengthofthepotentialandkinetic energies.MinimizingthefunctionF x in2wehave F x 0 = )]TJ/F24 11.9552 Tf 9.299 0 Td [(g )]TJ/F24 11.9552 Tf 11.955 0 Td [(e )]TJ/F25 7.9701 Tf 6.586 0 Td [(x 0 )]TJ/F24 11.9552 Tf 11.955 0 Td [(x 0 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(x 0 41

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where x 0 isthevalueof x whichminimizesF x .Thetermintheparenthesesin2 isalwayspositiveandwendaboundstateforallvaluesof g .Welookattwolimitsof g toexaminetheenergyandlengthscalebehaviorforthegroundstate. For g 0 weareinthequantumlimit.Inthislimitweexpandtheexponential integralforsmall x 0 : Ei x 0 ')]TJ/F24 11.9552 Tf 21.918 0 Td [( )]TJ/F15 11.9552 Tf 11.955 0 Td [(ln x 0 + x 0 + O x 0 2 where =0 : 57721 :: istheEuler'sconstant.Forsmall x 0 F x 0 = )]TJ/F24 11.9552 Tf 9.299 0 Td [(gx 0 2 : Hence,afterreinstatingunits,thegroundstateenergy E g is E g = )]TJ/F24 11.9552 Tf 9.298 0 Td [(V 0 b r c 2 e )]TJ/F22 7.9701 Tf 6.587 0 Td [(2 e )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 =g andthelengthscaleofthewavefunctionis a =2 r c e e 1 =g : Sothegroundstateenergyisexponentiallysmall,whilethelengthscalegrowsexponentially,afeatureofthepathologicalnatureofthepotential. For g !1 weapproachtheclassicallimit.ExpandingF x 0 forlarge x 0 and reinstatingunitsasbefore,wehave E g = )]TJ/F24 11.9552 Tf 9.298 0 Td [(V 0 b r c 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(ln g g + ::: and a = 2 r c ln g : Asinthecaseofanedgedislocation,wehavecalculatedthebindingenergyfora 3 He impurityinthescrewdislocationelasticeldpotentialofsolid 4 He,bytakingphysical valuesofparameters.Thisisessentiallydoneasinthecaseofanedgedislocation,by 42

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treating 3 Heimpuritiesassubstitutionalentitiesina 4 Helattice.Theparameters,letsus knowthestrengthofthescrewdislocationpotential V 0 .Thecut-offdistance r c istaken tobealatticeconstantofsolid 4 Hewhichisabout4 A. Forthevariationalwavefunctionin2forthepotentialwithahardcut-off,as in2wegetabindingenergyintemperatureunitsof52.5mK.Wealsosought anexactnumericalvalueforthebindingenergyusingtheshootingmethod[73].The numericalvalueforthebindingenergyis61.1mK.Thesamecalculationrepeatedfor asoftcut-offgivesavariationalbindingenergyof27.8mKandanumericallyobtained bindingenergyof28.7mK.Atabulationoftheseresultsalongwiththosefortheedge dislocationisgiveninTable2-3. 43

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Table2-1:Summaryofgroundstateenergyestimatesoftheedgedislocationpotential.Energyisgiveninunitsof 2 mU 2 0 = h 2 .TabletakenfromK.Dasbiswas,D.Goswami, C.-D.YooandA.T.Dorsey,Phys.Rev.B 81 ,064516.Copyrightbythe AmericanPhysicalSocietyhttp://link.aps.org/doi/10.1103/PhysRevB.81.064516. ReferencesGroundstateestimate Landauer[46]-0.102 Emtage[48]-0.117 NabutovskiiandShapiro[51]-0.1014 SlyusarevandChishko[49]-0.1111 Dubrovskii[50]-0.1196 FarvacqueandFrancois[52]-0.1113 DorseyandToner[53]-0.1199 Dasbiswas etal. [54]-0.139 Table2-2:Comparisonofrstfewenergyeigenvaluesobtainedfromdifferentnumericalmethods.Energyunits: 2 mU 2 0 = h 2 n indicatesquantumnumberofthestate. TabletakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. n biconjugateJacobi-Arnoldi-Coulomb DavidsonLanczosbasis 1-0.14-0.13954-0.13952-0.09697 2-0.041-0.041480-0.041478-0.03281 3-0.023-0.023314-0.023314-0.022067 4-0.02-0.020086-0.020086-0.016744 5-0.012-0.012592-0.012594-0.011944 44

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Table2-3:Comparisonofresultsforedgeandscrewdislocations. EdgedislocationScrewdislocation Potential 4 3 1+ 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [( b a r 3 0 cos r = U 0 cos r )]TJ/F22 7.9701 Tf 12.612 4.707 Td [(1+ 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 a b 2 2 1 r 2 = )]TJ/F24 11.9552 Tf 9.298 0 Td [(V 0 b r 2 dipolarwithinlineartheorycentral,withnonlinearstrains Strengthofthe potential U 0 r 0 =3 : 8 K 4 V 0 =1 : 4 K Quantummechanicalapproach: 1withoutcut-off : Variational wavefunction r; = 2 AB C p )]TJ/F24 11.9552 Tf 11.956 0 Td [(r=BC p )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 B +2 B 2 e )]TJ/F26 5.9776 Tf 8.801 3.258 Td [(r C )]TJ 10.494 17.922 Td [(p 1 )]TJ/F24 11.9552 Tf 11.956 0 Td [(A 2 C 2 r 8 3 r cos e )]TJ/F26 5.9776 Tf 8.801 3.258 Td [(r C Variational parameters A =0 : 803 ;B = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 774 ;C =2 : 14 mU 0 2 h 2 Nodiscreteboundstates! Bindingenergy 0 : 856 K 2withhardcut-off: Potential )]TJ/F24 11.9552 Tf 9.299 0 Td [(V 0 b r 2 ;r>r c )]TJ/F24 11.9552 Tf 9.299 0 Td [(V 0 b r c 2 ;r
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Figure2-1:Schematicdiagramforanedgedislocation.Theatomsareshownassmall circles.Notetheextrahalf-planeofatomsinthecenterwhichformsthedislocation. L showstheclosedcontourandtheBurger'svector b isthelinejoining A to C .Figure partiallytakenfromwww.materialseducation.org. Figure2-2:Plotofgroundstateenergyversuscut-offparameter. 46

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Figure2-3:Comparisonofeigenvaluesobtainedfromdifferentmethods.Theplotis onalog-logscale.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T. Dorsey,Phys.Rev.B 81 ,064516.CopyrightbytheAmericanPhysical Societyhttp://link.aps.org/doi/10.1103/PhysRevB.81.064516. 47

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Figure2-4:FitfortheeigenvaluespectrumobtainedfromJADAMILUusing f x = a x )]TJ/F24 11.9552 Tf 9.299 0 Td [(b c .Fitvaluesare )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 06 ; 0 : 61 and 0 : 96 for a;b and c respectively.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. 48

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Figure2-5:Eigenfunctionforthegroundstate.Theplotshowsaprojectionof theeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunction getsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. Figure2-6:Eigenfunctionforthe1stexcitedstate.Theplotshowsaprojection oftheeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunctiongetsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. 49

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Figure2-7:Eigenfunctionforthe2ndexcitedstate.Theplotshowsaprojection oftheeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunctiongetsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. Figure2-8:Eigenfunctionforthe3rdexcitedstate.Theplotshowsaprojection oftheeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunctiongetsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. 50

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Figure2-9:Eigenfunctionforthe4thexcitedstate.Theplotshowsaprojection oftheeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunctiongetsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. Figure2-10:Eigenfunctionforthe10thexcitedstate.Theplotshowsaprojectionoftheeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunctiongetsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. 51

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Figure2-11:Eigenfunctionforthe23rdexcitedstate.Theplotshowsaprojectionoftheeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunctiongetsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. Figure2-12:Eigenfunctionforthe24thexcitedstate.Theplotshowsaprojectionoftheeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunctiongetsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. 52

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Figure2-13:Eigenfunctionforthe50thexcitedstate.Theplotshowsaprojectionoftheeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunctiongetsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. Figure2-14:Eigenfunctionforthe100thexcitedstate.Theplotshowsaprojectionoftheeigenfunctiononthe x;y plane.Forthecolorschemeshown,thefunctiongetsincreasinglypositivefromthebluetotheredendofthespectrum.FiguretakenfromK.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys. Rev.B 81 ,064516.CopyrightbytheAmericanPhysicalSociety http://link.aps.org/doi/10.1103/PhysRevB.81.064516. 53

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CHAPTER3 HEATCAPACITYOFSOLID 4 HE 3.1ExperimentalMeasurementoftheAnomalousHeatCapacityofSolid 4 He Experimentalevidenceforapossiblesupersolidtransitioninsolid 4 Hecouldappear inaspecicheatexperiment.Forinstance,forasecondorderphasetransitionlike thesuperuidtransitionfromthenormaluidtosuperuid,asingularitycalledthe -anomalyappearsintheheatcapacityatthetransitiontemperature.Dorsey etal. showedthatthesupersolidtransitionliesinthesameuniversalityclassasthesuperuid transition[28]predictinga -anomalyintheheatcapacityofthesupersolidtransition. Theyalsofoundthatinhomogeneousstressproduceslocalvariationsofthesuperuid transitiontemperatureandbroadensthepeakintheheatcapacity. Wenowdiscussahighprecisionspecicheatmeasurementofsolid 4 He,conductedbythePennStategroup[25].Inthisexperimenttheyobserveaheatcapacity peakthatcoincideswiththeonsettemperaturefortheperioddropinthetorsionaloscillatorexperiments.Lin etal .usedana.c-calorimetrymethodat0.1Hztostudysolid 4 Hesamplesgrownatconstantvolume.Thoughtheheatcapacityofsolid 4 Hehadbeen longstudiedtheresolutionbelow100mKislimitedbytheheatcapacityofthesample cells.RealizingthisLin etal .usedasamplecellconsistingofundopedsiliconforits smallheatcapacityandhighthermalconductivityatlowtemperatures.Thesamples weregrownoveraperiodof20hours,whichmadethesamplesinsensitivetofurther annealing. Theheatcapacityoftheemptycellandsolid 4 Heforvariousconcentrationsof 3 He isshowninFig.3-1.Theemptycellbackgroundhadbeensubtractedfromthedata. TheyndanexcessheatcapacityinadditiontotheDebyetermatlowtemperatures.On subtractingthe T 3 phononcontribution,abroadpeakinthespecicheatisobserved. Thepeakiscentrednear75mKforthe1ppb,0.3ppmand10ppmsamplesandthepeak heightinallthesethreecasesis 20 Jmol )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 K )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 asshowninFig.3-2. 54

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Lin etal .donotrelatethepeaktothephaseseparationofthemixtureof 3 He4 He,becauseitisnothystereticandthemagnitudeisindependentoftheimpurity concentrationof 3 Heimpuritiesuptofourordersofmagnitude.Thepeakat75mKalso appearstobetheonsettemperaturefortheNCRIin1ppbsolidHeliumsamples[75]. Lin etal .speculatethattheonsettemperatureforthesamplesofotherconcentrations of 3 Hemightalsobecloseto75mK.Theyinterpretedtheirresultsasathermodynamic signatureofthesupersolidtransitioncomplementarytotheNCRI. However,afollowupexperimentfromthesamegroupreportstohaveseen evidenceforanisotopicphaseseparationof 3 He4 He[76],alongwiththerecovery ofaheatcapacitypeak.Theheatcapacitypeakisalsoobservedtobedependenton thesamplepreparationprotocol:blockedcapillaryBCsampleswith1ppband0.3 ppm 3 Heconcentrationshowingapeakof 50 Jmol )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ,whileconstantpressure CPandsolid-liquidcoexistencesamplesshowingapeakof 5 Jmol )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 .Theheat capacitypeakpositionalsochangeswiththepreparationmethod,withBCsamples exhibitingthepeakattemperaturesabove100mKwhileCPandsolid-liquidcoexistence samplesshowingthesignaturecloseto60mKSeeFig.3-3.Thefactthatthespecic heatpeakpersistsevenforasolid-liquidcoexistencesample,whichpossiblyhas thelowestpossiblestrain,accordingtoLin etal. supportstheirinterpretationofa thermodynamicsignatureofthesupersolidtransition. 3.2LatticeGasModelfor 3 HeImpuritiesinSolid 4 He Inthissectionwewouldliketoinvestigateanalternativeexplanationfortheheat capacitypeakobservedbyLin etal .,intermsoftheabsorptionordesorptionof 3 He impuritiesfromdislocationsinsolid 4 He.Theideahereistoseekanon-supersolid interpretationfortheheatcapacitypeakandshowthattheoriginofthepeakobservedis aSchottkyanomaly,typicallyexpectedinamodeltwo-levelsystem. Theestimateofthebindingenergyofa 3 Heimpuritytoadislocationinsolid 4 He isthebasicmotivationalsteptodevelopourmodel.Theboundstatespectrumforan 55

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edgedislocationpotentialandthegroundstateforascrewdislocationpotential,within linearelasticitytheory,hasbeenworkedoutindetailinChapter2.Specically,wehave estimatedinSection2.1.2,therangeofbindingenergiesfora 3 Heimpuritytoanedge dislocationinsolid 4 He,forvaryingcut-offvaluestothepotential.FromFig.2-2we furthernotethatforacut-offvaluecorrespondingtoalatticeconstantof 4 Hewegeta bindingenergyof70mK,whichisclosetothetemperature 75mKatwhichtheheat capacitypeakhasbeenobservedinRef.[25].Thismotivatestheinterpretationthatthe heatcapacitypeakmightbeduetotheunbindingof 3 Heimpuritiesfromdislocationsin solid 4 He. Wemodelthe 3 Heimpuritiesinsolid 4 Heasacombinationofaone-dimensional systemconsistingofimpuritiesboundtothedislocationandathree-dimensionalsystem inthebackgroundofaslowlyvaryingpotentialofsolid 4 Helattice.Theformersystem isaone-dimensionallatticegaswith M numberofbindingsiteseachhavingabinding energy .Thetwosystemsareassumedtobeinchemicalequilibrium,i.etheyhavethe samechemicalpotential.Itisrelevanttomentioninthiscontextofamodeldeveloped byAntsygina etal. [77]tounderstandtheheatcapacityofsolidsolutionsof 3 Hein 4 He. However,theytreatedtheentiresystemasalatticegas. WerstdiscussageneralizedlatticegasmodelwhichisananalogoftheIsing model.TheIsingmodelisawellknownmodelinstatisticalphysicswhichwasdevised asamodelofaferromagnetonalattice.Itconsidersclassicalspinvariablesatthe latticesites,placedinanexternalmagneticeldandinteractingthroughexchange interactions.If S i = 1 isthespincorrespondingtospinupordown, H ext isa uniformexternalmagneticeldand J isthenearest-neighborinteraction,thentheIsing Hamiltonianis H = )]TJ/F24 11.9552 Tf 9.298 0 Td [(H ext N X i =1 S i )]TJ/F24 11.9552 Tf 11.956 0 Td [(J X h ij i S i S j ; 56

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where N isthenumberoflatticesitesand h ij i denotesthenearestneighbors.Signicantly,althoughtheone-dimensionalIsingmodeldoesnotexhibitanitetemperature phasetransitionithasaheatcapacitypeakwithrespecttotemperature. Inalatticegasmodelwhichwasdevisedasamodelofliquid-gastransitions,we dividethespaceintocellseachhavinganoccupationnumber n i =0 ; 1 .Thismaybe achievedfromtheIsingmodelmakingthesimpletransformation n i = S i +1 = 2 .Thus the S i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 spindownand S i =+1 spinupstatesintheIsingmodelrespectively correspondtotheabsenceorpresenceofaparticleinthe i thcellofthelatticegas.If U 0 denotestheenergyofoccupancyinthecellsand U nn denotesthenearest-neighbour interactionsthenthelatticegasHamiltonianmaybewrittenas H = U 0 N X i =1 n i + U nn X h ij i n i n j ; where N isthenumberoflatticesitesand h ij i denotesthenearestneighbours.Again, theparameters U 0 and U nn areanalogoustotheexternalmagneticeld H ext andthe exchangeterm J uptosomeconstantsinrelationtotheIsingmodel. Wefocusonasimpliednon-interactinglatticegasmodelforourproblemby neglectingnearest-neighbourinteractions U nn .ThisreducesthelatticegasHamiltonian to H = U 0 N X i =1 n i ; where U 0 isthebindingenergyof 3 Heimpuritiestoanedgedislocationinsolid 4 He. 3.3StatisticalMechanicsoftheLatticeGasModel ThespecicheatmeasurementinRef.[25]usesaxedamountofsolid 4 He samplewithacontrolledconcentrationof 3 Heimpuritiesinasamplecellofaparticular volume.Thesystemcanbeconsideredclosedinthatheattransferispossibletothe cellbutthevolumeandnumberofboth 3 Heand 4 Heatomshaverelativelywell-known xedvaluesforaparticularexperiment.TheFermitemperatureforthe 3 Heimpuritons fortypicaldensitiesisfoundtobemuchlowerthanourbindingenergyestimatesSee 57

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AppendixCfordetails.ThissuggeststhatclassicalBoltzmannstatisticsshouldbe sufcienttomodelthebinding-unbindingprocessandtheeffectofspincanbeignored. Also,weignoreeffectslikenearestneighborinteractionsastheessentialphysicsofthe modeliscapturedeveninthiscompletelynon-interactingtwo-levelpicture.Thetotal internalenergyofthesystemissimplythesumofenergiesofthetwosubsystems: E = h N 3D ih E 3D i)]TJ/F24 11.9552 Tf 19.261 0 Td [(M h N site i ; where h N 3D i and h N site i aretheequilibriumnumbersof 3 Heatomsinthethreedimensionalbulkandboundtotheone-dimensionalsystemofdefectspersiterespectivelyand h E 3D i istheaverageenergyper-particleinthethree-dimensionalbulk givenintermsofthesingle-particlepartitionfunctionforthe3Dsubsystem: Z 1 as h E 3D i = )]TJ/F25 7.9701 Tf 13.075 4.707 Td [(@ @ ln Z 1 ,where =1 =k B T k B beingtheBoltzmannconstant.There beingnointeractionsamongtheparticlesorsitesitisconvenienttoworkinterms of Z 1 ,asthenthetotalpartitionfunctionforthe3Dsubsytemisrelatedtothisas: Z = Z 1 N 3D =N 3D .Thechemicalpotential forthethree-dimensionalsubsystem isrelatedtothenumberdensityandtemperatureas h N 3D i = e Z 1 : Thebindingstatisticsoftheone-dimensionalsystemisgivenbythestandardexpression h N site i = e + 1+ e + : Theabovetwosubsystemsareinchemicalequilibriumandsoshouldhavetheir chemicalpotentialsequal.This,alongwiththerequirementthatthetotalnumberof 3 He atomsintheentiresystemisconstant, N tot = h N 3D i + M h N site i ; 58

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letsuseliminatethechemicalpotential,andderiveexpressionsforaveragenumberof impuritiesineithersubsystem.Theaveragenumberinthethree-dimensionalbulkis h N 3D i = 1 2 N tot )]TJ/F24 11.9552 Tf 11.955 0 Td [(M )]TJ/F24 11.9552 Tf 11.956 0 Td [(p T + 1 2 q N tot )]TJ/F24 11.9552 Tf 11.955 0 Td [(M )]TJ/F24 11.9552 Tf 11.955 0 Td [(p T 2 +4 N tot p T ; where N tot isthetotalnumberof 3 Heimpurities,and p T = e )]TJ/F25 7.9701 Tf 6.586 0 Td [(=k B T Z 1 : Itcanbeeasilycheckedthatinthehightemperaturelimitcomparedtothebinding energy h N 3D i tendsto N tot whileinthelowtemperaturelimititgoesto0or N tot )]TJ/F24 11.9552 Tf 12.374 0 Td [(M dependingonwhether N tot islessthanorgreaterthan M ,respectively.Thisistotally accordingtoourexpectationsthatalltheimpuritiesarefreeathightemperaturesandall availablebindingsitesareoccupiedatlowtemperature. Thetotalheatcapacitycanbewrittenas: C = h N 3D i d h E 3D i dT + + h E 3D i d h N 3D i dT : Thespecicheathastwodistinctcontributions:oneabackgroundtermduetothe3D systemandasecondpartfromthebinding-unbindingtothe1Dsystemofdefectswhich isresponsibleforthebump. Weinvokeatight-bindingmodeltosolveforthethermodynamicpropertiesofthe three-dimensionalsubsystem.Thesingleparticlepartitionfunctionisgivenby Z 1 = V Z d 3 k 3 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(" ~ k ; wheretheintegrationisovertherstBrillouinzone.Theexactenergyspectrumwill dependonthelatticegeometry. Herewechooseasimplecubiclatticeforsimplicity,butitcanbearguedthat thegeneralfeaturesofthemodeldonotdependonthespecicsofthelattice.The 59

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tight-bindingenergyforasimplecubiclatticeisgivenby ~ k = E 0 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 t cos k x a +cos k y a +cos k z a ; where a isthelatticeconstantand t isthetransferintegral.Forthischoiceoflattice structure,thepartitionfunctionis Z 1 = V I 0 t a 3 ; where I 0 x isthemodiedBesselfunctionoftherstkind.Fromthepartitionfunction wecanobtainexpressionsforvariousthermodynamicquantitiesinastandardway. ThetotalspecicheatcannowbeobtainedbydifferentiatingEq.3withrespectto temperatureforaconstantvolumeandconstanttotalnumberof 3 Heimpurities.Details ofthecalculationfortheheatcapacityforthe3DsystemareincludedinAppendixD. Thetight-bindingexchangetermfor 3 Heismuchsmallerthanthebindingenergyas thebandwidthisknowntobearound1mK[78],whichmeansthatthecontributionfrom h E 3D i isnegligibleincomparisonto inEq.3.Thismeansthatthespecicheat expressionsimpliesintoasumofthetwocontributions,the3Dtermbeingimportant onlyatlowtemperaturesofaroundamKbutdyingoutas 1 =T 2 athighertemperatures seeAppendixDfordetails,whiletheothertermdependsonthechangeinnumberof boundimpuritonswithtemperatureandisexpectedtoproducetheSchottkypeak. Thespecicheatcapacityexpressedaspermoleof 4 HeisplottedagainsttemperatureinFig.3-4,fordifferentconcentrationsof 3 Hewithdefectconcentrationandbinding energyvalueschosenreasonablytoproducepeakinthespecicheatwithmagnitudes andpositionsclosetotheLin etal. experiment,atleastforsomeconcentrations. 3.4DiscussionofResultsObtainedFromtheModel FromtheLin etal. experimentandalsofromourbindingenergyestimateswe expectthepeaktooccurattemperaturesmuchhigherthanamKthisisevidentin Fig.3-4,whichmeansweareinthehightemperaturelimitofthetightbindingmodel 60

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andthespecicsofthelatticestructuredonotmatterasfarasthispeakisconcerned. Thisalsoletsusestimatethepeakposition.For T t = k B T ln h N 3D i )]TJ/F24 11.9552 Tf 11.955 0 Td [(k B T ln C + O T )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ; wheretheco-efcient C isrelatedtothedensityofstatesfunctionas R g d andis thereforethetotalallowednumberofstates.Thiswhenputintotheeffectivecondition forSchottky-likepeak, k B T + ,givesustheapproximatescalingofpeak temperaturewith 3 Heconcentration: T k B 1 )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(ln h N 3D T i C )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 : Thisisaself-consistentdependence.Forahighlysaturatedcaseweexpect h N 3D i to bealmostequalto N tot irrespectiveoftemperatureandthepositionscalingbehavior simpliesfurther.Thisshowsthatincreasing 3 Heconcentrationshouldcauseincrease inpeaktemperaturewhichiswhatisobservedinthespecicheatplotinFig.3-4. Thepeakisgeneratedbythebinding-unbindingof 3 Heimpuritiesfromthedefect sitesandsoitsmagnitudeisdeterminedbythenumberofparticlesactuallyparticipating inthebinding-unbindingprocess.Thusinthesaturatedcase,thepeakheightis determinedmorebythenumberofdefectsites M increaseswithincreasing M andfor theunsaturatedcaseitvariesstronglywiththetotalnumberofimpurities N tot increases withincreasing N tot .Thereisalsoanobservedweakerreversedependenceofpeak heightonthenumberofimpuritiesinthesaturatedcaseincreasesfordecreasing N tot Thisisnotexpected apriori butcomesoutofthestatistics. Furthermore,asseeninFig.3-4thepeakheightdecreaseswithincreasing 3 He concentration.ThisisdifferentfromtheexperimentalresultsseenbyLin etal. wherethe peakheightdoesnotdependonthe 3 Heconcentration.Althoughseveralconcentrations yieldpeaksthatcanbearguedtobeofsimilarmagnitudewithinexperimentalerror, the1ppbconcentrationclearlyshowsamuchsmallerpeakthansaya1ppmone. 61

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Thisdiscrepancyisonlytobeexpectedsincetheroleofimpuritiesiscentraltoour modelwhereasLin etal. 'sworkseemstosuggestaspecicheatpeakevenwithout anysignicantimpurityconcentration.Thedefectconcentration M alsoturnsoutto beadecidingparameterfortheheightofthepeakforaxedconcentrationof 3 He. Thedefectdensitycanhoweveronlyberoughlyestimatedfromultrasoundattenuation experiments.Thevaluesof requiredtoproducepeaksat75mKasseenbyLin et al. needtobeoftheorderofakelvin,whichagreeswithourestimateforanedge dislocationwithoutacutoffseeChapter2aswellasthequantumMonteCarlo estimatesprovidedbyCorboz etal. [79].Howeverourestimateswithcutoffsarean orderofmagnitudelower.TheLin etal. resultsalsodifferfromourmodelinhavingvery broadpeakscenteredaround75mK.Itishoweverpossibletohavebroadpeakswithin ourmodelifweworkwithsomedistributionofbindingenergiesinsteadofasinglevalue of .Thiswouldalsobephysicallymorereasonablesinceinarealsamplethereare likelytobedifferentkindsofdislocationseachwithitsownbindingenergytowithinan orderofmagnitudeorsooftheestimatedvalue. 62

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Figure3-1:Heatcapacityoffoursamplescontainingdifferentamountsof 3 Heimpurities.Thegreenlineistheexpected T 3 behaviorforaDebyesolid.Blackopentriangles, 30ppm;purpleopensquares,10ppm;bluetriangles,0.3ppm;redcircles,1ppb.The emptycellbackgroundisshownasblackopencircles.Inset,diagramofthesiliconcell. ReprintedbypermissionfromMacmillanPublishersLtd:Nature[X.Lin,A.C.Clarkand M.H.W.Chan,Nature 449 ,1025],copyright. 63

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Figure3-2:Specicheatpeaksofthe1ppb,0.3ppmand10ppmsamples.Purpleopen squares,10ppm;bluetriangles,0.3ppm;redcircles,1ppb.Thestandarddeviationofthe 1ppbdatasetisshownasreddashedlines.Inset,comparisonofthespecicheatofthe threesamplesbeforesubtractionoftheconstanttermofthe10ppmsampledottedline at59 Jmol )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 K )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 .ReprintedbypermissionfromMacmillanPublishersLtd:Nature[X. Lin,A.C.ClarkandM.H.W.Chan,Nature 449 ,1025],copyright. 64

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Figure3-3:Hysteresisinthemolarheatcapacityof10ppmand500ppmsamples.The insetshowstheexcessnon-phononcontributiontothespecicheat.Thehysteresisand upturnsignatureoftheheatcapacityareattributedtoisotopicphaseseparationof 3 He4 Hemixture.ReprintedFig.3withpermissionfromX.Lin,A.C.Clark,Z.G.Chengand M.H.W.Chan,Phys.Rev.Lett. 102 ,125302,CopyrightbytheAmerican PhysicalSociety.http://link.aps.org/doi/10.1103/PhysRevLett.102.125302 65

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Figure3-4:Specicheatcapacityforvariousconcentrationsof 3 Heobtainedfromthe model.Theplotsareshownforabindingenergy =0 : 8 Kandadefectconcentration m =30 ppb. 66

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CHAPTER4 DISLOCATIONINDUCEDSUPERFLUIDITYINAMODELSUPERSOLID 4.1Introduction Iwouldrstliketomentionthatmostofthischapterisinpreprintform[74]andis currentlyunderreviewforpublicationinPhysicalReviewB. Asdiscussedinearlierchapters,sampledefects,likelyintheformofdislocations, playanimportantroleinexplainingdisparaterecentexperimentalresultsonsolid 4 He[19].Mostexistingtheoreticalworkshavefocusedonsingledislocations;for instance,quantumMonteCarlostudies[80]haveshowntheexistenceofsuperuidity alongthecoreofscrewdislocationsinamodel 4 Hecrystal.Inaphenomenological approach,Landaumodelsshowthatsuperuiditynucleatesrstonedgedislocations inasuperconductororaBosesolid[30,54,81,82]andShevchenko[31,32],Toner [30],BouchaudandBiroli[83]haveworkedoutsomeofthepropertiesofanetworkof dislocationsinaquantumcrystal.Inthischapterwecontinueandextendtheworkof ShevchenkoandToner,bysystematicallyderivinganeffectiverandom-bond XY model foranetworkofsuperuiddislocations. Thischapterisorganizedasfollows.Werstintroduceourmodelforsuperuidityin thepresenceofasingleedgedislocation,bycouplingtheelasticstraineldduetothe dislocationtothesuperuidorderparameter[28].Then,usingalinearstabilityanalysis, weshowthatsuperuidityalwaysnucleatesrstonthedislocation.Asystematic, weaklynonlinearanalysisisdevelopedandusedtoderivetheone-dimensionalLandau theoryforsuperuidityalongthedislocationaxis.Finally,weincorporatethermal uctuationsintoourdescriptionofthesuperuidity,anddeterminetheeffectivecoupling betweentwositesalongasingledislocation.Thelastresultmotivatesaneffective descriptionofthenetworksuperuidityintermsofarandom-bond XY model,and wearguethatsuchamodelhasanorderingtemperatureexponentiallysensitiveto thedislocationdensity.Aseriesofappendicesfurtherdevelopstheweaklynonlinear 67

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analysis:AppendixEshowshowtoimprovethenaiveweaklynonlinearanalysisusing themethodofstrainedcoordinates;AppendixFdemonstratestheefcacyofthe analysiswithsimplevariantoftheedgedislocationmodel;andAppendixGextendsthe equilibriumresultstothetime-dependentGinzburg-Landautheory. 4.2Superuid-DislocationCouplinginLandauTheory FollowingDorsey,Goldbart,andTonerDGT[28]weanalyzetheorderedphase usingaLandautheoryinwhichthesuperuidorderparameter x couplestothe straintensor u ij inducedbyaquencheddislocation.Tosimplifytheanalysisweassume thesolidisisotropic,sothesuperuidcouplestothetraceofthestraintensori.e., divergenceofthedisplacementeld,whichisthefractionallocalvolumechangeofthe solid.Forasingleedgedislocationalongthe z -axis,withBurger'svector b alongthe y axis,thetraceofthestraintensoris[84] u ii = 4 2 + b cos r ; where and aretheLam eelasticconstants,wherewehaveintroducedthecoordinates x = r ;z ,with r inthe x )]TJ/F24 11.9552 Tf 12.036 0 Td [(y plane[ r; arepolarcoordinatesinthe x )]TJ/F24 11.9552 Tf 12.037 0 Td [(y plane]. TheLandaufreeenergyfunctionalfortheisotropicsupersolidisthen[28,30] F = Z d 3 x c 2 jr j 2 + 1 2 a r j j 2 + 1 4 u j j 4 ; with a r = a 0 t 0 + B cos r : Here a 0 c ,and u arephenomenologicalparametersallpositive; t 0 = T )]TJ/F24 11.9552 Tf 12.498 0 Td [(T 0 =T 0 is thereducedtemperature,with T 0 themean-eldcriticaltemperatureforthesupersolid transitioninthe absence ofthedislocation t 0 > 0 isthenormalsolid,and t 0 < 0 the supersolid;and B isacouplingconstantintowhichwehaveabsorbedtheelastic constants.Castinthisform,thecouplingbetweenthedislocationandthesuperuid 68

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orderparametercanbethoughtofasalocalchangeinthecriticaltemperaturedueto localchangesinthespecicvolumeofthesolid. Tosimplifythesubsequentanalysis,weintroducethecharacteristiclengthscale l = c=a 0 B ,orderparameterscale = a 0 B= p cu ,andfreeenergyscale F 0 = a 0 Bc= 2 u andthedimensionlessprimedquantities x 0 = x =l 0 = = ,and F = F=F 0 ;intermsof thedimensionlessquantitiesthefreeenergybecomes F = Z d 3 x jr j 2 +[ V r )]TJ/F24 11.9552 Tf 11.956 0 Td [(E ] j j 2 + 1 2 j j 4 ; where V r =cos =r E )]TJ/F24 11.9552 Tf 22.13 0 Td [(ct 0 =a 0 B 2 ,andwehavedroppedtheprimesonallquantities forclarityofpresentation. 4.3LinearStabilityAnalysis Beforeproceedingfurther,weanalyzethebehaviorinthehigh-temperature,normal phase.Inthisphasewecanneglectthequarticterminthefreeenergy;theresulting quadraticfreeenergyis F 0 = Z d 3 x jr j 2 +[ V r )]TJ/F24 11.9552 Tf 11.955 0 Td [(E ] j j 2 = Z d 3 x ^ H )]TJ/F24 11.9552 Tf 11.955 0 Td [(E ; wheretheHermitianlinearoperator ^ H isgivenby ^ H = r 2 + V r : Wecandiagonalizethefreeenergybyintroducingacompletesetoforthonormal eigenfunctions n x of ^ H ^ H n = E n n ; where n labelsthestates,andweassumethattheeigenvalues E n areorderedsuch that E 0
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eigenfunctionswereobtainedwithextensivenumericalworkinRef.[54].Expandingthe orderparameterintermsoftheeigenfunctions, x = X n A n n x ; withexpansioncoefcients A n ,andsubstitutingintothefreeenergy,afterusingthe orthogonalitypropertiesoftheeigenfunctionsweobtain F 0 = X n E n )]TJ/F24 11.9552 Tf 11.955 0 Td [(E j A n j 2 : Thefreeenergyispositiveaslongas E n >E ,forall n .Recallthat E )]TJ/F15 11.9552 Tf 22.406 0 Td [( c=a 0 B 2 T )]TJ/F24 11.9552 Tf -451.064 -23.908 Td [(T 0 =T 0 with a 0 and c bothpositive,sohightemperatures T correspondtolarge,negativevaluesof E .Aswedecrease T E increases,untileventuallywehita condensation temperature T cond atwhich E T cond = E 0 ;belowthistemperaturethequadraticfree energy F 0 becomesunstablenegative.Rearrangingabit,wehave T cond )]TJ/F24 11.9552 Tf 11.955 0 Td [(T 0 T 0 = )]TJ/F24 11.9552 Tf 9.298 0 Td [(E 0 a 0 B 2 c : If ^ H hasnegativeeigenvaluesi.e.,iftheequivalentSchr odingerequationhasbound statesthen T cond >T 0 ,andthedislocationinducessuperuidity above thebulkordering temperature.AsemphasizedinRefs.[30,54,81],thedipolarpotential cos =r ,which hasanattractiveregionirrespectiveofthecouplingconstants, always hasanegative energyboundstate.Wearethusleadtothesurprisingandimportantconclusion [81,30]thatsuperuidityrstnucleatesaroundtheedgedislocationbeforeappearingin thebulkofthematerial. Justbelowthecondensationtemperature T cond ,thenucleatedorderparameterhas theform = A 0 0 r ; where 0 isthenormalizedgroundstatewavefunctionand A 0 isanamplitudethatis xedbythenonlineartermsinthefreeenergy[85].SubstitutingEq.4intothe 70

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dimensionlessfreeenergy,Eq.4,weobtain F = E 0 )]TJ/F24 11.9552 Tf 11.955 0 Td [(E j A 0 j 2 + 1 2 g j A 0 j 4 ; wherethecouplingconstant g isgivenby g = Z d 2 r 4 0 r : Minimizingthefreeenergywithrespectto A 0 ,weobtain A 0 = p E )]TJ/F24 11.9552 Tf 11.955 0 Td [(E 0 =g; andtheminimumvalueofthefreeenergyis F min = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [( E )]TJ/F24 11.9552 Tf 11.955 0 Td [(E 0 2 2 g : FromtheextensivenumericalworkofDasbiswas etal. [54]weknowthatforthedipole potentialthegroundstateenergyis E 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 139 withtheenergyoftherstexcited state E 1 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 0414 ,andthecouplingconstant g =0 : 0194 Torecapfollowingtheworkofpreviousauthors[30,81]wehaveshownthat superuidityalwaysnucleates rst onedgedislocations,andwehavecalculatedthe formoftheorderparameternear E 0 ,thethresholdvalueof E i.e.,attemperatures justbelowthecondensationtemperature.Physically,weimagineacylindricaltube ofsuperuid,witharadiusequaltothetransversecorrelationlengthoforder1inour dimensionlessunits,thatencirclesthedislocationshownschematicallyinFig.4-1. However,thisnaivemean-eldpictureignoresthethermaluctuationswhichdestroy theone-dimensionalsuperuidityonlonglengthscales.Whatisneededisaneffective one-dimensionalmodelforthesuperuid,capableofcapturingnontrivialuctuation effects.Inthenextsectionwederivethisone-dimensionalmodelusingasystematic, weakly-nonlinearanalysisnearthethreshold E 0 71

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4.4WeaklyNonlinearAnalysisNearThreshold Withinthemean-eldLandautheory,theorderparametercongurationsthat minimizethefreeenergyaresolutionstotheEuler-Lagrangeequation F =0= r 2 +[ V r )]TJ/F24 11.9552 Tf 11.955 0 Td [(E ] + j j 2 : Thisnonlineareldequationisdifculttosolve,evennumerically.Instead,weresorttoa weaklynonlinearanalysis [86]nearthethresholdforthelinearinstability.Ourgoalisto integrateoutthemodestransversetothedislocationandobtainaneffectivemodelfor theone-dimensionalsuperuidnucleatedalongthedislocation.Wethentreatthermal uctuationsofthisone-dimensionalsuperuidinthenextSection. Westartbyintroducingacontrolparameter E )]TJ/F24 11.9552 Tf 11.956 0 Td [(E 0 thatmeasuresdistancefromthelinearinstability.Fromtheanalysisinthepreceding Section,weseethattheorderparameternearthresholdscalesas 1 = 2 ,whichsuggests arescalingoftheorderparameter = 1 = 2 ; with aquantitywhoseamplitudeis O .Next,notethatifwehadincludedtheplane wavebehavioralongthe z -axisintheanalysisinSec.4.3,thecoefcientof j A n j 2 in thequadraticfreeenergy,Eq.4,wouldbe )]TJ/F24 11.9552 Tf 9.299 0 Td [( + k 2 .Thissuggeststheimportant uctuationsalongthe z -axisoccuratwavenumbers k 1 = 2 ,oratlengthscalesoforder )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = 2 i.e.,longwavelengthuctuationsareimportantclosetothreshold;therequired rescalingis z = )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = 2 : SubstitutingthesevariablechangesintoEq.4,andwriting E = E 0 + ,weobtain ^ L = @ 2 + )-222(j j 2 ; 72

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wheretheHermitianlinearoperator ^ L isgivenby ^ L = r 2 ? + V r )]TJ/F24 11.9552 Tf 11.955 0 Td [(E 0 ; with r 2 ? theLaplacianindimensionstransverseto z .Next,weexpand inpowersof = 0 + 1 + 2 2 + :::: Collectingterms,weobtainthefollowinghierarchyofequations: O : ^ L 0 =0 ; O : ^ L 1 = @ 2 0 + 0 )-222(j 0 j 2 0 ; O 2 : ^ L 2 = )]TJ/F15 11.9552 Tf 11.956 0 Td [(3 2 0 1 : Thesolutionofthe O equationisthenormalizedgroundstateeigenfunction, 0 r ;thereisanoverallintegrationconstant A 0 ,sothat 0 = A 0 0 r : Substitutethisintotherighthandsideofthe O equation, ^ L 1 = 0 @ 2 A 0 + 0 A 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 3 0 j A 0 j 2 A 0 : Wecandetermine A 0 byleftmultiplyingthisequationby 0 ,integratingon d 2 r ,and usingthefactthat ^ L isHermitian,tond @ 2 A 0 + A 0 )]TJ/F24 11.9552 Tf 11.955 0 Td [(g j A 0 j 2 A 0 =0 ; where g isdenedinEq.4.Thisisthe solvabilitycondition fortheexistence ofnontrivialsolutionsofthe O equation.Inprinciplewecouldsolvethisequation for A 0 ,substitutebackintotherighthandsideofthe O ,andsolvetheresulting 73

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inhomogeneousequationtoobtain 2 .Inpracticethisisdifcultforthedipolepotential, sowewillstopatthisleveloftheperturbationtheory. WecanrecastEq.4intermsof = 1 = 2 z and A 0 = )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = 2 as @ 2 z z + )]TJ/F24 11.9552 Tf 11.955 0 Td [(g j j 2 =0 ; whichistheEuler-Lagrangeequationforthefreeenergyfunctional F = Z dz 1 2 j @ z j 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 j j 2 + g 4 j j 4 : Reinstatingthedimensions,weobtain F = Z dz c 2 j @ z j 2 + a 2 j j 2 + b 4 j j 4 ; where a = a 0 t t = T )]TJ/F24 11.9552 Tf 12.015 0 Td [(T cond =T cond isthereducedtemperaturemeasuredrelativetothe condensation temperature,and b = gu .Torecap,wehaveintegratedoutthetransverse degreesoffreedomtheuctuationsofwhichhaveanonzeroenergygapandreduced thefullthree-dimensionalproblemtoaneffectiveone-dimensionalmodel.Inthenext sectionwewillstudytheuctuationsofthisone-dimensionalmodel,andderivean effectivephase-onlymodelforadislocationnetwork. Adeciencyofourperturbativeapproachistheappearanceoftheterm 0 on therighthandsideofthe O equationinEq.4.Since 0 isazeromodeof ^ L ,it becomesasecularterminsubsequentordersoftheperturbativeexpansion.Forsome systemse.g.,anonlinearoscillatorsuchseculartermsleadtounboundedsolutions thatrequireregularizationusingamoresophisticatedmultiplescaleanalysis[87].In ourapplicationtheseculartermsarebenign,asthesolutiondecaysatinnitybutwith thewrongdecayrate.However,inAppendixEweshowthattheseculartermscanbe eliminatedusingthe methodofstrainedcoordinates [87]withresultsthat,onthewhole, areequivalenttotheresultsabovebutwithanimprovedapproximationtotheorder parameter.InAppendixFweapplyourperturbativemethodstothetwo-dimensional 74

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Coulombpotential V r = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 =r ,forwhichthecalculationscanbeexplicitlyworked outthrough O 2 ,andweshowthattheresultsareincloseagreementwithdetailed numericalsolutionsofthenonlineareldequation.Finally,inAppendixGweshow thatthesamemethodsmaybeusedtotreatdynamicequationsofmotionfortheorder parameter. 4.5DerivationoftheDislocationNetworkModel Sofarwehaveconsideredasingleedgedislocation,reducingthefullthreedimensionalLandautheorytoaneffectiveone-dimensionaltheoryforasuperuidtube localizednearthedislocationcore.However,a 4 Hecrystalwillconsistofatangleof dislocations,manyofwhichwillcrosswhentheycomewithinatransversecorrelation lengthofeachother.Conceptually,wecanmodelthisasarandomlatticeornetwork ofdislocations,withthecrossingpointsservingaslatticesites.Whilethermaluctuationsdestroyanylongrangeorderinasingle,one-dimensionaltube,thelatticeoftubes willgenerallyorderatatemperaturecharacteristicofthephasestiffnessbetweenadjacentlatticesites.ThisisthemotivationbehindthemodelsdevelopedbyShevchenko [31]andToner[30].InthisSectionwerevisittheShevchenkoandTonermodelswitha systematicapproachtocalculatingthecouplingbetweenadjacentlatticesitesinthenetworkmodel,andobtainnewresultsonthelengthscalingofthecouplingconstant.We concludewithsomeobservationsregardingimplicationsofourresultsforexperiments ontheputativesupersolidphaseof 4 He. Westartbyconsideringthecorrelationsbetweentwopointsalongasinglesuperuidtube,separatedbyadistance L .Usingthenotationoftheprevioussection, = 1 and L = 2 arethevaluesofthesuperuidorderparameterattwosites alongthetube.Thecorrelationsarecapturedbythepropagator K 2 ;' 1 ; L ,whichcan beobtainedfromthefunctionalintegral K 2 ;' 1 ; L =exp )]TJ/F24 11.9552 Tf 9.299 0 Td [(H e = Z L = 2 = 1 D ';' exp )]TJ/F24 11.9552 Tf 9.299 0 Td [( Z L 0 dz c 2 j @ z j 2 + a 2 j j 2 + b 4 j j 4 ; 75

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where a b and c aretheparametersoftheone-dimensionalLandautheoryderived intheprevioussection,and H e istheeffectiveHamiltonianthatcharacterizesthe couplingbetweenthelatticesites.Castinthisform,weseethatthefunctionalintegral fortheclassicalone-dimensionalsystemisequivalenttotheFeynmanpathintegralfor aquantumparticleinatwo-dimensionalquarticpotentialtwo-dimensionalbecausethe orderparameter iscomplex,with z intheclassicalproblemreplacedbytheimaginary time forthequantumsystem[88].Indeed,previousauthorshaveusedthisanalogy tostudytheeffectofthermaluctuationsontheresistivetransitioninone-dimensional superconductors[89,90]obtainingessentiallyexactresultsforthecorrelationlength andthermodynamicproperties.ConsistentwiththeMermin-Wagnertheorem[91]these authorsndnosingularitiesinthethermodynamicproperties,andaone-dimensional correlationlengththatgrowsasthetemperatureisreduced,butneverdiverges[90]. MostoftheseresultsareconvenientlycapturedusingasimpleHartreeapproximation [92]inwhichthequartictermisabsorbedintothequadratictermwiththequadratic coefcientredenedas a = a + 1 2 b hj j 2 i ; where a = a 0 T )]TJ/F24 11.9552 Tf 11.961 0 Td [(T cond =T cond and hj j 2 i astatisticalaveragewithrespecttotheeffective quadratictheory.Carryingouttheaveraging,weobtain[92] hj j 2 i = k B T c 2 ? ; where = c= a 1 = 2 isthecorrelationlengthfortheone-dimensionalsystemalongthe superuidtube,and ? isthecross-sectionaldimensionofthe1Dsuperuidregionthe transversecorrelationlength,asshowninFig.4-1.InsertingthisresultintotheHartree expressionofEq.4,weobtainacubicequationforthecorrelationlength, 1 2 = 1 2 0 + k B Tb 2 c 2 2 ? ; 76

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where 0 = c=a 1 = 2 istheGaussiancorrelationlength.While 0 divergesat T cond remainsniteatalltemperaturesgrowingasthetemperatureislowered[90],reecting thelackoflong-rangeorderintheone-dimensionalsuperuidtube.Fig.4-2shows aqualitativeplotofthecorrelationlengthsobtainedfrommean-eldandHartree approximationinone-dimension. ContinuingwiththeHartreeapproximation,wecanndtheexplicitformofthe propagatorbyexploitingananalogywiththepartitionfunctionforatwo-dimensional quantumharmonicoscillator;theresultis[88] K 2 ;' 1 ; L = k L exp )]TJ/F24 11.9552 Tf 36.455 8.088 Td [(c 2 sinh L= j 2 j 2 + j 1 j 2 cosh L= )]TJ/F15 11.9552 Tf 9.903 0 Td [(2 j 2 jj 1 j cos 1 )]TJ/F24 11.9552 Tf 9.902 0 Td [( 2 ; wheretheprefactor k L = c= 2 sinh L= ,and 1 = j 1 j e i 1 and 2 = j 2 j e i 2 .The effectiveHamiltonianuptoanadditiveconstantisgivenby H e = c 2 coth L= j 2 j 2 + j 1 j 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [(J 12 L cos 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( 2 ; where J 12 L = c j 2 jj 1 j sinh L= = 8 > < > : c j 2 jj 1 j =L;L= 1; c j 2 jj 1 j = e )]TJ/F25 7.9701 Tf 6.587 0 Td [(L= ;L= 1 : ThelastterminEq.4istheoneofinterest,asitcouplesthephasesatthe neighboringsitesthroughaneffectiveferromagneticcouplingconstant J 12 .The behaviorof J 12 asafunctionoftheinter-siteseparation L isoneofourimportant resultsforsmall L= J 12 1 =L ,reproducingtheresultofToner[30]whereasforlarge L= J 12 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(L= .Since isniteforall T ,forasufcientlydilutenetworkofdislocations wewillalwayssatisfythelatterconditioni.e.,dilutenetworksofdislocationspossess couplingconstantsexponentiallysmallinthedislocationdensity. 77

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Weemphasizethataone-dimensionalsystemwithacontinuoussymmetrydoes notexhibitlongrangeorderorderparametercorrelationsdecayexponentiallyonthe scaleofthecorrelationlength .Asaresult,wecannotreplacethefullLandaufunctionalwithaphase-onlymodelinwhichoneassumesawell-formedorderparameter amplitude,andweneedtotreatthephaseandamplitudeuctuationsonthesame footing,inthespiritoftheworkbyScalapino etal. [90].Inthisrespectourresultsdiffer fromToner's[30],whousedaphase-onlytreatmentandfoundacouplingconstantthat scalesas 1 =L .Toner'sresult does applyatshortlengthscales L ,wherethereis localsuperuidorderandaphase-onlyapproximationcanbeused.Ontheotherhand, atlonglengthscales L theexponentialdecayofcorrelationsresultsinacoupling constantthatisexponentiallysmallin L .TheresultinEq.4correctlycapturesboth thesmallandlargedistancebehaviorofthecouplingconstant. SofarwehavesystematicallyderivedaneffectiveHamiltonianthatdescribesthe phasecouplingbetweentwopointslatticesitesonasingledislocation.Tomakethe conceptualleaptothedislocationnetwork,weproposethattheappropriatemodelfor thenetworkisarandombond XY modeloftheform H = )]TJ/F29 11.9552 Tf 11.291 11.358 Td [(X h ij i J ij cos i )]TJ/F24 11.9552 Tf 11.956 0 Td [( j ; where h ij i representsnearest-neighborlatticesites,and J ij isapositivecoupling betweenthesitesthatscalesas e )]TJ/F25 7.9701 Tf 6.587 0 Td [(L ij = forasufcientlydilutenetworkofdislocations. AsnotedbyToner[30]therandomnessisirrelevantintherenormalization-groupsense [93]andinthreedimensionsweexpectthesuperuidityinthenetworktoorderwhen thetemperatureisoforderthetypicalcouplingstrength [ J ij ] ;i.e., k B T c = O [ J ij ] .Again, foradilutenetworkofdislocationswitharealdislocationdensity n 1 =L 2 ,wewould nd T c e )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = n 1 = 2 ,aremarkablysensitivedependenceonthedislocationdensity. Itmightbenaturaltoaskhereastowhatextentthefeaturesofthemodelproposed herecorrelateswithcurrentexperimentsonsolid 4 He.Theemergingconsensusis 78

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thattheputativesupersolidresponsedependsonsamplequalityandpreparationfor instance,seeRittnerandReppy[11].Asanexample,theNCRIfractionshowsa sensitivedependenceonthesamplequalityvaryingfromasmuchas 20% forsamples preparedbytheblockedcapillarymethodandthushavingmoredisorderto 0 : 5% for thosepreparedunderconstantpressure[12],whiletheonsettemperatureitselfshows aweakdependenceondisorder.Thiscouldsuggestthatexistingexperimentsareinthe highdislocationdensityregime L wherethecouplingconstantandthereforethe criticaltemperaturescaleas 1 =L .Unfortunately,therehavebeennosystematicstudies thatcorrelatethedislocationdensityofsolid 4 Hesampleswiththeonsettemperaturefor supersolidsignal.Ultrasoundattenuationexperimentsonsolid 4 He[94,95]suggested dislocationdensitiesrangingfrom10 4 -10 6 cm )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 ;atpresentthedislocationdensityisnot evenknowntowithinanorderofmagnitude,whichmakesitdifculttopredict T c from ourtheory.Moreover,theLandautheoryparametersforsolid 4 Heareunknownunlike thesituationinconventionalsuperconductors,forexample,compoundingthedifculties indirectlycomparingourresultstotheexistingexperiments.However,thedislocation networkmodelstillservesasausefulconceptualstartingpointforunderstandingsome aspectsofdislocationsonBosecrystals,andourworkisimportantinestablishing theequivalencebetweenthenetworkmodelandamorefundamentalLandautheory. Finally,wenotethatourtheorydoesnotincludedynamicshowever,seeAppendixG forarudimentarydiscussionofdynamicsandwecannotcommentonexperimentsthat studytherateofsuperow,suchasbyRayandHallock[24].Howeverwebelievethat themodelproposedaboveservesasausefulstartingpointtonumericallysimulateand understandsomeoftheseeffects. 79

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Figure4-1:Schematicdiagramshowingthedislocationaxisalong z andthetubular superuidregionthatdevelopsaroundit.Theradiusofthecylinderisdeterminedbythe lengthscaleofthegroundstatewavefunction.Theaxisofthetubewillbeoffsetfrom thedislocationaxis. 80

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Figure4-2:Qualitativeplotof1Dcorrelationlengthobtainedfrommean-eldand Hartree-Fockapproximation.Whilethemeaneldcorrelationlength MF wrongly divergesatatransitiontemperature T C ,theHartree-Fockcorrelationlength HF remains nite,consistentwiththefactthatlong-rangeordercannotbeestablishedin1D. 81

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Figure4-3:Orderparameter r versus r for =0 : 09 .Thebluelineisthenumericalsolution,thegreenlineistheone-termperturbativeresult,andtheredlineisthetwo-term perturbativeresult. 82

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Figure4-4:Orderparameter r versus r for =0 : 43 .Thebluelineisthenumericalsolution,thegreenlineistheone-termperturbativeresult,andtheredlineisthetwo-term perturbativeresult.Thetwotermexpansionprovidesanexcellentapproximationtothe numericalresult,evenfor =0 : 43 83

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Figure4-5:Orderparameteramplitude asafunctionof .Thecrossesarethenumericalresults,thegreenlineistheone-termperturbativeresult,andtheredlineisthe two-termperturbativeresult.Theplotcomparesthenumericalresultswiththeperturbationtheoryforawiderangeof ;thetwo-termperturbativeresultprovidesanexcellent approximationevenforvaluesof aslargeas 0 : 8 84

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Figure4-6:Plotof = 2 1 = 2 asafunctionof .Thecrossesarethenumericalresults, thegreenlineistheone-termperturbativeresult,andtheredlineisthetwo-termperturbativeresult.Theplothighlightstheroleofthesecondordertermintheexpansion. 85

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Figure4-7:Comparisonofthenumericalshootingmethod,naiveperturbationtheory andstrainedcoordinatesperturbationtheoryfor =0 : 43 .Theseareshowninblue,red dashedandgreendashedlinesrespectively.Thelogscalehighlightstheasymptotic behavioroftheorderparameterthenumericalandstrainedcoordinatescalculations areindistinguishableforlarge r ,whilethenaiveperturbativeresultclearlydecaystoo rapidly. 86

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CHAPTER5 CONCLUSION Wehavebrieyoverviewedthecurrentstatusoftheoreticalandexperimental researchintheputativesupersolidphaseofmatter.Theroleofcrystaldefectssuch asdislocationsandgrainboundarieshadbeendiscussedandfoundtobeofprime importance.Inthiswork,wehavestudiedboundstatesoftopologicallinedefectsalong withdefect-basedanalyticalmodelstounderstandthesupersolidstate. First,usingavariationalapproachwehaveestimatedthegroundstateenergyfor the2Ddipolepotentialemergingfromlinearelasticitytheoryduetoanedgedislocation. Thishadbeenfollowedbynumericalestimatesoftheboundstatespectrumofthe potential,bysolvingthe2DSchr odingerequation,usingmethodsofexactdiagonalizationandCoulombbasisexpansionmethods.Weobtainthegroundstateenergyin dimensionlessunitstobe-0.139.Acomparativestudyofthesevariousmethodsare discussedandwehavefoundtheresultstobeconsistent.Aclosematchoftheenergy eigenvalueswithsemiclassicalanalysisisalsoobtained.Certainfeaturesoftheeigenfunctionsshowsignaturesofchaoticdynamicsandareopenforfurtherinvestigation. Wehavealsodiscussedthequantummechanicsoftheinversesquarepotentialdueto theelasticeldofascrewdislocationandseenthatthepotentialdoesnotallowbound states.However,boundstatesarepossibleonintroductionofacut-offtothepotential. Wehavenumericallyestimatedthegroundstateenergyforthispotentialwiththehard andsoftcut-offschemes. Wethenproposedamodeltounderstandthespecicheatpeakobtainedin theexperimentsofLin etal. [25].Themajormotivationofthemodelhasbeenour calculationofthebindingenergyof 3 Heimpuritiestoedgedislocationsinsolid 4 He whichhappenstobeclosetothetemperatureofabout75mkatwhichtheheatcapacity peakhadbeenobserved.Wehadthensetupalatticegasmodelof 3 Heimpurities bindingorunbindingfromone-dimensionaldislocationsinsolid 4 Heinthebackground 87

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ofthesolid 4 Helattice.Theconsiderationthattheone-dimensionalsubsystemandthe threedimensionalbeinchemicalequilibriumhasintroducedachemicalpotentialinto ourmodel.Wehadworkedthestatisticalmechanicsofthemodelanddeterminedthe specicheatforvariousconcentrationsof 3 Heimpuritiesanddislocationdensities.We havefoundthatapartfromthebindingenergythechemicalpotentialtobeacrucial parameterwhichdeterminesthetemperatureatwhichtheheatcapacitypeakoccurs. Forsuitablechoicesofbindingenergiesanddefectdensitieswehadfoundaclose matchwiththeexperimentaldata,thussuggestingtheheatcapacitypeaktobea Schottkyanomalytypicallyexpectedintwo-levelsystems. Inthenalsegmentofthework,wehaveconstructedamodelforasupersolid basedonsuperuidityinducedalonganetworkofdislocations.Startingfroma Ginzburg-Landautheoryforthebulksolid,wehadshownthatsuperuiditynucleatesaroundanedgedislocationbycouplingthestraineldofanedgedislocationtothe superuidorderparameter.Wethenderiveaone-dimensionalequationdescribingsuperuidityalongasingledislocationbyimplementingasystematicperturbationanalysis. Wehadextendedthepicturetoanetworkofdislocationsandtheeffectofoverlapof thesestrandsof1Dsuperuid,thatgiveusbacksuperuidbehaviorforthebulk.Within aHartreeapproximation,wehavederivedan XY likemodelforthedislocationnetwork. Dependingontheinternodalseparationofthenetworkweobtainedapowerlawor exponentialdependenceofthetransitiontemperatureonthedislocationdensity.The naiveperturbativeapproachhadbeenrenedusingthemethodofstrainedcoordinates tocapturetherightasymptoticshasbeenshown.Theefcacyoftheperturbativestudy ispresentedfortheexactlysolvablecaseofa2DCoulombpotentialandpossibilitiesof includingdynamicsintothemodelhadbeenbrieydiscussed. Wewouldliketoconcludewithsomepossibilitiesforfuturework.First,thenumericalmethodstostudyboundstatesforanedgedislocationcouldbeextendedtothe caseofagrainboundary.Thegrainboundarypotentialwithinlinearelasticitytheoryis 88

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periodicalongtheplaneand,exponential,transversetotheplaneofthegrainboundary [43].AbasisexpansionschemesimilartotheCoulombbasismethodfortheedge dislocationproblemcouldbedeveloped,withthebasisstatestakenastheeigenstates alongandnormaltotheplaneoftheboundary. Itisknownthatalow-anglegrainboundarycouldbethoughtofasanarrayofedge dislocations.Ourphenomenologicalmodelshowsthatsuperuiditynucleatesalong adislocation,thusopeningupthepossibilitythataseriesofsuchdislocationsmight leadtoamodelofgrainboundarysuperuidity.Thisproblemhascloseanalogieswith ultracoldatomstrappedinarraysofone-dimensionalopticallattices.Onecouldstudy thepossiblethermalandquantumphasetransitionsinsuchasystemandthismay relatetosomeoftheyetunexplainedanomalousfeaturesobservedintheexperiments onsolid 4 He. Theperturbativeandweaklynon-linearanalysisdevelopedinthisworkwithin aLandautheorymightbeextendedtothecaseofaGross-Pitaevskiidescriptionof coldatomsinalow-densitylimit.Anotherscopeforfuturestudy,couldbetointroduce dynamicsintothenetworkmodel,somethingwhichisroughlydiscussedinthiswork. ThesuperclimbmodelproposedbyS oyler etal. [36],seemstobeemergingasa possiblemodeltounderstandtheRayandHallockexperiments.Itmightfurtherbe usefultostudythedetailedmechanismofthismodelwithintheframeworkofquantum dissipation. 89

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APPENDIXA ELASTICFIELDOFANEDGEDISLOCATION a b FigureA-1:Figureshowingthevectors b and foraageneraldislocationlinebfor astraightedge-dislocationalongthe z -axis. InthefollowingwefollowtheformalisminRef.[100]closely. If isatwo-dimensionalpositionvectorfromthedislocationaxistoapointonthe planeperpendiculartothetangentvector tothedislocationaxisasinFig.A-1a,then theequationforthedisplacementeld u is 1 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 rr u + r 2 u = b ; A where isthePoisson'sratioand b istheBurger'svector.Forastraightedgedislocationorientedalongthe z -axis: = ^ z ,theBurger'svectorisinthe x )]TJ/F24 11.9552 Tf 12.472 0 Td [(y plane, whichwechoosetobe b = )]TJ/F24 11.9552 Tf 9.298 0 Td [(b ^ x asinFig.A-1b.Hence, b = )]TJ/F24 11.9552 Tf 9.299 0 Td [(b ^ y : A EquationAgives, r 2 u + 1 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 rr u = )]TJ/F24 11.9552 Tf 9.299 0 Td [(b ^ y r : A 90

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Wewrite u = u 0 + w ,where u 0 ismulti-valuedand w issingle-valued.For u 0 wetake u ox = b 2 tan )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 y=x ;u oy = b 2 ln p x 2 + y 2 ; A sothat r u 0 =0 ; r 2 u 0 = )]TJ/F24 11.9552 Tf 9.298 0 Td [(b ^ y r : A SubstitutingEq.AinEq.Aweobtain r 2 w + 1 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 rr w = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 b ^ y r : A Tosolvethisequationfor w weusetheGreen'sfunction G ij denedthrough r 2 G ij x )]TJ/F35 11.9552 Tf 11.955 0 Td [(y + 1 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 @ i @ k G kj x )]TJ/F35 11.9552 Tf 11.955 0 Td [(y = ij x )]TJ/F35 11.9552 Tf 11.955 0 Td [(y : A Fouriertransforming,weobtain )]TJ/F24 11.9552 Tf 11.956 0 Td [(k 2 G ij )]TJ/F15 11.9552 Tf 26.924 8.088 Td [(1 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k i k k G kj = ij : A Wethendecompose G ij intolongitudinalandtransverseparts G ij = k i k j k 2 G L + ij )]TJ/F24 11.9552 Tf 13.15 8.088 Td [(k i k j k 2 G T : A PuttingthisinEq.Aandcomparingcoefcientsweobtain G ij = )]TJ/F15 11.9552 Tf 13.179 8.088 Td [(1 k 2 ij + 1 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k i k j k 4 : A UponinverseFourier-transforming,wegettheGreen'sfunctioninrealspaceas G ij x = )]TJ/F15 11.9552 Tf 14.028 8.088 Td [(1 4 ij j x j )]TJ/F15 11.9552 Tf 31.204 8.088 Td [(1 4 )]TJ/F24 11.9552 Tf 11.955 0 Td [( @ i @ j j x j # : A Havingobtained G ij x )]TJ/F35 11.9552 Tf 11.955 0 Td [(y ,wecansolveEq.A: w i x = b 2 Z dz 0 iy R )]TJ/F15 11.9552 Tf 31.205 8.088 Td [(1 4 )]TJ/F24 11.9552 Tf 11.955 0 Td [( @ i @ y R # ; A 91

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with R = j x )]TJ/F24 11.9552 Tf 11.955 0 Td [(z 0 ^ z j = p x 2 + y 2 + z )]TJ/F24 11.9552 Tf 11.955 0 Td [(z 0 2 .Oncalculatingthecomponents, w x = b 4 )]TJ/F24 11.9552 Tf 11.955 0 Td [( xy x 2 + y 2 ; A w y = )]TJ/F24 11.9552 Tf 14.467 8.088 Td [(b 2 3 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 2 )]TJ/F24 11.9552 Tf 11.956 0 Td [( ln p x 2 + y 2 L + b 4 )]TJ/F24 11.9552 Tf 11.955 0 Td [( y 2 x 2 + y 2 : A Toobtainthefullsolution u ,weadd u 0 toabove,toobtain u x = b 2 tan )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 y=x + xy 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( x 2 + y 2 # ; A u y = )]TJ/F24 11.9552 Tf 14.467 8.088 Td [(b 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( ln p x 2 + y 2 =r 0 + 1 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( x 2 x 2 + y 2 # ; A wheretheconstantcontributionsareabsorbedinthe ln r 0 term. Inpolarcoordinates, u r = b 2 cos )]TJ/F15 11.9552 Tf 17.703 8.088 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin ln r=r 0 # ; A u = )]TJ/F24 11.9552 Tf 14.467 8.087 Td [(b 2 sin + cos 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( + 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( cos ln r=r 0 # : A Thusfortheedge-dislocation r u = )]TJ/F24 11.9552 Tf 14.467 8.088 Td [(b 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin r : A Thelocalvolumechangeassociatedwithanedge-dislocationis V V = r u = b 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin r : A Wecanfurthercalculatethelinearstraintensor U ij givenby U ij = 1 2 @ i u j + @ j u i ; A wherethe i;j indicesreferto x;y inCartesiancoordinatesand r; inpolarcoordinates. 92

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Inpolarcoordinatesthecomponentsofthestraintensoraregivenby U rr = U = )]TJ/F24 11.9552 Tf 10.706 8.088 Td [(b 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin r ; A U r = b 4 )]TJ/F24 11.9552 Tf 11.955 0 Td [( cos r : A Thecomponentsofthestresstensor ij canbeobtainedusing ij = ij U ii +2 U ij ; A where and refertotheL amecoefcients.Theseareexpressedinpolarcoordinates as rr = = )]TJ/F24 11.9552 Tf 28.999 8.087 Td [(b 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin r ; A r = b 2 )]TJ/F24 11.9552 Tf 11.955 0 Td [( cos r ; A zz = rr + = )]TJ/F24 11.9552 Tf 22.803 8.088 Td [(b )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin r ; A rz = z =0 : A Notethatthoughthereisnostraincomponent U zz ,westillhaveanon-zerostress component zz .Finally,weobtainthehydrostaticpressureas p = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 3 ii = b 3 1+ 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [( sin r : A 93

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APPENDIXB ELASTICFIELDOFASCREWDISLOCATION Inthecaseofascrewdislocation,thedislocationlineandtheBurger'svectorare alongthesamedirection.Thus,forastraightscrewdislocationwiththedislocationaxis orientedalongthe ^ z -axis,thetangentvector = ^ z andtheBurger'svector b = )]TJ/F24 11.9552 Tf 9.298 0 Td [(b ^ z ,so that b =0 Theonlycomponentofthedisplacementeldisalongthedislocationaxis,so u = u z ^ z .Furthermore,duetocylindricalsymmetrywecanwrite u z = u z x;y Therefore, r u =0 andEq.Areducesto r 2 u z =0 : B Thesolutionthathasthenecessarysingularbehaviorinpolarcoordinatesis u z = C; B with C beingaconstant.Theconstantisdeterminedusingthequantizationcondition I L d u = b ; B where L isacontoursurroundingthelinedirectionofthescrewdislocation.Thisgives theconstant C = b 2 andhence u z = b 2 = b 2 tan )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 y x : B Werecallheretheexpressionforthestraintensorseeforeg.[84] U ij = 1 2 @ i u j + @ j u i + @ i u k @ j u k : B Forlinearstrains,thenon-zerocomponentsofthestraintensorinCartesiancoordinates are U xz = )]TJ/F24 11.9552 Tf 14.466 8.087 Td [(b 4 y x 2 + y 2 ; B 94

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U yz = b 4 y x 2 + y 2 : B Thecomponentsofthestresstensor ij canbeobtainedusing ij = ij U ii +2 U ij ; B where and refertotheL amecoefcients.Thus,inCartesiancoordinatesthenonzerocomponentsofthestresstensorare xz = )]TJ/F24 11.9552 Tf 10.945 8.087 Td [(b 2 y x 2 + y 2 ; B yz = b 2 x x 2 + y 2 : B Notethatinderivingtheabovestrainandstresstensorswehaveonlyconsideredlinear strains.However,linearstrainsdonotgiverisetoanyelasticinteractionbetweena soluteatomandthescrewdislocationeldofthesolventlattice,asthehydrostatic pressure p = )]TJ/F22 7.9701 Tf 10.494 4.707 Td [(1 3 ii iszero.Hence,weneedtoconsidernon-linearstrainstoseekany possibleelasticinteraction. Consideringnon-linearstrains,fromEq.BandB,wehave, U xx = 1 2 @u z @x 2 = 1 2 b 2 2 y 2 x 2 + y 2 2 ; B and U yy = 1 2 @u z @y 2 = 1 2 b 2 2 x 2 x 2 + y 2 2 : B Hence,thetraceofthestraintensoris U ii = U xx + U yy = 1 2 b 2 2 1 x 2 + y 2 : B Withinlinear,isotropicelasticitytheorythetraceofthestresstensorcorrespondingto thetraceofthestraintensorinEq.Baboveis ii = +2 U ii = 1 2 +2 b 2 2 1 r 2 ; B 95

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where r istheradialpartinpolarcoordinates.Notethatthoughthestrainsarenonlinearthescopeoftheelasticitytheoryisstilllinear.Thehydrostaticpressureisthen givenby, p = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 3 ii = )]TJ/F24 11.9552 Tf 10.494 8.088 Td [( 3 1+ 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 b 2 2 1 r 2 ; B where isthePoisson'sratio. 96

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APPENDIXC FERMITEMPERATUREOF 3 HEPARTICLESINTHE1DLATTICEGAS HereweestimatetheFermitemperatureof 3 Heimpuritiesadsorbedalongonedimensionaldislocationlinesinasolid 4 Helattice.Thenumberofdislocationssites is N D = Totallengthofadislocation L c ; C where c isthelengthalongthe c -axisdirectionofhcpsolid 4 He.Thelineardensityof 3 Heparticlesonadislocation,assumingalldislocationsitestobeoccupiedis n = N D L = 1 c : C Also,if k F istheFermiwavevector,thelineardensityof 3 Heimpuritiesis n = k F : C Hence,theFermienergyisgivenby E F = h 2 k F 2 2 m = n 2 2 h 2 2 m ; C where m isthemassofa 3 Heatom.ThisgivestheFermitemperatureas T F = E F k B = n 2 2 h 2 2 mk B ; C where k B istheBoltzmannconstant.Using c =5 : 9 Aweget T F =0 : 226 mK. 97

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APPENDIXD HEATCAPACITYFOR3DGASOF 3 HEPARTICLESIN 4 HELATTICE Herewederiveanexpressionfortheheatcapacitycontributionfromthethree dimensionalgasof 3 Heparticles,asdiscussedinthemodelforheatcapacityinChapter3.Weconsiderthe 3 Heparticlestobeinthebackgroundofa 4 Helatticeandtreat thesystemwithinatightbindingscheme. Thesingleparticlepartitionfunctioninonedimensionisgivenby Z 1 = Z a )]TJ/F26 5.9776 Tf 7.782 3.258 Td [( a dk 2 e )]TJ/F22 7.9701 Tf 6.586 0 Td [(2 t cos ka D = I 0 t ; D where =1 =k B T t isthetransferintegral, a isthelatticespacingof 4 Heand I 0 x isthe modiedBesselfunctionoftherstkind.Forasimplecubiclattice,thepartitionfunction foragasof N particles N beingthesameas N 3 D inChapter3is Z N = 1 N [ I 0 t ] 3 N : D Theinternalenergyofthesystemisthengivenby h E 3 D i = )]TJ/F24 11.9552 Tf 10.494 8.087 Td [(@ ln Z N @ D = )]TJ/F24 11.9552 Tf 14.13 8.087 Td [(@ @ ln 1 N [ I 0 t ] 3 N D = )]TJ/F15 11.9552 Tf 9.299 0 Td [(6 Nt I 1 t I 0 t ; D wherewehaveusedtheStirling'sapproximation ln N != N ln N + N andtheproperty ofmodiedBesselfunctionsthat I 1 x = dI 0 =dx .Toobtainthespecicheat,wesimply differentiatetheinternalenergywithrespecttotemperature.Thisgives C =12 N k B t 2 T 2 I 0 I 0 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(I 1 I 0 2 I 0 2 ; D 98

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where I 0 I 0 t I 1 I 1 t andprimedenotesderivative.Usingtherecursion relation I 0 x = xI 0 x )]TJ/F24 11.9552 Tf 11.955 0 Td [(I 1 x D thespecicheatinEq.Dcanberewrittenas C =12 N t k B T 2 1 )]TJ/F15 11.9552 Tf 18.9 8.088 Td [(1 2 t I 1 I 0 )]TJ/F29 11.9552 Tf 11.956 20.444 Td [( I 1 I 0 2 k B : D Fromtheasymptoticbehaviorofthe I 0 x and I 1 x itfollowsthatfor t T C 1 =T 2 Alsofor T 0 C goestotheclassicalvalueof 3 2 Nk B asexpectedforBoltzmann statistics.ThegurebelowshowsaplotfortheheatcapacityobtainedinEq.D. FigureD-1:Heatcapacityof 3 Heparticlesina 4 Helattice,obtainedfromEq.D. Notethattheheatcapacitygoestotheclassicalvalueas T 0 .Thebumpintheheat capacityisprobablyafeaturearisingfromthetightbindingdensityofstates. 99

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APPENDIXE METHODOFSTRAINEDCOORDINATES InthisAppendixweshowhowtoimproveuponthenaiveperturbationtheory developedinChapter4,usingthe methodofstrainedcoordinates .Westartwiththe Euler-LagrangeequationseeEq.4 )-222(r 2 + V x )]TJ/F24 11.9552 Tf 11.956 0 Td [(E + j j 2 =0 ; E where E = E 0 + ,with E 0 thelowesteigenvalueof r 2 + V x thegroundstate energy.Assuming V 0 as j x j!1 ,theboundsolutionsofthisequationbehave as exp )]TJ 9.298 9.624 Td [(p )]TJ/F24 11.9552 Tf 9.298 0 Td [(E j x j for j x j!1 ;thereisacharacteristiclengthscale 1 = p )]TJ/F24 11.9552 Tf 9.298 0 Td [(E .To accountforthisscale,denenewcoordinates X i = x i =l ,where l =1 = p 1+ =E 0 .In termsofthesenewcoordinates,wehave )]TJ/F15 11.9552 Tf 11.955 0 Td [(+ =E 0 r 2 X + V l X )]TJ/F24 11.9552 Tf 11.955 0 Td [(E + j j 2 =0 : E Forsmall thelengthscaleis l =1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(= 2 E 0 + = 8 =E 0 2 + ::: ;thepotentialisthen V l X V X )]TJ/F24 11.9552 Tf 20.413 8.088 Td [( 2 E 0 X r X V + O 2 : E Introducingthelinearoperator ^ L = r 2 X + V X )]TJ/F24 11.9552 Tf 11.955 0 Td [(E 0 ; E andrearrangingthingsabit,wehave ^ L = )]TJ/F24 11.9552 Tf 14.83 8.088 Td [( E 0 ^ L + E 0 1 2 X r V + V )]TJ/F24 11.9552 Tf 11.955 0 Td [( j j 2 + O 2 : E Nowexpand : = 0 + 1 + O 2 : E 100

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SubstitutingintoEq.E,weobtain: O : ^ L 0 =0 ; E O : ^ L 1 = 1 E 0 1 2 X r V + V 0 )-222(j 0 j 2 0 : Noticethattherighthandsidedoesnotcontain 0 byitselftherearenosecularterms, unlikethenaiveversionoftheperturbationtheory. Thesolutiontothe O equationis 0 = A 0 X ,where 0 isthenormalized groundstateeigenfunctionand A isanamplitude.Substitutingthisintothe O equation, ^ L 1 = A 1 E 0 1 2 X r V + V 0 )]TJ/F24 11.9552 Tf 11.955 0 Td [(A 3 3 0 : E Nowleftmultiplyby 0 ,andusethefactthat ^ L isHermitian: h 0 ; ^ L 1 i =0= A 1 E 0 1 2 h X r V i 0 + h V i 0 )]TJ/F24 11.9552 Tf 11.955 0 Td [(gA 3 ; E where g = Z d 2 X 4 0 : E Notethat X r V = )]TJ/F32 11.9552 Tf 9.298 0 Td [(X F ,with F theforceassociatedwiththepotential V .The quantumversionofthevirialtheorem[96]statesthat h X F i 0 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 h T i 0 ,with T the kineticenergy;therefore,thequantityinsidetheparenthesesinEq.Eistheground stateenergy E 0 = h T i 0 + h V i 0 ,andwehavesimply 0= A )]TJ/F24 11.9552 Tf 11.955 0 Td [(gA 3 ; E or A =1 = p g Pullingalloftheresultstogether,wehave x = g )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = 2 0 x =l ; E 101

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where l =1 = p 1+ =E 0 = p E 0 =E .Thestrainedcoordinatecalculationintroducesan dependenceintotheorderparameteritselfthenaiveperturbationtheorygavethe sameresult,butwith l =1 ,whichguaranteesthecorrectasymptoticbehaviorofthe orderparameter. 102

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APPENDIXF ANALYSISOFALANDAUMODELWITHACOULOMBPOTENTIAL InthisAppendixwesolveasimpliedversionofthedipolepotential,replacing cos =r bytheattractivetwo-dimensionalCoulombpotential V r = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 =r .Thesetwo potentialssharethesamelengthscaling;however,theCoulombpotentialisrotationally symmetricandthelinearproblemcanbesolvedexactly.Thedetailsoftheperturbation calculationfollowthegeneralschemedevelopedinSec.4.4.Wecomparetheresultsof theperturbationtheorywithnumericalsolutionsofthenonlineareldequation,andnd closeagreementforawiderangeof TheenergyeigenvaluesfortheHamiltonian ^ H = r 2 ? )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 =r aregivenby[66,97] E n = )]TJ/F15 11.9552 Tf 31.042 8.088 Td [(1 n +1 2 ;n =0 ; 1 ; 2 ;:::; F so E 0 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ,withagroundstateeigenfunction 0 r = p 2 =e )]TJ/F25 7.9701 Tf 6.586 0 Td [(r .Therelatedlinear operator[seeEq.4]is ^ L = r 2 ? )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 r +1 : F Expanding asbefore, = 0 + 1 + 2 2 + :::; F wehave ^ L 0 =0 ; F withthesolution 0 r = A 0 2 1 = 2 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(r F notethatwewillignorethe z -dependenceinthisAppendix.Substitutingintotheleft handsideofthe O equation, ^ L 1 = 0 )-222(j 0 j 2 0 = A 0 2 1 = 2 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(r )]TJ/F24 11.9552 Tf 11.955 0 Td [(A 3 0 2 3 = 2 e )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 r : F 103

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Weleftmultiplythisequationby p 2 =e )]TJ/F25 7.9701 Tf 6.587 0 Td [(r andintegrateon d 2 r toobtain 0= A 0 )]TJ/F15 11.9552 Tf -423.378 -23.908 Td [( = 2 A 3 0 ,sothat A 0 = p 2 .SubstitutingbackintoEq.F,andassuming 1 has cylindricalsymmetry,weobtainaninhomogeneousequationfor 1 : )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 r d dr r d 1 dr )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 r 1 + 1 =2 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(r )]TJ/F15 11.9552 Tf 11.955 0 Td [(8 e )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 r : F Theexplicitsolutionofthisequationthatdecaysto0forlarge r is 1 r = ce )]TJ/F25 7.9701 Tf 6.587 0 Td [(r + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 r + re )]TJ/F25 7.9701 Tf 6.587 0 Td [(r + 1 2 e )]TJ/F25 7.9701 Tf 6.586 0 Td [(r ln r + Z 1 2 r e )]TJ/F25 7.9701 Tf 6.586 0 Td [(t t dt ; F where c isanintegrationconstant.Wesubstitute 1 r intotherighthandsideofthe O 2 equationtoobtain ^ L 2 = )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 2 0 1 : F Weagainleftmultiplyby p 2 =e )]TJ/F25 7.9701 Tf 6.587 0 Td [(r ,integrateon d 2 r andusethefactthat ^ L isHermitian toobtainthesolvabilityconditionfortheconstant c ,withtheresult c = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(11 12 + 2 +ln )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(3 4 ln= )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 75887 ; F where =0 : 577210 istheEuler-Mascheroniconstant.Wehavenowexplicitlycalculated twotermsintheperturbationexpansion;bothtermsarenonzeroattheorigin,with 0 =2 and 1 = 1 12 +ln )]TJ/F22 7.9701 Tf 13.151 4.707 Td [(3 4 ln= )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 047478 Wehavesolvedthenonlineareldequationforthe )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 =r potentialnumericallyfora widerangeofvaluesof = E )]TJ/F24 11.9552 Tf 11.986 0 Td [(E 0 = E +1 ,usingtheshootingmethod[73].Theresults fortheorderparameterarepresentedinFig.4-3and4-4fortwodifferentvaluesof Thetwo-termperturbationtheorygivesanexcellentapproximationevenforafairlylarge valueof =0 : 43 104

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Togetasenseoftheefcacyoftheperturbationtheory,wecancalculatethe amplitudeoftheorderparameterattheorigin: = 1 = 2 0 + 3 = 2 1 + ::: =2 1 = 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(0 : 047478 3 = 2 + :::: F ThisresultisplottedinFig.4-5and4-6,alongwithournumericalresults.Again,wesee theexcellentagreementbetweenthetwo-termperturbationtheoryandthenumerical results,evenforrelativelylargevaluesof Whilethenaiveperturbationtheorydoesanexcellentjobincapturingtheoverall amplitudeoftheorderparameter,itproducesthewrongasymptoticbehaviorofthe orderparameter,asdiscussedinAppendixD.Thisdeciencyisremediedusingthe methodofstrainedcoordinates,andforthesakeofcompletenesswereanalyzethe attractiveCoulombpotentialproblemfollowingtheprocedureoutlinedinAppendixE. Asnotedpreviously, E 0 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 ,sothecharacteristiclengthscale l = )]TJ/F24 11.9552 Tf 12.841 0 Td [( )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 = 2 = 1 )]TJ/F15 11.9552 Tf 12.236 0 Td [( = 2 + = 8 2 + ::: .Changingcoordinatesto X i = x i =l ,andcollectingterms,we haveat O ^ L 0 =0 ; F where ^ L = r 2 X )]TJ/F15 11.9552 Tf 14.728 8.088 Td [(1 R +1= )]TJ/F15 11.9552 Tf 12.071 8.088 Td [(1 R d dR R d dR )]TJ/F15 11.9552 Tf 14.728 8.088 Td [(1 R +1 : F Thesolutionis 0 R = A 0 p 2 =e )]TJ/F25 7.9701 Tf 6.586 0 Td [(R .At O ,wehave ^ L 1 = 1 2 R 0 R )]TJ/F24 11.9552 Tf 11.955 0 Td [( 3 0 R : F Substituting 0 intotherighthandside,leftmultiplyingby p 2 =e )]TJ/F25 7.9701 Tf 6.586 0 Td [(R ,andintegratingon d 2 R ,weobtain A 0 = p 2 ,so 0 R =2 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(R : F 105

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The O equationisthen )]TJ/F15 11.9552 Tf 14.729 8.088 Td [(1 R d dR R d 1 dR )]TJ/F15 11.9552 Tf 14.728 8.088 Td [(1 R 1 + 1 = e )]TJ/F25 7.9701 Tf 6.586 0 Td [(R R )]TJ/F15 11.9552 Tf 11.955 0 Td [(8 e )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 R : F Theexplicitsolutionthatdecaystozeroas R !1 is[onecanshowthatintegration constant c isgivenbyEq.Fasbefore] 1 R = ce )]TJ/F25 7.9701 Tf 6.586 0 Td [(R + e )]TJ/F22 7.9701 Tf 6.587 0 Td [(3 R + 1 2 e )]TJ/F25 7.9701 Tf 6.586 0 Td [(R ln R + Z 1 2 R e )]TJ/F25 7.9701 Tf 6.587 0 Td [(t t dt : F ThisisalmostthesameresultasforthenaiveperturbationtheoryobtainedinEq.F, where R = p 1 )]TJ/F24 11.9552 Tf 11.955 0 Td [(r .Thedifferenceisthethirdterm re )]TJ/F25 7.9701 Tf 6.587 0 Td [(r inEq.F;thisisthesecular termthatisgeneratedinthenaiveperturbationtheory.Thestrainedcoordinatesmethod hassubsumedthistermintotherstorderterm;i.e., 2 e )]TJ/F25 7.9701 Tf 6.586 0 Td [(R =2 e )]TJ 6.587 6.183 Td [(p 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(r 2 e )]TJ/F25 7.9701 Tf 6.587 0 Td [(r + re )]TJ/F25 7.9701 Tf 6.586 0 Td [(r + :::: F Togetasenseoftheimprovementincapturingthecorrectasymptoticbehaviorofthe orderparameter,inFig.4-7weplotthenumericalresultsobtainedusingtheshooting methodagainstthenaiveandstrainedcoordinateperturbationtheory.Thestrained coordinateresultisindistinguishablefromthenumericalresultinthislogplot;thenaive perturbationtheoryresultclearlydecaystoorapidlyforlarge r 106

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APPENDIXG ANALYSISOFATIME-DEPENDENTMODEL InthisAppendixwegeneralizetheresultsofSec.4.4toderiveaone-dimensional dynamicalmodelforthesuperuid.Forsimplicity,we'llassumethatthereareno conserveddensities,sothedynamicsaredescribedbymodelAoftenreferredtoas time-dependentGinzburg-LandautheoryintheHohenberg-Halperinclassication [98,99].Therelaxationalequationofmotionis @ @t = )]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(0 F + x ;t ; G where )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(0 isarelaxationrateand istheuctuatingnoisetermwithGaussianwhite noisecorrelations,i.e., h x ;t i =0 and h x ;t x 0 ;t 0 i =2 k B T )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(0 x )]TJ/F32 11.9552 Tf 12.838 0 Td [(x 0 t )]TJ/F24 11.9552 Tf -426.847 -23.908 Td [(t 0 .Asbefore,tofacilitatethereductiontoaone-dimensionalmodelweintroduce dimensionlessquantities,withatimescale ~ t =2 c= )]TJ/F24 11.9552 Tf 11.867 0 Td [(a 2 0 B 2 =2 l 2 = )]TJ/F24 11.9552 Tf 11.867 0 Td [(c .Intermsofthe dimensionlessvariables, @ @t = r 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [([ V r )]TJ/F24 11.9552 Tf 11.955 0 Td [(E ] )-222(j j 2 + x ;t ; G wherethenoisecorrelationsaregivenby h x ;t x 0 ;t 0 i =2 k B T=F 0 x )]TJ/F32 11.9552 Tf 11.955 0 Td [(x 0 t )]TJ/F24 11.9552 Tf 11.956 0 Td [(t 0 : G Asbefore,weintroducethesmallparameter = E )]TJ/F24 11.9552 Tf 12.534 0 Td [(E 0 ,with = 1 = 2 z = )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 = 2 t = )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 ,and = 3 = 2 ~ ,toobtain ^ L = )]TJ/F24 11.9552 Tf 9.299 0 Td [(@ + @ 2 + )-222(j j 2 +~ : G Weexpand inpowersof ,andat O wehave ^ L 0 =0 ,thesolutionofwhichis 0 = A 0 ; 0 r : G 107

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Substitutingthisintotherighthandsideofthe O equation,leftmultiplyingby 0 ,and thenintegratingon d 2 r ,thesolvabilityconditionyields @ A 0 = @ 2 A 0 + A 0 )]TJ/F24 11.9552 Tf 11.956 0 Td [(g j A 0 j 2 A 0 + ; G where istheone-dimensionaluctuatingnoisetermthethree-dimensionaltermwith thetransversedimensionsprojectedout, ; = Z d 2 r 0 r ~ r ;; : G Usingthefactthat 0 isnormalizedtoone,itisstraightforwardtoshowthat has Gaussianwhitenoisecorrelations.Undoingthe scalings,weobtainournalonedimensionaltime-dependentGinzburg-Landautheory, @ t = @ 2 z + )]TJ/F24 11.9552 Tf 11.955 0 Td [(g j j 2 + : G 108

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[85]Notethatduetothetranslationalinvariancealongthe z -axis,thewavefunction shouldalsocontainafactorof e ikz ,whichcontributes k 2 totheenergyeigenvalues.However,thegroundstateisalwaysobtainedfor k =0 ,sowesuppressthe z -dependenceinthewavefunctions. [86]P.G.DrazinandW.H.Reid, HydrodynamicStability,SecondEdition Cambridge UniversityPress,Cambridge,UK,2004. [87]J.K.KevorkianandJ.D.Cole, MultipleScaleandSingularPerturbationMethods Springer-Verlag,NewYork,1996. [88]R.P.FeynmanandA.R.Hibbs, QuantumMechanicsandPathIntegrals McGraw-Hill,NewYork,1965. [89]L.W.GruenbergandL.Gunther,Phys.Letters 38A ,463. [90]D.J.Scalapino,M.Sears,andR.A.Ferrell,Phys.Rev.B 6 ,3409. [91]N.D.MerminandH.Wagner,Phys.Rev.Lett. 17 ,11331136. [92]J.R.TuckerandB.I.Halperin,Phys.Rev.B 3 ,3768. [93]A.B.Harris,J.Phys.C 7 ,1671. [94]I.Iwasa,K.Araki,andH.Suzuki,J.Phys.Soc.Japan 46 ,1119. [95]I.IwasaandH.Suzuki,J.Phys.Soc.Japan 49 ,1722. [96]Forinstance,seeR.Shankar, PrinciplesofQuantumMechanics,SecondEdition Springer,NewYork,1994. [97]B.ZaslowandM.E.Zandler,Am.J.Phys. 35 ,1118. [98]P.C.HohenbergandB.I.Halperin,Rev.Mod.Phys. 49 ,435. [99]Whileweconcentrateonrelaxationaldynamics,thegeneralmethodoutlined inAppendixFcanbeappliedtootherdynamicalmodels,e.g.,thereversible Gross-PitaevskiidynamicsofaBosecondensate. [100]G.F.Mazenko, Fluctuations,Order,andDefects JohnWileyandSons,NewYork, 2003. 113

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BIOGRAPHICALSKETCH DebajitGoswamiwasbornandraisedinthecityofKolkata,India.Aftercompleting hisbachelor'sdegreefromtheUniversityofCalcutta,hemovedtotheJawaharlalNehru University,NewDelhitoobtainamaster'sdegreeinphysics.HecametotheUniversity ofFloridainthefallof2005andjoinedProf.AlanDorsey'sgroupintheoreticalcondensedmatterphysicsinthesummerof2006,forpursuinghisdoctoralstudies.He receivedhisPh.D.inthesummerof2011. 114