Defects in Novel Superfluids

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Defects in Novel Superfluids Supersolid Helium and Cold Gases
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Dasbiswas, Kinjal
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Physics
Committee Chair:
Dorsey, Alan T
Committee Members:
Hirschfeld, Peter J
Cheng, Hai Ping
Biswas, Amlan
Sinnott, Susan B

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bec -- dislocation -- superfluid -- supersolid -- vortex
Physics -- Dissertations, Academic -- UF
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Abstract:
We investigate the role played by various topological defects, especially crystal dislocations and super?uid vortices, in some novel super?uids - such as the putative supersolid phase in solid 4 He, and in dilute Bose-Einstein condensates (BEC) in traps. The ?rst part of this thesis addresses experimental ?ndings in solid helium, such as the period shift in resonant oscillators, that have been interpreted as a signature of super?uidity coexisting with crystalline order in solid helium. We establish using Landau’s phenomenological theory of phase transitions that crystal defects such as dislocation lines and grain boundaries can induce local super?uid order, and show that a network of dislocation lines can give rise to bulk super?uid order within a crystal. Our ?ndings are also relevant to other phase transitions in the presence of crystal defects. The second part concerns the stability and dynamics of a single vortex in a rotating trap of BEC, and the possibility of the macroscopic quantum tunneling of such a vortex from a metastable minimum at the center of the trap. The complete dynamics of such a vortex is derived by integrating out the phonon modes from a hydrodynamic action, and estimates for the tunneling rate are obtained using a variety of semiclassical methods. This is analogous to the problem of tunneling of a charged particle in a very high magnetic ?eld, the Magnus force on the vortex being analogous to the Lorentz force on a charge. We conclude that the vortex action has a complicated nonlocal form and further, that the Magnus-dominated dynamics of the vortex tends to suppress tunneling.
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by Kinjal Dasbiswas.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
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Adviser: Dorsey, Alan T.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-02-28

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DEFECTSINNOVELSUPERFLUIDS:SUPERSOLIDHELIUMANDCOLDGASESByKINJALDASBISWASATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012KinjalDasbiswas 2

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Tomyparents,bothphysicians,whohavealwayssupportedandnurturedmyambitionstobeaphysicist 3

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ACKNOWLEDGMENTS Iamgratefultomyadviser,ProfessorAlanT.Dorsey,forthepivotalrolehehasplayedasamentorinmyPhDHeintroducedmetomanytopicsincondensedmatterphysics,particularlythenoveltopicofsupersolidhelium.Ihavelearnedanumberoftechniquesthroughworkingwithhim,butequallyimportantly,hehastaughtmetochoosemyproblemscarefullyandtopresentmyresultsinacoherentmanner.IwouldliketothankProfessorsPeterJ.Hirschfeld,AmlanBiswas,H.-P.ChengandSusanB.Sinnottforusefuldiscussionsandforservingonmycommittee.IwouldalsoliketoacknowledgeProfessorsY.-S.LeeandPradeepKumarfortheirvaluableinputsaboutthesupersolidproblem,ProfessorsD.L.Maslov,RichardWoodard,S.L.Shabanov,andK.A.Muttalibfortheirinspiringteaching,andtheMaxPlanckInstituteinDresdenforgivingmetheopportunitytoattendasummerschoolonBose-Einsteincondensates.Iamindebtedtosomeofmygraduatestudentcolleaguesforstimulatingdiscussionsandforensuringacollegialenvironmentinthedepartment.IamgratefultoK.NicholaandP.Marlinfortheirgenerousassistancewiththeadministrativeaspectsofmygraduateprogram.IwouldliketoacknowledgetheNationalScienceFoundationforthesupporttheyhaveextendedtomyresearch,andalsotheGraduateSchoolandDepartmentofPhysicsattheUniversityofFlorida,whohaveprovidedmewithanalumnifellowshipandotherformsofsupport.FinallyIwouldliketoexpressmygratitudetofriendsandfamily,whosupportedmeintheseveyears.Iamespeciallyindebtedtomyparentswhohaveinvestedmucheffortandaffectioninmyupbringingandeducation. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 1.1Superuids:HeliumandDiluteBEC ..................... 11 1.2TopologicalDefects:Dislocations,Vortices,etc. ............... 18 1.3TheSupersolidConundrum .......................... 20 1.4VorticesinRotatingBose-EinsteinCondensates .............. 24 2DISLOCATION-INDUCEDSUPERFLUIDITY ................... 27 2.1LandauTheoryfortheSupersolidPhaseTransition ............. 28 2.2Superuid-DislocationCoupling ........................ 30 2.3EffectiveLandauTheoryforSuperuidityalongaDislocation ....... 33 2.4QuantitativeEstimatesoftheShiftinCriticalTemperature ......... 38 2.4.1TheQuantumDipoleProblemin2D ................. 38 2.4.2NumericalMethodsUsed ....................... 39 2.4.3VariationalCalculation ......................... 39 2.4.4RealSpaceDiagonalizationMethod ................. 40 2.4.5CoulombBasisMethod ........................ 43 2.4.6SemiclassicalAnalysis ......................... 45 3SUPERFLUIDITYINTHEPRESENCEOFMULTIPLEDISLOCATIONS .... 50 3.1GrainBoundarySuperuidity ......................... 50 3.2Berezinskii-Kosterlitz-ThoulessSuperuidinaGrainBoundary ...... 57 3.3NetworkofDislocationLinesandaModelSupersolid ............ 63 4VORTICESINTRAPPED,DILUTEBOSE-EINSTEINCONDENSATE ..... 69 4.1Gross-PitaevskiiFormulation ......................... 69 4.1.1WeaklynonlinearAnalysisforSmallCondensates ......... 72 4.1.2ThomasFermiAnalysis ........................ 74 4.2VortexEnergeticsandStability ........................ 75 4.2.1EnergyofaVortexattheCenteroftheTrap ............. 77 4.2.2EnergyofanOff-centerVortex .................... 79 4.2.3VortexStabilizationinaRotatingTrap ................ 80 4.2.4VortexinaSmallCondensate ..................... 84 5

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4.3VortexDynamicsintheCo-rotatingFrame .................. 88 4.3.1TheMagnusForce ........................... 92 4.3.2VortexMass ............................... 94 5MACROSCOPICQUANTUMTUNNELINGOFAVORTEXINAROTATINGBOSEGAS ...................................... 103 5.1ClassicalMechanicsofaVortex ....................... 104 5.2SemiclassicalEstimatesofVortexTunneling ................. 110 5.2.1SchrodingerEquationforaChargeinaMagneticField ....... 110 5.2.2WKBAnalysisofTunneling ...................... 111 5.2.3TheMethodofInstantonsorBounceTrajectories ........ 115 5.3ExperimentalDiscussion ........................... 118 6CONCLUSION .................................... 121 APPENDIX ASTRAINFIELDFORANEDGEDISLOCATION .................. 125 BANALYSISOFALANDAUMODELWITHA1=rPOTENTIAL .......... 126 CANALYSISOFATIME-DEPENDENTMODEL .................. 130 DVORTICESINATWO-DIMENSIONALXYMODEL ................ 132 REFERENCES ....................................... 136 BIOGRAPHICALSKETCH ................................ 143 6

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LISTOFTABLES Table page 2-1Summaryofgroundstateenergyestimatesoftheedgedislocationpotential.Energyisgiveninunitsof2mp2=~2. ........................ 39 2-2Comparisonofrstfewenergyeigenvaluesobtainedfromdifferentmethods.Energyunits:2mp2=~2.nindicatesquantumnumberofthestate. ....... 43 7

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LISTOFFIGURES Figure page 1-1TorsionaloscillatorsetupusedbyAndronikashvili ................. 14 1-2Thetwobasictypesofdislocations ......................... 19 1-3TorsionaloscillatorsetupusedbyKimandChan ................. 22 1-4Periodshiftsinatorsionoscillatorexperiment ................... 23 1-5Shearhardeninginsolid4He ............................ 24 1-6ImagesofaBose-Einsteincondensatestirredwithalaserbeam ........ 26 2-1Singledislocationlinewithsuperuid ........................ 32 2-2Comparisonofeigenvaluesofthe2Dquantumdipoleproblem,obtainedbydifferentmethods ................................... 43 2-3Fitfortheeigenvaluespectrumof2Dquantumdipolepotential ......... 46 2-4Eigenfunctionsoftherstvestatesofthe2Dquantumdipoleproblem .... 48 2-5Eigenfunctionsofvehigherexcitedstatesofthe2Dquantumdipoleproblem 49 3-1Symmetrictiltgrainboundary ............................ 51 3-2Networkofdislocationlines ............................. 64 3-3Correlationlengthvs.temperature ......................... 66 4-1Condensatewavefunction (r)versusrbydifferentmethodsinlow,andhighdensitysituations ................................... 76 4-2Condensatewavefunction (r)versusrforthel=1vortexstateinasmallcondensate ...................................... 86 4-3Magnusforceonarotatingobjectinaowinguid ................ 93 5-1EnergycostassociatedwithasinglyquantizedstraightvortexinarotatingtrapintheTFlimit .................................. 107 B-1Orderparameter (r)versusrbydifferentmethods ............... 128 B-2Orderparameteramplitude (0)asafunctionof,bydifferentmethods .... 129 8

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDEFECTSINNOVELSUPERFLUIDS:SUPERSOLIDHELIUMANDCOLDGASESByKinjalDasbiswasAugust2012Chair:AlanT.DorseyMajor:Physics Weinvestigatetheroleplayedbyvarioustopologicaldefects,especiallycrystaldislocationsandsuperuidvortices,insomenovelsuperuids-suchastheputativesupersolidphaseinsolidhelium-4(4He)andindiluteBose-Einsteincondensates(BEC)intraps. Therstpartofthisworkaddressesrecentexperimentalndingsinsolidhelium,suchastheperiodshiftinresonantoscillatorsthathasbeeninterpretedtobeasignatureofsuperuiditycoexistingwithcrystallineorderinsolidhelium.WeuseLandau'sphenomenologicaltheoryforphasetransitionstoestablishthatcrystaldefectssuchasdislocationlinesandgrainboundariescaninducelocalsuperuidorderandshowthatanetworkofdislocationlinescangiverisetobulksuperuidorderwithinacrystal.Ourndingsarealsorelevanttootherphasetransitionsinthepresenceofcrystaldefects. ThesecondpartconcernsthestabilityanddynamicsofasinglevortexinarotatingtrapofaBose-Einsteincondensate(BEC)andthepossibilityofthemacroscopicquantumtunnelingofsuchavortexfromametastableminimumatthetrapcenter.Thecompletedynamicsofsuchavortexisderivedbyintegratingoutthephononmodesfromahydrodynamicaction,andestimatesforthetunnelingrateareobtainedusingavarietyofsemiclassicalmethods.Thisisanalogoustotheproblemoftunnelingofachargedparticlethroughapotentialbarrierinthepresenceofaveryhighmagneticeld, 9

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theMagnusforceonthevortexbeinganalogoustotheLorentzforceonacharge.Weconcludethatthevortexactionhasacomplicatednonlocalformandfurther,thattheMagnus-dominateddynamicsofthevortextendstosuppresstunneling. 10

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CHAPTER1INTRODUCTION 1.1Superuids:HeliumandDiluteBEC Thehistoryofthemodernstudyofthepropertiesofmaterialsatverylowtemperaturesdatesbacktothedevelopmentoftechnologytoliquefyheliumandtheconsequentdiscoveryofsuperconductivityjustoveracenturyago.Thephenomenaseenatlowtemperaturesareoftennovelandatoddswithintuitionbecausequantumeffectsbecomemoreimportantasthethermalvibrationsofatomsinmatterarereduced.Theoretically,manyofthesephenomenaareunderstoodtobeconsequencesofBose-Einsteincondensation,anoutcomeofthelawsofquantumstatistics,whereallparticlesfollowingBosestatisticsinhabitthesamegroundstatewhencooleddowntoabsolutezero.Asidefromthefundamentalinsightsintothelawsofquantummechanicsitprovides,thestudyofmatteratverylowtemperatureshasalsoyieldedpracticalapplicationsofsocialsignicance:forexampletheuseofsuperconductorstocreatethelargemagneticeldsneededforthediagnosticmedicalprocedureofMRI. KamerlinghOnnesrstliqueedhelium-4in1908.Soonafter(1911),hediscoveredsuperconductivityinmercurywhenmeasuringtheabruptdisappearanceofresistanceinmercuryat4.19K[ 1 ].KamerlinghOnnesandKeesomcollectedmanyhintsofaphasetransitioninheliumintheLeidenlaboratory,includingtheremarkableanomalyintheheatcapacitywithafamousshapeatatemperatureofT2.17K.Inmanyways,heliumwastheidealcandidateforobservingquantumphenomenaatlowtemperaturesinliquidsbecauseitremainsliquidtillveryclosetoabsolutezero(unlikeothermaterialsthatfreezeintosolid).Thisisbecauseofitssmallmassthatfavorsquantumuctuations,andbecauseunliketheevenlighterhydrogenthattendstoformmolecules,itischemicallyinertandhasweakerinteractions.KeesomdistinguishedthehighandlowtemperaturephasesasliquidheliumIandII,respectively.JohnF.AllenandDonMisenerrstshowedtheinviscidnatureofheliumIIin1937whentheyexamineditsow 11

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throughverysmallcapillaries[ 2 ].SimilarresultswerefoundsimultaneouslybyPyotrKapitza,leadinghimtosuggestthatheliumIIwasinfactasuperuid[ 3 ]. Aphenomenologicaltwo-uidmodelforheliumIIwasdevelopedsoonafterbyTisza[ 4 ],Landau[ 5 ],andothersbasedonthenotionthatpartofliquidheliumbelowTturnedsuperuid,buttheotherpartremainednormal,i.e.wasdissipative(hadniteviscosity)andcouldtransportentropylikeordinaryuids.ThiscouldbeintuitivelyunderstoodfromLandau'stheoryofBoseandFermiliquids(whichwasalsodevelopedataboutthistime),andtheattendantideathatlowenergyexcitationsfromthegroundstateofaweaklyinteractingbosonicsystemcouldbecharacterizedasquasiparticleswithdenitemomentumandenergy.Thesuperuidcomponentoftwo-uidheliumIIcouldthenbeidentiedwiththegroundstateoftheBoseliquidwhilethenormalcomponentcorrespondedtothequasiparticles.LandauidentiedthequasiparticlesofaBoseliquidasbeingoftwotypes:quantizedsoundwavesorphonons,whoseenergydependslinearlyonmomentumpas=cp,wherecisthespeedofsound,androtonsorquantaofrotationalmotionwiththespectrum(p)=+(p)]TJ /F5 11.955 Tf 12.15 0 Td[(p0)2=2m.ThisphenomenologicalmodelprovedtottheexcitationspectrumobtainedfromneutronscatteringexperimentsonheliumII[ 6 ]verywellbutaclearerandmoremicroscopicunderstandingofthesequasiparticleswasnotobtaineduntil1956,whenFeynmandevelopedavariationalAnsatztodescribetheatomicinteractionsinliquidhelium[ 7 ].Wellbeforethat,Landauhadmadeseveralremarkablepredictionsbasedonhistwo-uidhydrodynamics,whichtothisdayremainthesmokinggunsignaturesofthesuperuidphase: Thereisacertaincriticalvelocityvcofasuperuid,suchthatitlosesitssuperuidityifmadetoowfasterthanthat.Thisissobecauseowsfasterthanvc=(p)=pexcitequasiparticlesintheliquidthatdestroythesuperuidphase. Itispossibletosetthenormalandsuperuidpartsintooscillationsthatareoutofphasewitheachother.(ThismodeofoscillationhadbeensuggestedearlierbyTiszaandcametobeknownassecondsound.) 12

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Whenacontaineroftheliquidisrotated,onlythenormalpartoftheliquidrotatesalongwithit.Thesuperuidcomponentdecouplesfromtherotationandsothereisanapparentdecreaseinthemomentofinertiaastheliquidiscooleddownpastits-point. ThelastofthesewasveriedsoonafterbyAndronikashvili[ 8 ]usingatorsionaloscillatorsetupshowninFig. 1-1 ,thathasspecialbearingonsomeoftheideastobeintroducedlaterinthisdissertation.Astackofconcentricdiskswassuspendedinliquidheliumbyatorsionrodthatcouldbetwistedtooscillatethestack.Ifthespacingbetweendiskswassmallenough,thenormalcomponentoftheuidwasdraggedwiththeoscillatingdisks.Asliquidheliumiscooledbelowits-point,thesuperuidcomponentstopsrotating.Thisdecouplingleadstoachangeinthemomentofinertiaoftheoscillator,andaconsequentmeasurableshiftintheresonantfrequencyofoscillations.Additionally,theQ-factoroftheoscillatorprovidesanestimateofthedissipationinvolved. ItwashoweverFritzLondonin1938[ 9 ]whorstrealizedthatthephenomenonofBose-Einsteincondensationunderlaythecuriouspropertiesofsuperuidhelium.Althoughacceptedunhesitatinglynow,theideaofaBose-Einsteincondensationhistoricallytooktimetogetestablishedasarealeffect,anditsconnectionwithsuperuiditywascertainlynotimmediatelyapparent.Einsteinhadrstrealizedthepossibilityofamacroscopicfractionofatomscondensingtotheirgroundstateatlowtemperatures[ 10 ],basedonthestatisticsworkedoutbyS.N.Bosein1924[ 11 ].HisargumentforcondensationinanoninteractinggasofatomsobeyingBosestatisticsisdeceptivelysimple.ThemeanoccupationnumberofasingleparticlestateinsuchasystemisgivenbytheBosedistributionfunction f()=1 e()]TJ /F10 7.97 Tf 6.59 0 Td[())]TJ /F3 11.955 Tf 11.95 0 Td[(1, where1=kBT,andisthechemicalpotentialthatensuresthatthenumberofparticlesinthesystemisxed. 13

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Figure1-1. AschematicofthetorsionaloscillatorsetupusedbyAndronikashviliin1946(Keesomisalsocreditedwithasimilarexperiment)todemonstratethedecouplingofthesuperuidfromrotation.Astackofconcentricdiskswassuspendedinliquidheliumbyatorsionrodthatcouldbetwistedtosetthestackintooscillations.Thenormalcomponentoftheuidisdraggedalongwiththeoscillatingdisks,butthesuperuidpartdoesnotrotate.Theresultingestimatesofthedensityofthenormalandsuperuidcomponentsasthetemperatureisvaried,isshownintheinset.IllustrationbyAlanStonebrakerandreprintedwithpermissionfromIacopoCarusotto,Physics3,5(2010),cAPS,2010. Thedensityofstatesforanoninteracting,freegasin3Dis g()=Vm3 2 21 22~31 2. ItisaninterestingmathematicalpropertyoftheBosefunctionabovethatinthreeorhigherdimensions1thereisanupperboundonthenumberofatomsthatcanbeaccommodatedintheexcitedstatesatagiventemperature.Toseethis,wewritethe 1TheargumentsketchedhereisforfreeBosegasinabox.Onincludingtheconningpotentialfromaharmonictrap,itispossibletogetcondensationin1or2D. 14

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expressionforthetotalnumberofatomsintheexcitedstatesfora3DBosegas,andscaleoutthetemperaturedependenceoftheresultingintegralas Nex=Z10dg()f()=c(kBT)3 2. Theconstantcisgivenby(in3D) c=Vm3 2 21 22~3\(3=2)(3=2), where)]TJ /F1 11.955 Tf 10.09 0 Td[(andaretheGammafunctionandRiemannzetafunctionrespectively.Thelatterisniteat3=2andsowehaveanitenumberofatomsintheexcitedstates.Thissuggeststhattheremainingatoms,ifany,havetobeaccommodatedinthegroundstate,whichcouldthenbemacroscopicallyoccupied.Thecriticaltemperatureforthiscondensationisthenthepointwherethenumberofparticlesthatcouldbeaccommodatedintheexcitedstatesequalsthetotalnumberofparticles: c(kBT)3 2N. Thisstraightforwardresultwashoweverhistoricallyheldtobeapathologicalartifactofthetheoryofnoninteractingbosons,whichwouldbeabsentwheninteractionsweretakenintoaccount.Londonestimatedthecriticaltemperatureforagasofheliumatoms,andfoundittobe3.3K,remarkablyclosetotheT=2.17Kmeasuredinliquidhelium.HethensuggestedthataBosecondensationwasresponsibleforthe-transitionandthatthecriticaltemperaturewasreducedasaresultoftheinteractions.ThiscondensatedescriptionhelpedexplainmanyexperimentalfactsknownaboutheliumII.Ifexactlyonestatewasmacroscopicallyoccupied,thentheentiresystemcouldbecharacterizedbythewavefunctionofthatsinglestate (r,t)=j 0(r,t)jexp[i(r,t)].Wethendenethesuperuidvelocityas vs(~=m)r.(1) 15

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Thisshowsthatthesuperuidvelocityisirrotational.Alsothereisnoinformationassociatedwiththesinglestate,andsothesuperuidfractioncarriesnoentropy,aspostulatedinthetwo-uidmodel. WeshouldpointoutherethatthetheoryofBose-Einsteincondensationisunresolvedinseveralways.ThereisnorigorousproofforthemacroscopicoccupationofasinglestateinageneralinteractingsystemofbosonsatT=0.Theonlyproofavailable,theoneprovidedoriginallybyEinstein,isfornoninteractingsystems.Thereisnothingtoassureusthatinteractionsdonotdepletethecondensatephase,orthattheydonotleadtocondensationinmultiplestates.Thesituationisevenlessclearfornonequilibriumsystems.Helium,beingastronglyinteractingliquid,isfarfromtheidealBosegasassumedbyEinstein-althoughevidenceformacroscopicoccupationofthegroundstatewasfoundthroughneutronscatteringexperiments.ThesearchforBosecondensationinothersystems,particularlythosewithweakinteractions,wasthereforeanimportantmissinglinkinthehistoryoflowtemperaturephysics. AnewerabeganwhenaBose-Einsteincondensate(BEC)ofdilute,atomicgaseswasrstpreparedin1995usinglasercoolinginmagneto-opticaltraps,bythegroupsofWiemannandCornellatJILA,Boulder[ 12 ],andKetterleatMIT[ 13 ].Theycapitalizedonaseriesofadvancesinexperimentaltechniquesinvolvingtheuseoflasersfollowedbymagneticevaporativecooling.Thechallengelayintrappingalargeenoughnumberofatomstoachieveequilibriumandhaveanobservablecondensate,andincoolingthemdowntothedegeneracytemperature.Theinitialexperimentswereperformedonweaklyinteractingvaporsofalkaliatomssuchasrubidium,sodiumandlithium.Sincethen,condensationhasbeenachievedinfermions,moleculesandparticleswithlong-rangedipolarinteractions.AmajoradvantageofworkingwithsuchweaklyinteractingatomicBECintrapsliesintheprecisecontrolwehaveoversuchsystems.ThestrengthoftheinteractionsbetweentheseatomscanbetunedwithFeshbachresonances[ 103 ].Theseatomscanbearrangedintoarticiallatticesbyapplying 16

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anopticallygeneratedperiodicpotential[ 14 ].Thisanalogybetweenatomsinopticallatticesandelectronsinioniclatticescanbeusedtocarryoutaprogramofquantumsimulation,wheremanyimportantnaturallyoccuringsystemsincondensedmattercanbemimickedbytrappedultracoldatomstailoredinthelaboratory. ThephysicsofdiluteBECsdiffersinseveralimportantrespectsfromthatofhelium.Themostimportantdistinctionliesintheroleofinteractions.Theinter-particleseparationinadiluteBECistypicallymuchlargerthanthescatteringlengthcharacterizingitsinter-particleinteraction.Heliumatomsontheotherhandarecloselypackedandinteractstrongly,theinteratomicseparationbeingofthesameorderasthelengthscaleofthevanderWaalsforcesamongthem.AnestimateofthedensityofatomsinBECandliquidheliumwouldputmattersintoperspective.ThetypicaldensityofaBECatthecenterofthetrapisaround1015cm)]TJ /F6 7.97 Tf 6.58 0 Td[(3,whilethatofliquidheliumisaround1022cm)]TJ /F6 7.97 Tf 6.58 0 Td[(3.ThepronouncedroleofinteractionsinheliummeansthatsomeofthecondensateisdepletedevennearT=0anditposesatheoreticallymorechallengingproblem.Forexample,BECscanbequiteaccuratelydescribedbyaHartreeFockdescriptionintermsofthesingle-particlestates,whileanysuchattemptforheliumislikelytofailbecauseofthestrongcorrelations.ThecriticaltemperatureforcondensateformationdependsonthedensityasTcn2=3,andsooneneedstocooldiluteBosegasestonanoKelvinstoachievecondensation,whereasheliumundergoesasuperuidtransitionat2.17K.AnadditionalfeatureoftheBECsrealizedbylasercoolingistheimportantroleoftheconningtrappotential.Thisexternalpotentialisusuallyharmonicandspatiallyconnesthecondensatewhilealsorenderingitnon-uniform.Therelevanttheoreticalframeworkforauniformdilutegaswasworkedoutinthe1950sand60sbutthepresenceoftheexternalharmonicpotentialcangiverisetonewandinterestingfeatures,suchasnewcollectivemodesorstabilityissues.Theboundaryconditionsandsurfacepropertieshavetobemodiedaccordingtothetrap.Anotherqualitativedistinctioncanbemadebetweenthetwosystems:heliumhasbeentheidealcandidate 17

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forobservationofsuperow,butthecondensationitselfisnotsoapparent.DiluteBECsontheotherhandaremanifestlycondensates,whosecondensatefractioncanbeeasilymeasuredintime-of-ightexperiments,butsuperowishardertoachieveintheseconnedsystems. 1.2TopologicalDefects:Dislocations,Vortices,etc. Defectsareubiquitousincondensedmattersystemsandplayanimportantroleindeterminingmanyaspectsoftheirproperties.Forexample,vacanciesandinterstitialscatalyzediffusionofparticlesinsolids,dislocationsdeterminethestrengthofcrystallinematerials,andvortexmotioncontrolstheresistivityofsuperconductors.Manymaterialpropertiesofpracticalinterestdependondefects.Vorticesassumeparticularsignicancein2Dsystemswheretheycanmediatephasetransitionsthroughtheirthermalunbinding[ 17 18 ].Ourworkhastwomainthemes:theeffectofdislocationsonthesuperuidphasetransitionandthebehaviorofvorticesinsuperuidsystems. WendforexamplethatwithinthescopeofaLandautheoryforthesuperuidphasetransition,dislocationscouldenhanceasuperuidphasetransition[ 48 78 ].TwodimensionaldefectslikegrainboundariesarealsocapableofsustainingKosterlitz-Thoulesstransitions.Weareparticularlyinterestedinprovidingaplausibledislocation-basedmodelfortheobservedsupersolidityofsolid4He[ 15 16 ],whichwearguecouldessentiallybesuperuidorderexistingalongdefectstructuresinthecrystal. Aparticularlyimportantclassofdefectsaretopologicaldefects,whichoccurinsystemswithsomebrokensymmetry,andarecharacterizedbyacoreregionwhereorderislocallydestroyed,andafareldbehaviorwhereelasticvariablesdescribingtheorderedstateslowlychange.Anexamplewouldbethesuperconductingorderparameterinthepresenceofavortex.Thetopologicaldefectisimpossibletogetridofbylocalalterationsoftheorderparametereld,andresembleelectricchargesinthattheireffectcanbedeterminedbyobservationoftheireffectonanysurfaceenclosingthem. 18

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Adislocationisaone-dimensionaltopologicaldefectintheregularcrystallinearrangementofatomsinasolid.ThesimplestwaytovisualizeadislocationisthroughtheVolterraconstruction.Imaginetheperfectcrystalbeingdividedintotwohalvesbyaslipplaneandthendisplacingonehalfrelativetotheotherbysomevector.Thisso-calledBurger'svectorbischaracteristicofadislocationlineandisanintegralmultipleofthelatticeparameter.IftheBurger'svectorisperpendiculartothedirection Figure1-2. Cartoonshowingboththebasictypesofdislocations.TheBurger'svectorbisshownbytheredarrow.Reprintedwithpermissionfrom http://courses.eas.ualberta.ca/eas421 ,accessedonApril2010. ofthedislocationline(b?t),theresultingstructureisanedgedislocation.Thiscanbecreatedbyforcinginanextrahalfplaneofatomswithitsedgelyingalongthedislocationline.Theothertype(bkt)isascrewdislocation.Arealdislocationisusuallyacombinationofthesetwobasictypes. Vorticesareeasilyvisualizedinuids,wheretheyimplyacoreregionwithsomesingularbehavior,aroundwhichtheuidrotates.Usuallythisrotationalvelocityfallsawayas1=r,whereristheradialdistancefromthecore.Thiscanbecontrastedwithrigidrotationwherethevelocityproleincreasesasr.Quantumvorticesoccurinawide 19

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varietyofphysicalsystems-fromsuperconductors,superuidhelium,andBECintrapstomagneticthinlms,andextremelydensequantumplasmassuchasfoundinneutronstars.Theyareoftenknowntohaveadramaticimpactonthephysicalpropertiesofthesystemwheretheyarepresent.Themotionofvorticesinsuperconductorscansuppresssuperconductivitybyinducingpotentialgradientsandsopracticalapplicationsinvolvingsuperconductivityoftenrequirethesevorticestobepinnedbyimpurities. Aquantumvortexisapointsingularityinthephaseeld(x)oftheorderparameterinsomesystemwithcontinuouslybrokensymmetry(suchasanXYmodel).Itsatisesthefollowingbasictopologicalconstraint: Id=2ev,(1) wheretheintegralistakenaroundanyloopenclosingthevortex,andevisanintegercalledthechargeorwindingnumberofthevortex.ThisiscalledtheFeynman-Onsageridentityinthecontextofsuperuidhelium,whichalongwithsuperconductors,weretherstsystemsforwhichtheideaofaquantizedvortexwasposited.Thedensityoftheorderparametergoestozeroatthevortexcoretopreventasingularityintheenergy.Theenergyofthevortexhastwoparts,acoreenergyEccorrespondingtothevanishingoftheorderparameteratthecenterofthevortex,andanelasticenergyEelassociatedwiththeslowspatialvariationofthephaseeldfarawayfromthevortexcore.Theactualphasecongurationfarawayfromthevortexcoreisdeterminedbytheminimizationoftheelasticenergyandthetopologicalconstraintorquantizationconditionmentionedabove.Theexactdynamicsofavortexanditsinertiaareasyetsomewhatcontroversialtopics,aswewilldiscusslaterinthisthesis,butareimportanttounderstandbecauseoftheirrelevancetomaterialproperties. 1.3TheSupersolidConundrum Asolidcanwithstandshear,i.e.itisdeformedbutdoesnotowwhensubjecttoashearingstress.Acrystallinesolidparticularly,ischaracterizedbyaregulararray 20

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ofatomsthatarelocalizedintheirrespectivepositions.Bosecondensation,ontheotherhand,impliesalargenumberofidentical,delocalizedparticles.Coexistenceofsuperuidityandcrystallinecharactermaythereforeseemparadoxical.Indeed,PenroseandOnsager[ 19 ],whowerearguablythersttoconsiderthepossibilityofasupersolid(in1956),concludedthatBosecondensationwasnotpossibleinacommensuratecrystal2.However,in1969itwaspositedbyAndreevandLifshitz[ 20 ],andindependentlybyThouless[ 21 ]andChester[ 22 ],thatzero-pointvacanciespresentinthegroundstateofaquantumcrystallikesolid4Hecandelocalizeandcausemasstransport.ThesevacancieswouldundergoBose-Einsteincondensationbelowacertaintemperature,thusgivingrisetosuperuid-likebehavior. Recentstudiesin2004byKimandChan[ 23 24 ]foundsomeevidenceforthisphenomenoninso-calledtorsionaloscillatorexperiments,whereasampleofsolid4Heisplacedinacontainerthatistorsionallyrotatedbysomeexternaldrivingmechanism[seeFig.( 1-3 )].Atatemperatureofaround100mK,theyobserveashiftintheresonantperiodoftheoscillatorasshowninFig.( 1-4 ).Thiswasattributedtothedecouplingofacertainfraction(nonclassicalrotationinertiaorNCRI)ofthemassofsolid4Hefromtherotation.TheperiodTofoscillationsisrelatedtothemomentofinertiaIandelasticconstantKasT=2p I=K,andsoareductionofinertiawouldleadtoadecreasedresonantperiod.Thisisexactlywhatwasobserved.Thisnonclassicalfractionisseentogoupwithdecreasingtemperature,anddownwithincreasingvelocity,suggestingasuperuidcomponent. Subsequentexperimentshaveconrmedthatanomaliesexistnotonlyintherotationalresponseofsolid4He,butalsoinitselasticandthermodynamicresponseatsimilartemperatures.However,theinterpretationoftheseresultsismorecomplicated 2Thisimpliesaperfectcrystalwithoutanyvacancies,inwhicheachunitcellcontainsanintegernumberofatoms. 21

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Figure1-3. ThetorsionaloscillatorsetupusedbyKimandChanintheir2004experiment.ReprintedwithpermissionfromRef.[ 23 ][E.KimandM.H.WChan,Nature427,225(2004)],cMacmillanPublishersLtd:Nature,2004. thanwasimagined.Ithascometobeacceptedthatcrystallinedefectssuchasgrainboundaries,dislocationsandimpuritiesplayamajorroleintheseeffects.RittnerandReppy[ 25 26 ]ndthattheNCRIincreaseswithdisorder,andissuppressedonannealingthesample.Linetal.[ 27 ]reportapeakinthespecicheatataroundthesametemperaturewhereNCRIsetsin,butthefeaturesofthepeakstronglydependonsamplequalityand3Heimpurityconcentration.Astrikingonsetofshearhardening[ 28 ]hasbeenfoundbyDayandBeamishwherethesampleseemstostiffenastemperatureisreduced,thetemperaturedependenceoftheshearmodulusbeingverysimilartothatoftheNCRI[seeFig.( 1-5 )].Suchstiffeninghasbeenattributedtothepinningofdislocationsby3Heatoms[ 28 ],oralternativelybyquantumrougheningofthedislocationlines[ 29 ],thelattermechanismbeingindependentof3Heconcentration. 22

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Figure1-4. Periodshiftsinatorsionoscillatorexperiment.Theshiftsareseentogodownwithincreasingrimvelocity,andarenotpresentfortheemptycell.ReprintedwithpermissionfromRef.[ 23 ][E.KimandM.H.WChan,Nature427,225(2004)],cMacmillanPublishersLtd:Nature,2004. ThesimplevacancybasedmodelofsupersolidityhasbeenruledoutbyquantumMonteCarlostudies[ 30 ],whichndtoolargeanenergycostassociatedwithvacancyformationinapurecrystallinesampleofsolid4He.Sasakietal.[ 31 ]demonstratemassowthroughgrainboundaries.Allthistakentogethermotivatesourquestforadislocation-basedmodelforsupersolidity,wherewebuildonpreviousworkbyDorseyetal.[ 32 ]andToner[ 78 ]basedontheLandautheoryforphasetransitions,toshowhowanetworkofdislocationscouldinducesuperuidorderwithinsolid4He. Whilethereisnoquestionthatthedisorderenhancesthesupersolidphenomenon,thereissomedebateaboutwhethersupersolidityisanintrinsicpropertyof4Hecrystalsasoriginallysupposed,orifitissolelybroughtaboutbydefectspresentin 23

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Figure1-5. Shearhardeninginsolid4He,shownfordifferent3Heconcentrations.ReprintedwithpermissionfromRef.[ 28 ][J.DayandJ.Beamish,Nature450,853(2007).],cAPS,2007. suchsamples.Therearemanyexperimentstosuggestthatsuperowreallytakesplace.TherewasasuggestionthattheshiftintheresonantpeakinthetorsionaloscillatorexperimentscouldentirelybeduetothestiffeningoftheshearmodulusseenbyBeamish.Recentstudies[ 34 ]withaveryhard-walledcontainerwherestiffeningof4Hewouldnotaffecttheelasticmodulusoftheoverallsystemhaveruledthisout.Signicantly,nosuchshiftinperiodisobservedintorsionaloscillatorexperimentsonsolid3He[ 34 ],makingastrongcaseforthepresenceofaBosecondensate.Ourmodelishowevermoregeneralandwouldservetoexplaintheenhancementoftheeffectbroughtaboutbydislocations,evenifthisweretobeanintrinsiceffect. 1.4VorticesinRotatingBose-EinsteinCondensates Therotationofaquantumuidstrikinglyillustratestheconstraintssetbyquantummechanicsonthevelocityeldofaquantummacroscopicobject.VorticesinsuperuidheliumwerersttheoreticallypredictedbyFeynmanandOnsager,andweresoonexperimentallydetectedbyVinen,Packardandothers[ 35 ].EversinceBECswererealizedin1995,theefforttocreateanddetectvorticesinthemwason.Thereisonepracticaldifcultywhichhadtobesurmountedinthisquest.Unlikeliquidheliumina 24

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rotatingcontainerwheretheroughnessinthewallsisbelievedtonucleatevortices,agBECisinamagneto-opticaltrapwheresuchamechanismisnotavailable. TwobasicexperimentalapproacheshavebeenusedtocreateandstudyvorticesinaBEC.Therst,carriedoutinJILAin1999[ 36 ],usesamixtureoftwohypernecomponentsof87Rb,spinningoneupwithrespecttotheotherbyapplyinganexternalelectromagneticbeam.ThesecondwasperformedinENS,Parisin2000[ 37 ],andisanalogoustotherotating-bucketmethodofconventionallow-temperaturephysics.Heretheaxisymmetricmagnetictrapinwhichtheatomsareconnedisdeformedbyanoff-centerlaserbeam,whicheffectivelystirsandrotatesthetrapbyapplyingadipolarforceoneachcondensateatom. ThevorticesinadiluteBECaretypicallylargerthaninheliumandtheirsizecanbecontrolledbyvaryingthedensityorspeciesofthetrappedatoms.Theyarealsoeasiertoobserveoncecreated.Thisistypicallydoneinthetime-of-ightexperiments,wheretheBECtrapisrotatedwhilealsobeingcooleddownbelowitscondensationtemperature,andoncethevorticesarethoughttohaveformed,thetrappingpotentialisreleased.Thecondensateexpandsintheabsenceofatrapandthevortexsizealsoincreasestoseveralm.Thecondensateisprobedbyopticalmeans,i.e.byshiningalaserbeamthatistunedtotheexcitationfrequencyoftheRbatoms,suchthattheactualdensitycanbeprobed.Thevorticesthenshowupasholesinthecondensatedensity.TheseexperimentswereabletoconrmcertaintheoreticalpredictionsmadeaboutvorticesinBEC[ 38 39 ],suchastheexistenceofacriticalvelocityoftraprotation,c,abovewhichavortexisstabilized. 25

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Figure1-6. TransverseabsorptionimagesofaBose-Einsteincondensatestirredwithalaserbeam.Forallveimages,thecondensatenumberisN105,andthetemperatureisbelow80nK.Therotationfrequency=(2)isrespectively(c)145Hz,(d)152Hz,(e)169Hz,(f)163Hz,(g)168Hz.In(a)and(b)weplotthevariationoftheopticalthicknessofthecloudalongthehorizontaltransverseaxisfortheimages(c)(0vortex)and(d)(1vortex).ReprintedwithpermissionfromRef.[ 37 ][K.W.Madison,F.Chevy,W.Wohlleben,andJ.Dalibard,Phys.Rev.Lett.84,806(2000)],cAPS,2000. 26

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CHAPTER2DISLOCATION-INDUCEDSUPERFLUIDITY Thereisnocompletelysatisfactorymicroscopicmodelfortheexistenceofsuperuidityinacrystalwhichcanexplainthevariousanomalouseffectsdiscoveredrecentlyinsolidhelium[ 23 ].However,ifarealthermodynamicphasetransitiontakesplaceinsolidheliumataroundthesetemperatures,itcouldbedescribedbyLandau'stheoryforsecondorderphasetransitions[ 40 ].Landauassumedthatanorderedphasecouldbecharacterizedbyanitefunctioncalledtheorderparameter, (x),whichwouldbezerowhentheorderislost,i.e.inthedisorderedphase.Nearthetransitiontemperature,onecouldexpandthefreeenergyofthesystemintermsofthisorderparameter,inawaythatrespectstheinherentsymmetriesofthesystem.Fortheprototypicalsecondorderphasetransition,e.g.theferromagneticorsuperconductingtransition,thiswouldbewrittenas F=Zd3x1 2cj@ j2+1 2a(T)j j2+1 4uj j4. Herea(T)isasmoothfunctionoftemperature,a=a0(T)]TJ /F5 11.955 Tf 12.68 0 Td[(T0)=T0whereT0isthecriticaltemperaturebelowwhichorderrstappears.Thequantity~=p 2ma(T)isthecoherencelength,alengthscaleoverwhichorderprevailsinthesystem.NearT0,thecoherencelengthdivergesas1=p (T)]TJ /F5 11.955 Tf 11.95 0 Td[(T0).Howevertheconclusionsofthistheory,suchasthecriticalexponentof=)]TJ /F3 11.955 Tf 9.3 0 Td[(1=2foundhere,andthecriticaltemperatureT0,aremodiedbytheeffectofuctuations,whichbecomeespeciallyimportantnearcriticality.Suchaphenomenologicaldescriptionisexpectedtoholdverygenerallyirrespectiveofthedetailedinteractionsinthesystem,andisaminimalmodelthatcancomeinhandyindescribingacomplexsystemsuchassolidhelium. Partsofthischapterarereproducedfromthepublishedarticles:K.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys.Rev.B81,64516(2010)andD.Goswami,K.Dasbiswas,C.-D.YooandA.T.Dorsey,Phys.Rev.B84,054523(2011). 27

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Historically,suchamodelwasappliedwithgreatsuccesstosuperconductorsbyLandauandGinzburg[ 41 ],andisknownastheGinzburg-Landautheoryinthiscontext.Theanalogoustheoryforneutralsuperuids(oftencalledtheGinzburg-Pitaevskiimodel[ 42 ],butwhichwegenericallycallaLandautheoryinthischapter)wasnotsosuccessfulindescribingthe-transitioninliquidhelium,presumablybecauseofthelargeroleplayedbyuctuations. Inthischapter,werstintroducetheLandautheoryforasupersolidtransitionbycouplingthesuperuidorderparametertotheelasticstrainsofthecrystallattice.Suchaphenomenologicaldescriptionisusefulevenifthereisnobulksupersolidtransition,andinfactweusethistopredictthatthesuperuidtransitiontemperatureisenhancedbythepresenceofadislocationlineinthecrystal.Weintroduceaformalismtodescribethisdislocation-inducedone-dimensionalsuperuidandthenprovidequantitativeestimatesofthiseffect. 2.1LandauTheoryfortheSupersolidPhaseTransition Thetransitionfromnormaluidtosuperuid(NF-SF)in4Heisastandardexampleofacontinuousphasetransition,andcanbedescribedbyaLandautheory.FollowingDorsey,GoldbartandToner[ 32 ],weassumethatthephasetransitionfromnormalsolidtosupersolid(NS-SS)isalsocontinuous.ALandautheoryofthetransitioncanthenbedevelopedbycouplingthesuperuidorderparametertotheelasticityofthecrystalline4Helattice.ThemainconsequencesofthistheoryaspredictedbyDorseyetal.areanomaliesintheelasticcoefcientsnearthetransitionandconverselylocalalterationsinthetransitiontemperaturebroughtaboutbyinhomogeneousstrainsinthelattice.Onecommonwayinwhichthesestrainscanariseinrealsolidsisthroughthepresenceofcrystaldefectssuchasvacancies,interstitials,dislocations,orgrainboundaries[ 43 ].Itwillbeshownsubsequentlythatedgedislocationsinparticularcaninducesuperuidityintheirvicinityattemperatureshigherthaninthebulk. 28

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AsinthecaseofsuperuidsandotherinstancesoforderedphaseswithcontinuouslybrokenU(1)symmetry,wecharacterizethesupersolidphasebyanorderparameterthatisacomplexscalareld (x).Themicroscopicinterpretationofthisorderparameteristhatitcorrespondstothequantumwavefunctionofthemanybodygroundstatetowhichallthe4HeatomsBosecondenseatzerotemperature.TheuniversalpropertiesofthistransitioncanbeobtainedirrespectiveofthemicroscopicdetailsofthesystembyconsideringanexpressionoftheLandaufreeenergyintermsofthelowestrelevantpowersoftheorderparameteranditsspatialgradients,thatsatisfyallthesymmetrypropertiesofthesystem, F=Zd3x1 2cij@i @j +1 2a(T)j j2+1 4uj j4.(2) Herea(T)isasmoothfunctionoftemperature,whichforaregularsuperuidtransition(suchasforNF-SFin4He)is,a=a0(T)]TJ /F5 11.955 Tf 13.41 0 Td[(T0)=T0whereT0isthebulkcriticaltemperaturebelowwhichsuperuidityrstappears.Theactualtransitiontemperaturecandifferfromthisbecauseoftheeffectofthermaluctuations.Thecoefcientscijcorrespondtotheanisotropyinheritedfromthecrystal.Foranisotropicsuperuidorcubiccrystal,cij=cij.a0,canduareallphenomenologicalparameters(allpositive)thatarisefromtheexpansionofthefreeenergyinthisLandautheory. However,inthepresenceofacrystalweshouldtakeintoaccountthephononmodesthatarisebecauseofthecrystallinityofthesystem.Theseareassociatedwithdisplacementsofatomsu(x)inthelatticefromtheirmeanpositionsbecauseofstraineldsinthecrystal.AstheLandaufreeenergyabovehastobeinvariantunderrotationsandtranslations,thedisplacementeldu(x)cancoupletothesuperuidorderparameteronlythroughthesymmetricstraintensor,denedasuij1 2(@iuj+@jui+@iuk@juk).TothelowestorderthecouplingwithcrystallinityaffectsonlythetermintheLandaufreeenergythatisquadraticintheorderparameter,andthecoefcienta(T)isnowafunctionofspatialcoordinates,a(T)=a0(T)]TJ /F5 11.955 Tf 11.96 0 Td[(T0)=T0+uijaij. 29

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Solid4Hehasahcpcrystallattice(exceptforasmallregioninitsphasediagramwhereitisbcc),andthecouplingtensorscijandaijmusthavethatsamesymmetrystructure.Fortherestofourpresentationhere,wetreattheisotropiccaseforthesakeofclarity,withtheconvictionthatourkeyresultsaremoregeneralanddonotdependonthespeciclatticestructure.Underthissimplifyingassumption,theLandaufreeenergycanbewrittenas F=Zd3xc 2jr j2+1 2a(x)j j2+1 4uj j4,(2) wherethesuperuiditycouplestothecrystallinitythroughthetraceofthestraintensor,a(x)=a0(T)]TJ /F5 11.955 Tf 12.41 0 Td[(T0)=T0+a1uii(x).ThespatialdependenceofthequadraticcoefcientoftheLandaufreeenergysuggeststhatthecriticaltemperatureTcmaybemodiedbyelasticstrainspresentinthecrystal. 2.2Superuid-DislocationCoupling Acrystaldefectstructurelikeadislocationlinedisruptstheregulararrangementofatomsinitsneighborhood.Thisshiftingofthepositionsoftheatomsofthelatticefromtheirminimumenergycongurationsinduceselasticstrainsinthecrystal.TheanalysisoftheprevioussectionsuggeststhatthelocalcriticaltemperatureTcforapossiblesuperuidtransitioncanbemodiedthroughthecouplingofthesuperuidorderparameterwiththeelasticstraineld.Wehavearrivedatessentiallysimilarconclusionsusingdifferentmethodsinthecontextoftheputativesupersolidstateof4He[ 48 78 79 ].ThesameideahasalsobeenexploredthroughapproachesotherthanthephenomenologicalLandautheoreticdescriptionwepursue.QuantumMonteCarlosimulationsforexamplendsuperuidityalongthecoreofascrewdislocationinsolid4He[ 44 ].Inthissection,wepresenttheresultofcouplingthestraineldofasingle,quenched,edgedislocationtothesuperuidorderparameter.WepostponethediscussionofmorerealisticsituationsinvolvingmultipleormobiledislocationstoChapter3ofthisdissertation. 30

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ItisalsointerestingtonotethatsimilarstudiesonsuperconductorswithinthephenomenologicalLandautheoryhavereportedanincreaseintheTcintheneighborhoodofanedgedislocation[ 45 73 ].Thisissomewhatcounterintuitivefromamicroscopicpointofview,asthepresenceofdisorderandnonmagneticdefectsisusuallyexpectedtoinhibitpairingandtoreducethesuperconductinggap.However,pairingisnotimportantinsuperuid4He,whichisaBoseliquid.Wesuggestthenthatthesuperuidsignatureseenintorsionaloscillatorexperimentscanbeattributedtosuperuidityinducedlocallyalongdislocationlines. Withoutanylossofgenerality,weconsiderasinglestraightedgedislocationpointingalongthez-axisandpassingthroughtheoriginofourcoordinatesystem.TheBurger'svectorbisalongthey-axis.Thiscorrespondstoasituationwhereanextrahalfplaneofatomshasbeenintroducedwithedgelyingalongthex-axis.Ifisotropyisassumed,thequantityofinterestthatcouplestothesuperuidorderparameterinEq.( 4 )oftheprevioussectionisthetraceofthestraintensor,whichisphysicallythelocalfractionalvolumechangeofthecrystallattice.ThishasbeenderivedinAppendixAfromconsiderationsoflinearelasticitytheory[ 50 74 ]tobe uii=4 2+bcos r,(2) whereandaretheLameelasticconstants,andwehaveintroducedthecoordinatesx=(r,z),withrinthex)]TJ /F5 11.955 Tf 12.19 0 Td[(yplane[(r,)arepolarcoordinatesinthex)]TJ /F5 11.955 Tf 12.19 0 Td[(yplane].ThelocalchangeinTcisreectedinthetermquadraticintheorderparameterintheLandautheory, a(x)=a0t0+Bcos r.(2) Here,a0,banduarephenomenologicalparameters(allpositive);t0=(T)]TJ /F5 11.955 Tf 12.32 0 Td[(T0)=T0isthereducedtemperature,withT0themean-eldcriticaltemperatureforthesupersolidtransitionintheabsenceofthedislocations(t0>0isthenormalsolidandt0<0, 31

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thesupersolid);andBisacouplingconstant(intowhichwehaveabsorbedtheelasticconstants). Figure2-1. Schematicshowingthedislocationaxis(alongz)andthetubularsuperuidregionthatdevelopsaroundit.Theradiusofthecylinderisdeterminedbythelengthscaleofthegroundstatewavefunction.Theaxisofthetubewillbeoffsetfromthedislocationaxis.ReprintedwithpermissionfromRef.[ 48 ][D.Goswami,K.Dasbiswas,C.-D.YooandA.T.Dorsey,Phys.Rev.B84,054523(2011)],cAPS,2011. Tosimplifythesubsequentanalysis,weintroducethecharacteristicscalesoflengthl=c=a0B,orderparameter=a0B=p cu,andfreeenergyF0=a0Bc=2u,andthedimensionless(primed)quantitiesx0=x=l, 0= =,andF=F=F0;intermsofthedimensionlessquantitiesthefreeenergybecomes F=Zd3xjr j2+[V(r))]TJ /F5 11.955 Tf 11.95 0 Td[(E]j j2+1 2j j4,(2) whereV(r)=cos=r,E)]TJ /F5 11.955 Tf 22.07 0 Td[(ct0=a0B2,andwehavedroppedtheprimesonallquantitiesforclarityofpresentation. 32

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2.3EffectiveLandauTheoryforSuperuidityalongaDislocation SofarwehavesetuptheLandautheoryforasuperuidcoupledtoasingledislocationinasolid.Herewearguethattheeffectofthedislocationmustbetoincreasethesuperuidcriticaltemperaturearounditself.Beforeproceedingfurther,letusanalyzethebehaviorinthehigh-temperature,normalphase.Inthisphasewecanneglectthequarticterminthefreeenergy;theresultingquadraticfreeenergyis F0=Zd3xjr j2+[V(r))]TJ /F5 11.955 Tf 11.95 0 Td[(E]j j2=Zd3x (^H)]TJ /F5 11.955 Tf 11.95 0 Td[(E) (2) wheretheHermitianlinearoperator^Hisgivenby ^H=r2+V(r).(2) Wecandiagonalizethefreeenergybyintroducingacompletesetoforthonormaleigenfunctionsn(x)of^H, ^Hn=Enn,(2) wherenlabelsthestates,andweassumethattheeigenvaluesEnareorderedsuchthatE0
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ThefreeenergyispositiveaslongasEn>E,foralln.RecallthatE)]TJ /F3 11.955 Tf 22.4 0 Td[((c=a0B2)(T)]TJ /F5 11.955 Tf -451.98 -23.9 Td[(T0)=T0(witha0andcbothpositive),sohightemperaturesTcorrespondtolarge,negativevaluesofE.AswedecreaseT,Eincreases,untileventuallywehitaconden-sationtemperatureTcondatwhichE(Tcond)=E0;belowthistemperaturethequadraticfreeenergyF0becomesunstable.Rearrangingabit,wehave Tcond)]TJ /F5 11.955 Tf 11.95 0 Td[(T0 T0=)]TJ /F5 11.955 Tf 9.29 0 Td[(E0a0B2 c.(2) If^Hhasnegativeeigenvaluesi.e.,iftheequivalentSchrodingerequationhasboundstatesthenTcond>T0,andthedislocationinducessuperuidityabovethebulkorderingtemperature.AsemphasizedinRefs.[ 45 78 79 ],thedipolarpotentialcos=r,whichhasanattractiveregionirrespectiveofthecouplingconstants,alwayshasanegativeenergyboundstate.Wearethusleadtothesurprisingandimportantconclusion[ 45 78 ]thatsuperuidityrstnucleatesaroundtheedgedislocationbeforeappearinginthebulkofthematerial. JustbelowthecondensationtemperatureTcond,thenucleatedorderparameterhastheform =A00(r),(2) where0isthenormalizedgroundstatewavefunctionandA0isanamplitudethatisxedbythenonlineartermsinthefreeenergy[ 83 ].SubstitutingEq.( 2 )intothedimensionlessfreeenergy,Eq.( 4 ),weobtain F=(E0)]TJ /F5 11.955 Tf 11.96 0 Td[(E)jA0j2+1 2gjA0j4,(2) wherethecouplingconstantgisgivenby g=Zd2r40(r).(2) 34

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MinimizingthefreeenergywithrespecttoA0,weobtain A0=p (E)]TJ /F5 11.955 Tf 11.96 0 Td[(E0)=g.(2) FromtheextensivenumericalworkofDasbiswasetal.[ 79 ],presentedinSection2.4ofthisdissertation,weknowthatforthedipolepotentialthegroundstateenergyisE0=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.139(withtheenergyoftherstexcitedstateE1=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.0414),andthecouplingconstantg=0.0194. Torecap-followingtheworkofpreviousauthors[ 45 78 ]wehaveshownthatsuperuidityalwaysnucleatesrstonedgedislocations,andwehavecalculatedtheformoftheorderparameternearE0,thethresholdvalueofE(i.e.,attemperaturesjustbelowthecondensationtemperature).Physically,weimagineacylindricaltubeofsuperuid,witharadiusequaltothetransversecorrelationlength(oforder1inourdimensionlessunits),thatencirclesthedislocation.However,thisnaivemean-eldpictureignoresthethermaluctuationswhichdestroytheone-dimensionalsuperuidityonlonglengthscales.Whatisneededisaneffectiveone-dimensionalmodelforthesuperuid,capableofcapturingnontrivialuctuationeffects.Wenowderivethisone-dimensionalmodelusingasystematic,weakly-nonlinearanalysisnearthethresholdE0. Withinthemean-eldLandautheory,theorderparametercongurationsthatminimizethefreeenergyaresolutionstotheEuler-Lagrangeequation F =0=r2 +[V(r))]TJ /F5 11.955 Tf 11.95 0 Td[(E] +j j2 .(2) Thisnonlineareldequationisdifculttosolve,evennumerically.Instead,weresorttoaweaklynonlinearanalysis[ 84 ]nearthethresholdforthelinearinstability.Ourgoalistointegrateoutthemodestransversetothedislocationandobtainaneffectivemodelfortheone-dimensionalsuperuidnucleatedalongthedislocation.Wethentreatthermaluctuationsofthisone-dimensionalsuperuidinthenextSection. 35

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Westartbyintroducingacontrolparameter E)]TJ /F5 11.955 Tf 11.95 0 Td[(E0(2) thatmeasuresdistancefromthelinearinstability.FromtheanalysisintheprecedingSection,weseethattheorderparameternearthresholdscalesas1=2,whichsuggestsarescalingoftheorderparameter =1=2,(2) withaquantitywhoseamplitudeisO(1).Next,notethatifwehadincludedtheplanewavebehavioralongthez-axisintheearlieranalysis,thecoefcientofjAnj2inthequadraticfreeenergy,Eq.( 2 ),wouldbe)]TJ /F4 11.955 Tf 9.3 0 Td[(+k2.Thissuggeststhattheimportantuctuationsalongthez-axisoccuratwavenumbersk1=2,oratlengthscalesoforder)]TJ /F6 7.97 Tf 6.58 0 Td[(1=2(i.e.,longwavelengthuctuationsareimportantclosetothreshold).Thissuggestsanotherrescaling, z=)]TJ /F6 7.97 Tf 6.59 0 Td[(1=2.(2) SubstitutingthesevariablechangesintoEq.( 2 ),andwritingE=E0+,weobtain ^L=@2+)-222(jj2,(2) wheretheHermitianlinearoperator^Lisgivenby ^L=r2?+V(r))]TJ /F5 11.955 Tf 11.96 0 Td[(E0,(2) withr2?theLaplacianindimensionstransversetoz.Next,weexpandinpowersof, =0+1+22+....(2) Collectingterms,weobtainthefollowinghierarchyofequations: O(1):^L0=0, (2) O():^L1=@20+0)-221(j0j20, (2) 36

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O(2):^L2=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(320)1. (2) ThesolutionoftheO(1)equationisthenormalizedgroundstateeigenfunction,0(r);thereisanoverallintegrationconstant,whichcanbeafunctionof,whichwewillcallA0(): 0=A0()0(r).(2) SubstitutethisintotherighthandsideoftheO()equation, ^L1=0@2A0+0A0)]TJ /F3 11.955 Tf 11.96 0 Td[(30jA0j2A0.(2) WecandetermineA0byleftmultiplyingthisequationby0,integratingond2r,andusingthefactthat^LisHermitian,tond @2A0+A0)]TJ /F5 11.955 Tf 11.95 0 Td[(gjA0j2A0=0,(2) wheregisdenedinEq.( 2 ).ThisisthesolvabilityconditionfortheexistenceofnontrivialsolutionsoftheO()equation.InprinciplewecouldsolvethisequationforA0,substitutebackintotherighthandsideoftheO(),andsolvetheresultinginhomogeneousequationtoobtain2.Inpracticethisisdifcultforthedipolepotential,sowewillstopatthisleveloftheperturbationtheory.However,inAppendixBweshowthatforV(r)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1=r(atwo-dimensionalCoulombpotential),theperturbationcalculationcanbeexplicitlyworkedoutthroughO(2),andtheresultsareincloseagreementwithdetailednumericalsolutionsofthenonlineareldequation. WecanrecastEq.( 4 )intermsof=1=2zandA0=)]TJ /F6 7.97 Tf 6.58 0 Td[(1=2'as @2z'(z)+')]TJ /F5 11.955 Tf 11.95 0 Td[(gj'j2'=0,(2) whichistheEuler-Lagrangeequationforthefreeenergyfunctional F=Zdz1 2j@z'j2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2j'j2+g 4j'j4.(2) 37

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Reinstatingthedimensions,weobtain F=Zdzc 2j@z'j2+a 2j'j2+b 4j'j4,(2) wherea=a0t,t=(T)]TJ /F5 11.955 Tf 12.56 0 Td[(Tcond)=Tcondisthereducedtemperaturemeasuredrelativetothecondensationtemperature,andb=gu.Torecap,wehaveintegratedoutthetransversedegreesoffreedom(theuctuationsofwhichhaveanonzeroenergygap)andreducedthefullthree-dimensionalproblemtoaneffectiveone-dimensionalmodel.Inthenextsectionwewillstudytheuctuationsofthisone-dimensionalmodel,andderiveaneffectivephase-onlymodelforadislocationnetwork. 2.4QuantitativeEstimatesoftheShiftinCriticalTemperature 2.4.1TheQuantumDipoleProblemin2D HerewedescribethemethodsusedtosolvethelinearizedGinzburgLandauequationdescribedintheprevioussection.Thegroundstateeigenvalueisofparticularinteresttous,sinceitisrelatedtotheshiftinTcinthepresenceofadislocation.Theeigenfunctionisalsousedinderivingthenetworkmodelinthefollowingsection.Theequationtobesolvedisidenticaltoa2DSchrodingerequationforthedipolarpotential, )]TJ /F7 11.955 Tf 15.57 8.09 Td[(~2 2mr2 +pcos r =E .(2) Thispotentialisnon-centralandcannotbesolvedanalyticallyorbyusingWKBmethods.Itpermitsboundstatesbecausethepotentialisattractiveinonehalfplane.Theproblemhasalonghistory,especiallyinthecontextofboundandexcitedstatesforedgedislocationsinsemiconductors,andhasbeentackledusingavarietyofvariationalandotherkindsofapproximatemethods.Theseearlyefforts,andthecorrespondingnondimensionalizedgroundstateeigenvalues,arelistedinTable 2-1 .Wesolvetheproblemusingadirectnumericalapproachandndagroundstateeigenvalueof)]TJ /F3 11.955 Tf 9.3 0 Td[(0.139,whichislowerthanalltheupperboundsestablishedbythevariationalmethods[ 57 ].SomesampleeigenvaluesfoundbydifferentmethodsarelistedinTable 2-2 38

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Table2-1. Summaryofgroundstateenergyestimatesoftheedgedislocationpotential.Energyisgiveninunitsof2mp2=~2. ReferencesGroundstateestimate Landauer(1954)[ 51 ]-0.102Emtage(1967)[ 52 ]-0.117NabutovskiiandShapiro(1977)[ 53 ]-0.1014SlyusarevandChishko(1984)[ 54 ]-0.1111Dubrovskii(1997)[ 55 ]-0.1196FarvacqueandFrancois(2001)[ 56 ]-0.1113DorseyandToner[ 57 ]-0.1199Thiswork-0.139 2.4.2NumericalMethodsUsed Adetailednumericalsolutionofthetwo-dimensionalSchrodingerequationwiththedipolepotential,Eq.( 2 ),islikelytoprovidemoreaccurategroundstateeigenvaluesinadditiontodeterminingtherestoftheboundstateeigenvaluesandcorrespondingwavefunctions.Wedothisbothbyarealspacediagonalization,wheretheSchrodingerequationisdiscretizedonasquaregrid,andbyexpandinginthebasisoftheeigenfunctionsofthetwo-dimensionalCoulombpotentialproblem.Twospecialfeaturesofthisdipolepotentialmakeitanumericallydifcultproblem:thesingularityattheoriginandthelongrangebehaviorofthepotential.ItisexpectedthattheCoulombwavefunctionswouldbebettersuitedtocapturingthislongrangebehaviorandconvergencewouldconsequentlybefaster.OurresultsshowthattheCoulombbasismethodismoreaccurateforthehigherboundstates(whichareexpectedtoextendmoreinspace),astherealspacemethodsarelimitedbysizeissues.However,therealspacemethodworksbetterforthegroundstate.TheCoulombbasisresultshaverecentlybeenfurtherrenedbyAmore[ 58 ]. 2.4.3VariationalCalculation Ourinitialapproachtodeterminethegroundstateenergyhasbeenvariationalbecausethiscanbecarriedoutanalyticallyandprovidesaroughestimatewhichcanthenguideourmoreexplicitnumericalsolution.Givenanormalizedwavefunction 39

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(r,),weminimizedtheenergyfunctional, F[ ]=Zd2x~2 2mjr j2+pcos rj j2.(2) ThisfunctionalhasitsextremaatthesolutionsoftheSchrodingerequation,Eq.( 2 ).NotethatthelengthandenergyscaleswhichemergefromEq.( 2 )(ortheSchrodingerequation)forthisproblemare~2=2mpand2mp2=~2.Indimensionlessvariables,thenormalizedtrialwavefunctionusedinourcalculationis (r,)=2AB Cp (1)]TJ /F5 11.955 Tf 11.95 0 Td[(r=BC) p (3)]TJ /F3 11.955 Tf 11.96 0 Td[(4B+2B2)exp)]TJ /F5 11.955 Tf 15.11 8.09 Td[(r C)]TJ 10.49 17.97 Td[(p 1)]TJ /F5 11.955 Tf 11.95 0 Td[(A2 C2r 8 3rcosexp)]TJ /F5 11.955 Tf 15.11 8.09 Td[(r C, (2) whereA,BandCarevariationalparameters.Wechoosethetrialwavefunctionsoastoaccountfortheanisotropyofthepotential.Further,theasymptoticbehaviorofthepotentialiscapturedbytheexponentiallydecayingfactors.TheminimumexpectationvalueoftheenergyoccurswhenA=0.803,B=)]TJ /F3 11.955 Tf 9.29 0 Td[(0.774andC=2.14withagroundstateenergyof)]TJ /F3 11.955 Tf 9.29 0 Td[(0.1199whichwasfoundbyDorseyandToner[ 57 ].Thisvalueis2.5%lowerthanthepreviouslowestvariationalestimate()]TJ /F3 11.955 Tf 9.3 0 Td[(0.1196)obtainedbyDubrovskii[ 55 ]Inaddition,byusingthisnormalizedtrialwavefunctionasthe0(x,y)wendtheparameterg=Rdxdyj0(x,y)j4=0.017. 2.4.4RealSpaceDiagonalizationMethod FornumericalpurposestheSchrodingerequationisconvertedtoadifferenceequationonasquaregridofspacingh,withtheLaplacianapproximatedbyitsve-pointnitedifferenceform[ 59 ],resultinginablocktridiagonalmatrixofsizeN2N2,wherethegridhasdimensionsofNN.Eachdiagonalelementcorrespondstoagridpointandhasvaluesof4=h2+V(x,y),whereasthenonzerooffdiagonalelementsallequal)]TJ /F3 11.955 Tf 9.3 0 Td[(1=h2.Thematrixisthusverylargebutsparse.Weusethreedifferentnumericalmethodstodiagonalizethismatrix:thebiconjugategradientmethod[ 60 ],theJacobi-Davidsonalgorithm[ 61 ]andArnoldi-Lanczosalgorithm[ 62 ],withthe 40

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lattertwobeingmoresuitedtolargesparsematriceswhoseextremeeigenvaluesarerequired.Weusefreelyavailableopensourcepackages(JADAMILU[ 63 ]andARPACK[ 64 ])bothwritteninFORTRAN.AllthreeapproachesareprojectiveKrylovsubspacemethods,whichrelyonrepeatedmatrix-vectormultiplicationswhilesearchingforapproximationstotherequiredeigenvectorinasubspaceofincreasingdimensions.Reference[ 65 ]providesaconciseintroductiontotheJacobi-Davidsonmethod,togetherwithcomparisonstoothersimilarmethods.TheimplicitlyrestartedArnoldipackage(ARPACK)isdescribedingreatdetailinRef.[ 66 ].Somegeneralissuesabouttherealspacediagonalizationaswellassomespecicfeaturesofthethreemethodsusedforitarediscussedbelow. Theaccuracyoftherealspacediagonalizationmethodsiscontrolledbytwomainparameters:thegridspacinghandthetotalsizeofthegrid,whichisgivenbyNh.Thenitedifferenceapproximationtogetherwiththerapidvariationofthepotentialneartheoriginimplythatthesolutionofthepartialdifferentialequationwouldbemoreaccurateforasmallergridspacing.Weworkwithopenboundaryconditions,whichmeansthataboundstatewavefunctioncouldbecorrectlycapturedonlyifthetotalsizeofthegridweretobegreaterthanthenaturaldecaylengthofthewavefunction.Inotherwords,theeigenstatehastobegivenenoughspacetorelax.Thislimitsthenumberofboundstateswecancalculateaccuratelybecausealargegridsizetogetherwithsmallgridspacingscallsforalargenumberofgridpoints,thusquadraticallyincreasingthesizeofthematrixtobediagonalized.Computationalresourcesaswellasthelimitationsofthealgorithmsthemselvesplaceaneffectiveupperboundonthesizeofadiagonalizablematrix.Weexperimentedtondthata106106sizesparsematrixwasaboutthemaximumthatcouldbediagonalizedwithourcomputationalresources. Theoriginofthesquaregridissymmetricallyoffsetinbothxandydirectionstoavoidthe1=rsingularity.WersttestedtheaccuracyoftherealspacetechniquesforthecaseofthetwodimensionalCoulombpotential,thespectrumofwhichiscompletely 41

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known[ 96 ].WeobservethatforvariouslatticesizesthebiconjugatemethodcapturesatmosttherstfourstateswhereastheJacobi-Davidsonmethodreturns20eigenstates.Theeigenvaluesobtainedfrombothmethodsareaccuratetowithin2%oftheexactvalues[ 96 ]. Wehaveappliedthebiconjugatemethodtotheedgedislocationpotentialforvariouslatticesizes,varyingfrom1010to600600.Thenumberofeigenstatescapturedincreaseswiththesizeofthelattice,asexpected.Thegroundstateenergyisobservedtovaryfrom-0.134to-0.142.WealsoobservethatforthenumberofgridpointsexceedingN=2000weencounteranumericalinstabilityduetotheaccumulationofroundofferrors.Forthelargestrealspacegridsizeof600600(N=1200,h=0.5)weobtainseveneigenstateswithagroundstateenergyof-0.139. ThegroundstateenergyfromtheJacobi-Davidsonmethod,employedforthesamelatticesizegives-0.139,whichmatcheswellwithourexpectationsfromthevariationalcalculation.Weareabletoobtain20boundstateeigenvaluesinthismethodusingN=1000,h=0.5.Itischeckedthatthelow-lyingeigenvaluesarenotverysensitivetovaluesofhinthisregime,soarelativelylargevalueof0.5servesourpurpose.Aspointedoutearlier,theaccuracyofthismethodisdeterminedbythechoiceoflatticeparameters.Thevariationinthecalculatedgroundstateeigenvaluefordifferinglatticeparametersiswithin0.001,whichthereforeistheestimatederrorinthecalculationofeigenvalue. TheArnoldi-Lanczosmethodyieldsthesameeigenvaluestowithinestimatederror.Ittakesmoretimeandmemoryresourcestoconvergebutcancalculatemoreeigenvalues.Itprovides30boundstateeigenvaluesforthesamesetoflatticeparametersastheabove.Finally,aftercalculatingthegroundstatewavefunctionwendthatthecouplingconstantg=0.0194,slightlylargerthanthevariationalestimateofg=0.017. 42

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Figure2-2. Comparisonofeigenvaluesofthe2Dquantumdipoleproblem,obtainedbydifferentmethods.(Theplotisonalog-logscale.)ReprintedwithpermissionfromRef.[ 79 ][K.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys.Rev.B81,64516(2010)],cAPS,2010. Table2-2. Comparisonofrstfewenergyeigenvaluesobtainedfromdifferentmethods.Energyunits:2mp2=~2.nindicatesquantumnumberofthestate. nbiconjugateJacobi-DavidsonCoulomb/Arnoldi-Lanczosbasis 1-0.14-0.139-0.09702-0.041-0.0415-0.03283-0.023-0.0233-0.02214-0.02-0.0201-0.01675-0.012-0.0126-0.0119 2.4.5CoulombBasisMethod Wealsocalculatethespectrumnumericallybyusingthelinearvariationalmethodwiththebasisofthe2Dhydrogenatomwavefunctions[ 96 ].Therearetwoadvantagesofthismethodovertherealspacediagonalizationmethods.First,thelinearvariationalmethodiscapableofcapturingmoreexcitedstatesbecausethenumberofcalculatedboundstatesisnotlimitedbythesizeoftherealspacegridbutbythenumberoflong-rangebasisfunctions.Second,thesingularityattheoriginoftheedgedislocation 43

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potentialdoesnotposeaproblemanymorebecauseelementsoftheHamiltonianmatrixbecomeintegrable. NowwecalculatetheelementsoftheHamiltonianmatrixwitha2Dedgedislocationpotential.TheSchrodingerequationwiththe2DCoulombpotentialisanalyticallyworkedoutinRef.[ 96 ].Thenormalizedwavefunctionsofa2Dhydrogenatomaregivenby Hn,l(r,)=r 1 Rn,l(r)8>>>>>><>>>>>>:cos(l)for1ln,1 p 2forl=0,sin(l)for)]TJ /F5 11.955 Tf 9.3 0 Td[(nl)]TJ /F3 11.955 Tf 21.92 0 Td[(1,(2) where Rn,l(r)=n (2jlj)!s (n+jlj)]TJ /F3 11.955 Tf 17.94 0 Td[(1)! (2n)]TJ /F3 11.955 Tf 11.95 0 Td[(1)(n)-222(jlj)]TJ /F3 11.955 Tf 17.93 0 Td[(1)!(nr)jljexp)]TJ /F4 11.955 Tf 10.5 8.09 Td[(nr 21F1()]TJ /F5 11.955 Tf 9.29 0 Td[(n+jlj+1,2jlj+1,nr),(2) withn=2=(2n)]TJ /F3 11.955 Tf 12 0 Td[(1)and1F1beingtheconuenthypergeometricfunction.TheelementsoftheHamiltonianwiththe2Ddipolepotentialare Hn1,l1j)-222(r2j Hn2,l2=l1,l2Z10dr1)]TJ /F4 11.955 Tf 13.15 8.79 Td[(2n2 4rRn1,l1(r)Rn2,l2(r),(2) Hn1,l1cos r Hn2,l2=~VZ10drRn1,l1(r)Rn2,l2(r),(2) where~V=l1,l21=2ifbothl1andl2arelessorgreaterthan0,or~V=1=p 2ifl1is0andl2positiveorviceversa.ThespectraareobtainedforseveraltotalnumbersofbasisfunctionsNbasis.DuetothenumericalprecisionincalculatingelementsoftheHamiltonianmatrixNbasiscannotbeincreasedtomorethan400.ForNbasis=400weobtainabout150boundstatesandthegroundstateenergyof-0.0969.Thiscalculatedgroundstateenergyisnotreliableasitishigherthaneventheupperboundof-0.1199estimatedvariationallyearlier[ 57 ].Inordertoimprovethegroundstateenergy,we 44

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introduceanadditionaldecayingparameterinthebasisfunctions,andoptimizetheenergylevelsforacertainvalueofthisparameter.Withthedecayingparameterweobtainthebestvariationalestimateforthegroundstateenergyof-0.1257forNbasis=400. WeshowthersttwentyeigenvaluesobtainedfromdifferentmethodsinFig. 2-2 andtherstverepresentativeeigenvaluesinTable 2-2 .Asseenearlier,therealspacediagonalizationmethodsprovideabetterestimateofthegroundstateenergywhereastheCoulombbasismethodismoresuitableforhigherexcitedstates.TheeigenvaluesofboththeCoulombbasismethodandtherealspacediagonalizationmethodsarefoundtomatcheachotherforexcitedstates,andthentheybegintodeviateagain(seeFig. 2-2 ).Thiscanbeunderstoodbythefactthattheextentofwavefunctionsofthe2Dedgedislocationpotentialdoesnotalwaysincreaseasonegoestohigherexcitedstatesthewavefunctionsofsomeexcitedstatesextendlessthanthoseoflowerenergy.Therefore,thereareintermediateboundstatesthataremissedintherealspacecalculationbecausethesizeofgridusedincalculationisnotlargeenoughtocapturethem.Forexample,wendfourmoreboundstateswiththeCoulombbasiscalculationbetweenthe18thand19thexcitedstatesascalculatedfromtherealspacediagonalizationmethod.Thisfeaturealsoexplainstheabruptincreaseoftheeigenvalueofthe19thstatecalculatedbyusingtheArnoldi-Lanczosmethod(ARPACKroutine)inFig. 2-2 2.4.6SemiclassicalAnalysis Itisusuallyinsightfultoconsiderthesemiclassicalsolutionofaquantummechanicsproblem,sincethehigherenergyeigenstatestendtoapproachclassicalbehavior.AsemiclassicalestimateoftheenergyspectrumhasbeenprovidedinRef.[ 68 ].HerethetotalnumberofeigenstatesuptoavalueofenergyEisproportionaltothevolumeoccupiedbythesystemintheclassicalphasespace.ThisisexpressedbyWeyl's 45

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theorem[ 69 ]: n(E)=A 42mjEj ~2+O s ~2 2mp2jEj!,(2) whereAistheclassicallyaccessibleareainrealspaceandjEjtheabsolutevalueofenergyofthestate.Thehigherordercorrectionscanbeshowntobelessimportantforhigherexcitedstates,whichiswherethesemiclassicalpictureapplies.TondA,weneedtheclassicalturningpointsforthispotentialdeterminedbysettingE=V(r,).Thentheaccessibleareaistheinteriorofacirclegivenby(x)]TJ /F11 7.97 Tf 16.4 5.03 Td[(p 2E)2+y2=(p 2E)2,withareaA=(p=2E)2.Therefore,weobtain(writingthenondimensionalizedenergyinoursystemofunitsas): n()=)]TJ /F3 11.955 Tf 15.99 8.09 Td[(1 16,(2) wherenisthequantumnumberoftheeigenstateandthecorrespondingenergy.Notethatthedensityofstatesdn=dscalesas1=2. Figure2-3. Fitfortheeigenvaluespectrumof2DquantumdipolepotentialobtainedfromJADAMILUusingf(x)=a(x)]TJ /F5 11.955 Tf 11.95 0 Td[(b)c.Fitvaluesare)]TJ /F3 11.955 Tf 9.3 0 Td[(0.06,0.61and0.96fora,bandcrespectively.ReprintedwithpermissionfromRef.[ 79 ][K.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys.Rev.B81,64516(2010).],cAPS,2010 46

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Tocheckthisresultwetthenumericalspectrumwiththefollowingfunctionalform: (n)=a(n)]TJ /F5 11.955 Tf 11.95 0 Td[(b)c,(2) withthettingparametershavingvaluesa=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.06,b=0.5,c=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.98,eachcorrecttowithin5%.(Sincewearedealingwithboundstateshere,alltheenergyeigenvaluesarenegative,andthehigherexcitedstateshavelowerabsoluteeigenvalues.)WeshowthettothespectrumobtainedfromJADAMILUroutineinFig. 2-3 .Thesemiclassicallyderiveddependenceisfoundtocloselymatchwiththetfornumericallycalculatedenergyeigenstates,exceptfortheb=0.5factor.Inthelimitoflargenvaluesi.e.higherexcitedstates,thetrelationtendstothesemiclassicalresultasexpected. Theclassicaltrajectoriesforthispotentialbearthesignatureofchaoticdynamicsshowingspace-llingnatureandstrongdependenceoninitialconditions.However,forreasonsnotyetcleartous,theyarenotergodic,llinguponlyawedge-shapedregioninrealspaceinsteadofthefullclassicallyallowedcircle.Thequantummechanicalprobabilitydensityascalculatedfromtheeigenfunctionsalsoexhibitssuchwedge-shapedregions.Somesamplewavefunctionsobtainedfromournumericalcalculationshavebeenpresented.Figure. 2-4 showsthelowestveeigenstatesandFig. 2-5 showssomehigherexcitedstates.Asexpected,thewavefunctionsareconnedtothelefthalfplane,wherethepotentialisnegativeandboundstatesarepossible.Theparityofthepotentialshowsupinthewavefunctionsbeingeithersymmetricorantisymmetricaboutthex-axis,althoughstatesofsuchoddandevenparitydonotalwaysalternate.Forexample,the2ndand4thexcitedstatesareodd,andthe3rdexcitedstatecorrespondinglyeven,butthegroundstateand1stexcitedstatesarebotheven.Similarlythe10th,50thand100thexcitedstatesarealloddwhilethe23rdand24thareodd.Thespatialextentofthewavefunctionsisseentobegenerallyhigherforhigherexcitedstates,butthisisnotalwaysthecase.Theextentdoesnotscalemonotonicallywithquantumnumber.Somecasesarefoundwhereahigherexcitedstatehaslessspatial 47

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extentthanalowerone.ForexampleweseeinFig. 2-5 ,thatthe24thexcitedstateislessextendedinthex-directioncomparedtothe23rd.Wedonothaveanysatisfactoryexplanationyetfortheseirregularfeatures. AWavefunctionforthegroundstate BWavefunctionforthe1stexcitedstate CWavefunctionforthe2ndexcitedstate DWavefunctionforthe3rdexcitedstate EWavefunctionforthe4thexcitedstate Figure2-4. Eigenfunctionsoftherstvestatesofthe2Dquantumdipoleproblem.Forclarityofpresentation,therangeofthexandyaxeshavebeenincreasedforthehigherexcitedstates.ReprintedwithpermissionfromRef.[ 79 ][K.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys.Rev.B81,64516(2010)],cAPS,2010 48

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AWavefunctionforthe10thexcitedstate BWavefunctionforthe23rdexcitedstate CWavefunctionforthe24thexcitedstate DWavefunctionforthe50thexcitedstate EWavefunctionforthe100thexcitedstate Figure2-5. Eigenfunctionsofvehigherexcitedstatesofthe2Dquantumdipoleproblem.Forclarityofpresentation,therangeofthexandyaxeshavebeenincreasedforthehigherexcitedstates.ReprintedwithpermissionfromRef.[ 79 ][K.Dasbiswas,D.Goswami,C.-D.YooandA.T.Dorsey,Phys.Rev.B81,64516(2010)],cAPS,2010. 49

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CHAPTER3SUPERFLUIDITYINTHEPRESENCEOFMULTIPLEDISLOCATIONS InChapter2,weconsideredthesuperuidityinducedbyasingle,quenched,dislocationlinebycouplingtheelasticstrainsinducedbyittothesuperuidorderparameter.Howeverarealcrystalpreparedinthelaboratoryislikelytocontaingrainboundaries(whichcanbethoughtofasregulararraysofdislocations),oratanglednetworkofdislocationlines.Inthischapter,wefollowasimilarapproachtoderivetheLandautheoreticmodelforsuperuidityindislocationarraysandnetworksbyconsideringtheoverlapbetweensuperuidregionsinducedbyeachdislocationlineandndtheireffectonsuperuidcriticaltemperatureTc.Thissuperuidorderingingrainboundariesornetworksarequasilongrange(in2D)andlongrange(in3D)respectively,andthushavethethermodynamiccharacteristicsofaphasetransition,unlikethelocal1Dsuperuidorderinginducedbyasingledislocation.Thenetworkmodel,weargue,couldalsoberesponsibleforsomeoftheanomalouseffectsseenintorsionoscillatorexperimentsinsolid4He.Furtherquestionsthatcouldbeaddressedaretheeffectsoftheelasticinteractionsbetweendislocationlinesonthesuperuidity,andalsooftheirmotion. 3.1GrainBoundarySuperuidity Agrainboundaryisaninterfacebetweentworegionsofdifferentcrystalorientationinapolycrystal[ 50 70 71 ],wheretheatomsarelocallydisorderedbecauseofthecompetingeffectsofneighboringcrystallineregions.Theoreticalstudies,bothQuantumMonteCarlomethods[ 72 ]andLandautheoreticmodels[ 73 ],indicatethatsuperuiditycanbelocallyinducedinadefectstructuresuchasagrainboundary.Also,thereissomeexperimentalevidenceformasssuperowthroughgrainboundariesinsolid4He[ 31 ].Alowanglegrainboundarycanbeinterpretedasasurfacecontainingasequenceofdislocations[ 50 74 ].Inparticular,alowangletiltgrainboundaryisaperiodicarrayofedgedislocations[seeFig. 3-1 ],andwehaveshowninChapter2thateachsuch 50

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dislocationiscapableoflocallyinducingsuperuidityarounditself.Theselocalregionsofsuperuiditycanoverlapforming2Dsuperuidorderinthegrainboundary.Thissuperuiditywouldbeanisotropic,withdifferingpropertiesalongthedislocationlinesandtransversetothem.Thesuperuidcouplingbetweenneighboringdislocationlinesandconsequently,thecriticaltemperature(Tc)woulddependsensitivelyonthespacingbetweenthem.TheMermin-Wagnertheorem[ 75 ]forbidsthepresenceoflong-range Figure3-1. Asymmetrictiltgrainboundaryshowingthecrystalplanes.Eachhalfplaneontheleftcanbeconsideredtobeanextrahalfplaneinsertedbetweentwohalfplanesontheright,thusgivingrisetoanarrayofstraightdislocationlineseachofwhichpointalongthey-axis.Thespacingbetweendislocationslinesisdandthegrainboundaryangleis. orderina2Dsystembecauseofthelargeroleplayedbyuctuations,butquasi-orderingcanresultfromtheBerezinskii-Kosterlitz-Thouless(BKT)mechanism[ 17 18 ].InaBKTsuperuid,vorticesandantivorticesareboundintoorderedpairs.Thisordercanbelostaboveacertaincriticaltemperaturewhenthesevortex-antivortexpairsdissociate.IntherestofthissectionwediscussthisanisotropicgrainboundarysuperuidityandaddresssomeofitsBKTproperties. 51

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Consideratiltgrainboundarywithangleoftiltandspacingdbetweentheneighboringdislocationsthatmakeupthisgrainboundary[seeFig. 3-1 ].Letthedislocationlinesbeorientedalongtheydirectionandaseriesofthemplaceddistancedapart(inunitsoflatticespacing)alongthex-axis.Thegrainboundaryinterfaceisthenjustthex)]TJ /F5 11.955 Tf 12.74 0 Td[(yplane.TheBurger'svectorofeachdislocationisperpendiculartothisplane,bkx.Itfollowsfromthegeometryofthesystemthatthesequantitiesarerelatedas d=b 2sin(=2),(3) wherebistheBurger'svectorofeachdislocationline.ThisiscalledtheFrankformulainmetallurgyliterature[ 50 ].Inthelimitofsmallangles,thisbecomes d'b .(3) TheLandautheoryforgrainboundarysuperuidityinanisotropicsolidproceedsinthesamewayasforasingledislocationintroducedinChapter2, F=Zd3xc 2jr j2+1 2a(x)j j2+1 4uj j4,(3) wherethesuperuiditycouplestothecrystallinitythroughthetraceofthegrainboundarystraintensor,a(x)=a0(T)]TJ /F5 11.955 Tf 12.97 0 Td[(T0)=T0+a1uii(x,z).Asinthecaseofthesingledislocation,thegrainboundarystrainscanlocallyenhancethesuperuidcriticaltemperature(Tc),thechangeinTcbeingrelatedtothegroundstateeigenvalueofthecorrespondinglinearproblem. 52

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Thestressfromagrainboundarycanbecalculatedbyconsideringitasasumofthestressesgeneratedbyeachindividualdislocationintheperiodicarray1: xz(x,z)=bBx1Xn=z2)]TJ /F3 11.955 Tf 11.95 0 Td[((x)]TJ /F5 11.955 Tf 11.95 0 Td[(nd)2 [z2+(x)]TJ /F5 11.955 Tf 11.96 0 Td[(nd)2]2.(3) Thismaybewrittenintheconvenientform, xz(x,z)=)]TJ /F5 11.955 Tf 9.3 0 Td[(bB dJ(,)+@J(,) @,(3) whereJ(,)=P1n=1 2+()]TJ /F11 7.97 Tf 6.59 0 Td[(n)2,=z=dand=x=darethenondimensionaldistancesinthezandx-directionsrespectively.UsingthePoissonsummationformula, 1Xn=f(n)=1Xk=Z1dxf(x)e2ikx, wend J(,)=Z1d 2+2+2Re1Xk=1e2ikZ1de2ik 2+2= +1Xk=1e)]TJ /F6 7.97 Tf 6.59 0 Td[(2kcos(2k). (3) Thestressfarawayfromtheplaneofthegrainboundaryisobtainedbysetting>>1intheaboveexpression.Inthatcase,onlythersttermofthesumwouldmatter,andtheresultantfar-eldstressisofthesimpliedform xz(x,z)=42Bbz d2e)]TJ /F6 7.97 Tf 6.59 0 Td[(2z=dcos(2x=d).(3) Thisstressisperiodicinthex-directionasexpected.Thecrystalismorestressedforahigheranglegrainboundary,wherethedislocationsareplacedclosertogether.Farawayfromtheplaneofthegrainboundary,thestressdecaysexponentiallywith 1ThesubsequentderivationisprovidedinChapter4ofthetextTheoryofElasticitybyL.D.LandauandE.M.Lifshitz[ 74 ]. 53

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distancealongthez-direction.Thisimpliesthatalthoughthestresseldsfromindividualdislocationsarelongrange(decayasapowerlaw),therearecancellationsfromthedifferentdislocationsoftheperiodicarray,thusleadingtoashortrangeformofthestress(orstrain)eld.TheotherstresscomponentscanbefoundsimilarlybysummingovertherespectivedislocationstresscomponentsusingthepropertyofPoissonsummation.Particularly,thetraceofthefar-eldstresstensor(whichcouplestothesuperuidityinourLandautheoreticframework)is ii=)]TJ /F3 11.955 Tf 9.29 0 Td[(2Bb de)]TJ /F6 7.97 Tf 6.58 0 Td[(2z=dsin(2x=d).(3) ThelinearizedLandauequationcannowbewritteninanondimensionalizedformas )-221(r2 +[Vgb(x,z))]TJ /F5 11.955 Tf 11.96 0 Td[(E] =0,(3) whereEisproportionaltothereducedtemperature,andVgbisanondimensionalpotentialcorrespondingtothetraceofthegrainboundarystraineld[proportionaltothehydrostaticstressfoundinEq.( 3 )].WecannowseethattheseparationAnsatzusedforthedislocationsuperuidityproblemcannotbesimplycarriedovertothegrainboundarysituation.Thefastdegreeoffreedomoftheorderparameteristhez-direction,transversetothegrainboundary.Ifthezdirectionisseparatedoutfromtheorderparameteramplitude,weareleftwithanorderparameterdependentonxandy.TheeigenfunctionofthelinearproblemEq.( 3 )ishoweverafunctionofxandzandcannotbeusedtointegrateoutthefastdegreeaswasdonefordislocations.Thegrainboundarystraineldtakesasimpleformfarawayfromtheboundary,butweareinterestedintheorderparameterintheplaneofthegrainboundary.ThelinearequationEq.( 3 )isnotanalyticallytractableinthatplanebecauseofthecomplicatedformofthepotential.Wethereforeanalyzethisproblemapproximatelyintwodifferentlimits,wherethecoherencelengthismuchbigger(smaller)thanthedislocationseparation:d(d).Intheformercase,weexpectthesuperuidregionsfromneighboring 54

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dislocationstooverlapstrongly.Theorderparameterwillthenhavealargeuniformbackgroundpiecewithsmallamplitudecorrugationontopofit.Inthelattercasetheoverlapwillbeexponentiallysmall.WegivesomeheuristicargumentstoobtainthedependenceoftheTconthegrainboundaryangleineachcase. WerstanalyzethecaseoflongcoherencelengthsfollowingtheworkofGurevichandPashitskii[ 73 ].Thestraineld,orderparameterandallphysicalquantitiesareperiodicinthex-directionbecauseofthewaytheconstituentdislocationsareplaced.Inthislimitofstrongoverlapbetweenneighboringdislocations,theorderparameterhasalargeconstantpiecealongthegrainboundaryandrapidbutsmalluctuationsaboutthisaverage.Wecanthusdecomposetheorderparameterintoslowandfastpartsas (x,z)= 0(z)+ 1(x,z),(3) where 0(x)=h (x,z)ix.Herehixindicatesaveragingoveraperioddinthex-direction(thedirectionofperiodicity),i.e.hA(x,z)ix=1 dRd0dxA(x,z).WeseparateEq.( 3 )usingastandardtechniquedevelopedforrapidlyoscillatingsystems[ 76 ].FirstweconsideronlythequantitiesuptoleadingorderonbothsidesofEq.( 3 ).Inthelimitweareconsidering,thegradientsof 1(x,z)aremuchbiggerthanthatof 0(z),althoughtheamplitudeof 1(x,z)ismuchsmallerthanthatof 0(z).Therefore, r2 1=Vgb 0.(3) Nowweaverageovertherapidlyoscillatingdegreeoffreedom,i.e.thex)]TJ /F1 11.955 Tf 12.62 0 Td[(coordinatetoget )-222(r2 0)]TJ /F5 11.955 Tf 11.96 0 Td[(E 0+hVgb 1ix=0.(3) InordertoobtainthisresultwehaveusedhVgbix=h 1ix=0,whichisduetothepotentialandcorrugationontheorderparameterbeingbothsinusoidallyoscillatingfunctionsinthex)]TJ /F1 11.955 Tf 12.62 0 Td[(direction.Thecorrugationtermcanbeexpressedexplicitlyinterms 55

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ofthepotentialbyFouriertransformingEq.( 3 ),whichyields 1(k,G)=Vgb(k,G) 0 k2+G2,(3) wherethe(inverse)Fouriertransformofthepotentialisdenedas Vgb(x,y)=XG6=0Z1dk 2Vgb(k,G)eikx+iGy.(3) Thefunctionisperiodicinthex-directionandsoisrepresentedasaFourierseriesoverthereciprocallatticevectors,G=2n=d,andn=..,)]TJ /F3 11.955 Tf 9.3 0 Td[(2,)]TJ /F3 11.955 Tf 9.3 0 Td[(1,1,2,...Forasymmetricgrainboundary,thetermwithG=0isabsentbecauseeachdislocationcausesadipolestraineldwhichhaszeroaveragealongthex-direction.Wehaveshownpreviouslythatatdistancesfarawayfromthegrainboundarycomparedtothedislocationspacingd,thestressesdecayexponentially.Theorderparameter 1(x,z)isalsoexpectedtodecreaserapidlywithdistancetransversetothegrainboundary.Inthelimitofsmallspacingdweareconsidering,thefunction~V(z)=hVgb 1ixdescribesashortrangepotentialwell,~V(z)=V0(z).IntegratingoverzandusingtheFouriertransformedexpressionforVgb(x,z),weobtain V0=XG6=0Z1dk 2jVgb(k,G)j2 k2+G2.(3) ThelinearizedLandauequationforthelarge,slowpieceoftheorderparameter,Eq.( 3 ),thenreducestoa1DSchrodingerequationforaparticleinadeltapotentialwell: )]TJ /F3 11.955 Tf 11.96 0 Td[([r2+V0(z)] 0=E 0.(3) Thegroundstateeigenvalueofthisproblem,E0=V20=4,isrelatedtotheshiftintheTcofthesuperuidityinthegrainboundary.WecanderivethedependenceofthisshiftinTconthegrainboundaryangle(ordislocationspacing)withasimplescalinganalysisatthispoint.Thisistheessentialpointofthissection.Thegrainboundarypotentialisthetraceofthestraintensorandvarieswithdislocationspacingasb=d.Oncalculating 56

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itsFouriertransformandusingtheexpressioninEq.( 3 ),wend(afterscalingoutthedependenceond)thatV0d.TheshiftinTcgoesasthesquareofV0(asknownfromthegroundstateeigenvalue)andthereforeasaninversesquarepowerlawinthegrainboundaryangle.Fromdimensionalconsiderations,wehavederivedtherelativeincreaseinTcas Tc Tc=e 2,(3) whereeissomecharacteristicangledependingontheelasticpropertiesofthecrystalunderconsideration. Forthelimitwheredislocationspacingislargerthansuperuidcoherencelength,wehaveasituationofweakoverlap,andtheTcforgrainboundarysuperuidityisproportionaltotheJosephsonsuperuidcouplingbetweenneighboringdislocations,andisexponentiallysensitiveonthespacing,Tce)]TJ /F11 7.97 Tf 6.59 0 Td[(d=be)]TJ /F6 7.97 Tf 6.59 0 Td[(1=.Thusweuncovertheseparatepowerlawandexponentialdependenceofthecriticaltemperatureonthegrainboundaryangleindifferentlimits.Wepresumethelatterislikelierinthecaseofsolidhelium,becausethecoherencelengthsareknowntobeaboutassmallastheatomicspacing(afewAngstroms)insuperuidhelium.TheformersituationanalyzedbyGurevichandPashitskii[ 73 ]couldbefoundforexampleinsuperconductorswithgrainboundaries. 3.2Berezinskii-Kosterlitz-ThoulessSuperuidinaGrainBoundary Asmentionedearlier,thesuperuidityina2Dsystemlikeagrainboundarycanarisethroughthevortex-bindingBKTmechanism.InthisSection,wediscusssomeoftheBKTpropertiesofthegrainboundaryinducedsuperuiddescribedabove,especiallyfocusingonhowthesepropertiesareaffectedbytheinherentanisotropyofthegrain 57

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boundary.TounderstandtheBKTargument,werstneedtounderstandthebasicpropertiesofavortexandcalculateitsenergy2. Vorticesareubiquitoustopologicaldefectsfoundin2DsystemswithcontinuouslybrokenU(1)symmetry,thatarebestdescribedassingularitiesinthephaseoftheorderparameter.Itisoftenusefultothinkofthemasanalogoustopointchargesinelectrodynamicsproblems.Consider,forexample,anXYspinmodelin2D,whichbelongstothesameU(1)universalityclassasasuperuid.AsimpleAnsatzforthephaseina2Dsystemwithasinglevortexlocatedattheoriginofcoordinatesis v=ev,(3) whereevisanintegercalledthewindingnumberorchargeofthevortex,andrandarethepolarcoordinatesin2D.Thelongwavelengthorelasticenergyofthevortexisthengivenby Eel=1 2sZd2x)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rv2=1 2s(2)(ev)2Zrdr r2=(ev)2sln(R=a),(3) whereaisthecoreradiusofthevortex.andRthesystemsize.InadditionthereisanenergycostEcassociatedwiththecoreofthevortex.Thesuperuidphasestiffnesssisusuallyidentiedwiththesquareoftheamplitudeoftheorderparameter,butcanberenormalizedbyuctuationsofthisamplitude,asintroducedforexamplebyvortices. AsimpleheuristicargumentrstgivenbyKosterlitzandThouless[ 18 ]showshowvorticescanleadtoasecond-orderphasetransitionintheXYmodel.TheenergyofasinglychargedvortexinsampleoflineardimensionR,ascalculatedinEq.( 3 ),issln(R=a).Thevortexcorecouldlieanywhereinthesample,andsocarriesanentropyproportionaltothearea,ln(R=a)2.ThefreeenergyofanXY-systemwithasinglevortex 2MostofthedevelopmentinthissectioncloselyfollowsSection9.4inthetextPrinci-plesofCondensedMatterPhysicsbyP.M.ChaikinandT.C.Lubensky[ 70 ] 58

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isthen F=E)]TJ /F5 11.955 Tf 11.96 0 Td[(TS=(s)]TJ /F3 11.955 Tf 11.96 0 Td[(2T)ln(R=a).(3) Thusitisreasonabletoidentify Tc= 2s,(3) withthecriticaltemperatureabovewhichitbecomesenergeticallyfavorabletohaveafreevortex.Amoredetailedanalysisshowsthatthereexistsanalgebraically(quasilong-ranged)orderedphasebelowsuchaTcinwhichthevorticesareboundintopairswiththeirantivortices.ThespinwavestiffnessoftheXYmodel,s,ismodiedbythepresenceofthevortices,andthischangestheTcfromthebareformderivedhere. Vorticesaremobiledegreesoffreedomthatarrangethemselvessoastominimizethefreeenergyofanyimposedgradientofthephaseoftheorderparameter.Themacroscopicspinwavestiffness,Rs,isrenormalizedbyvortexuctuations.Itisexpressedasthedifferenceinfreeenergybetweenthesystemwithandwithoutanexternallyimposedowvelocity: F(v))]TJ /F5 11.955 Tf 11.96 0 Td[(F(0)=1 2VRsv2.(3) Herevs=r(x)istheuniformgradientofthephaseoftheXYmodel,tobeidentiedwiththesuperuidvelocityafterincorporatingasuitablydimensionfulfactorofh=m.Visthetotalvolumeofthesystem.ThefreeenergyisfoundbystatisticallyaveragingtheHamiltonianoverphaseuctuations(inthefollowingtreatmentwetaketheBoltzmannfactorkB=1), F(v)=)]TJ /F5 11.955 Tf 9.3 0 Td[(TlnTrexp)]TJ /F5 11.955 Tf 11.95 0 Td[(H(v)=T,(3) wheretheHamiltonianforlowenergyphaseuctuationsisgivenby H(v)=1 2sZd2x)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(r2.(3) 59

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Thephase(v)intheaboverelationscanbedecomposedintoasingular,transversepartfromthevortex,ananalytic,longitudinalandspin-wavepart,andaconstantpiececorrespondingtotheuniformgradientorvelocity: r=v?s+vjjs+v.(3) Afterexpandingintheuniformgradienttoquadraticorderandaveragingovertheuctuations,weobtainanexpressionfortherenormalizedstiffnessintermsofthebarestiffnessandacorrelationfunctionoftransversevelocities(associatedwiththevortices), Rs=s)]TJ /F4 11.955 Tf 13.15 8.09 Td[(2s TZd2xhv?s(x).v?s(0)i.(3) Usingstandardrelationsforvortexvelocitiesandsourcefunctions,andtheenergyofasystemofdilutevorticesatlowtemperature,itcanbeshownthattherenormalizedstiffness(seeappendixforderivation)isrelatedtothebarestiffnessandvortexfugacityf=e)]TJ /F11 7.97 Tf 6.58 0 Td[(Ec=Tas K)]TJ /F6 7.97 Tf 6.59 0 Td[(1R=K)]TJ /F6 7.97 Tf 6.58 0 Td[(1+43f2Z1adr ar a(3)]TJ /F6 7.97 Tf 6.59 0 Td[(2K).(3) Thenotationinthisequationhasbeenmadesimplerbydeningareducedspinstiffness,K==T.Bysimplepowercounting,theaboveintegralconvergeswhen(3)]TJ /F3 11.955 Tf 12.24 0 Td[(2K)<)]TJ /F3 11.955 Tf 9.3 0 Td[(1,i.e.attemperatureslowerthanTc= 2s(Tc).Whenthetemperatureishigherthanthisself-consistentlydeterminedTc,theintegraldivergesforlarger,andtheperturbationtheoryinfugacitybreaksdown.Thedifcultyassociatedwiththedivergencecanbehandledthrougharenormalizationprocedure[ 77 ].Insteadofgoingintothedetailsofthisrealspacerenormalizationprocedure,wementionsomeofthekeyresultsofthisrenormalizedXYmodelin2D.BelowtheTcwhichisthexedpointoftheRGow,thefugacityisfoundtobeirrelevant,whichqualitativelyimpliesthattherearenofreevortices.Anyvorticespresentareboundintopairs,andsotheirpresenceisnotfeltasonecoarsensthescale.AboveTC,thefugacityowstowardshighervaluesimplyingtheexistenceoffreevortices.Theseconclusionsareconsistentwith 60

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theexpressionfortheenergyofasystemofdilute,interactingvortices:theenergyofafreevortexactuallydivergeswithsystemsize.Soatlowtemperatures,itisenergeticallyfavorabletohaveonlyboundvortexpairswithtotalcharge,Pev=0. Modicationsforananisotropictheory.Wehaveremarkedbeforethatthe2Dgrainboundarysuperuidisexpectedtobeanisotropic,asthephasestiffnessalongthedislocationlines,they-direction,isexpectedtobestrongerthanthecouplingtransversetothem(thex-direction).ThesituationcanbedescribedbyananisotropicXYmodelwithdifferentphasestiffnessvaluesalongxandy.TheHamiltonianforthelowenergyphaseuctuationsofsuchamodelis H=1 2x(@x2+1 2y(@y2.(3) Theratioofthephasestiffnessesshouldvarywiththegrainboundaryangleas,xye)]TJ /F14 5.978 Tf 7.99 3.26 Td[(1 .Followingthesameanalysisasintheisotropiccase,therenormalizedstiffnessesarefoundtohaveformssimilartoEq.( 3 ). Rx=x)]TJ /F4 11.955 Tf 13.15 8.09 Td[(2x TZd2xhv?x(x).v?x(0)i,Ry=y)]TJ /F4 11.955 Tf 13.16 8.95 Td[(2y TZd2xhv?y(x).v?y(0)i. (3) Usinganexpressionforthetransversevelocityintermsofthevortexsourcefunctionm(x), v?(x)=rZd2xG(x)]TJ /F17 11.955 Tf 11.96 0 Td[(x0)m(x),(3) theexpressionforrenormalizedstiffnessasgiveninEq.( 3 )yield, Rx=x)]TJ /F3 11.955 Tf 11.95 0 Td[((2)22x 2TZd2xy2hn(x)n(0)i,Ry=y)]TJ /F3 11.955 Tf 11.95 0 Td[((2)22y 2TZd2xx2hn(x)n(0)i. (3) Theanisotropyisdealtwithbyrescalingthelengthscalestorendertheproblemisotropic.Itisusefultodenethefollowingquantities:aneffectivestiffnessep xy 61

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andananisotropyfactor=p x=y.TherescaledradialpolarcoordinateisthenR=p x2+2y2.Thecorrelationfunctionforvortexdensitiesforasmallfugacity(lowtemperature)canbederivedbystatisticallyaveragingovertheHamiltonianforadilutegasofinteractingvortices,inthesamewayasEq.( 3 ),giving KRx=Kx)]TJ /F3 11.955 Tf 11.96 0 Td[(43f2K2e 2TZdR aR a3)]TJ /F6 7.97 Tf 6.58 0 Td[(2Ke,KRy=Ky)]TJ /F3 11.955 Tf 11.96 0 Td[(43f2K2e 2TZdR aR a3)]TJ /F6 7.97 Tf 6.59 0 Td[(2Ke. (3) Kisasusualaquantitydenedtodenotethestiffnessdividedbytemperature,tomakethenotationintheRGproceduresimpler.Thex)]TJ /F1 11.955 Tf 12.62 0 Td[(andy)]TJ /F1 11.955 Tf 12.62 0 Td[(equationsinEq.( 3 )canbecombinedasasingleequationfortheeffectivestiffnessKebymultiplyingandexpandinguptoquadraticorderinthefugacity. (KRe))]TJ /F6 7.97 Tf 6.59 0 Td[(1=(Ke))]TJ /F6 7.97 Tf 6.59 0 Td[(1+43f2K2e 2TZ1adR aR a3)]TJ /F6 7.97 Tf 6.59 0 Td[(2Ke+O(f4).(3) ThisisessentiallythesamerenormalizedstiffnessaswasobtainedinEq.( 3 )fortheisotropiccase.SimilarlyanequationfortherenormalizedanisotropycanbeobtainedfromEq.( 3 )as R=q KRx(KRy))]TJ /F6 7.97 Tf 6.58 0 Td[(1=+O(f4).(3) Thisshowsthattheanisotropyisirrelevantandtheeffectivestiffnessdenedasthegeometricmeanofthetwostiffnessesinthex-andy-directionsisrenormalizedinexactlythesamewayasintheisotropicsuperuid.Sowedonotexpectanyqualitativedifferencesinthecoarse-grainedbehavioroftheanisotropicBKTsuperuidinagrainboundaryfromanisotropic2Dsuperuid. 62

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3.3NetworkofDislocationLinesandaModelSupersolid InChapter2,weconsideredasingleedgedislocation,reducingthefullthree-dimensionalLandautheorytoaneffectiveone-dimensionaltheoryforasuperuidtubelocalizednearthedislocationcore.Then,intheprecedingsection,weconsideredagrainboundarywhichcanbethoughtofasaregulararrayofdislocations.However,realcrystalsincludingsolid4He,willconsistofatangleofdislocationsmanyofwhichwillcrosswhentheycomewithinatransversecorrelationlengthofeachother.Conceptually,wecanmodelthisasarandomlattice(ornetwork)ofdislocationswiththecrossingpointsservingaslatticesitesasillustratedinFig. 3-2 .Whilethermaluctuationsdestroyanylongrangeorderinasingle,one-dimensionaltube,thelatticeoftubeswillgenerallyorderatatemperaturecharacteristicofthephasestiffnessbetweenadjacentlatticesitesofthisrandomnetworkofintersectingdislocationlines.ThisisthemotivationbehindthemodelsdevelopedbyShevchenko[ 80 ]andToner[ 78 ].InthisSectionwerevisittheShevchenkoandTonermodelswithasystematicapproachforcalculatingthecouplingbetweenadjacentlatticesitesinthenetworkmodel,andobtainnewresultsonthelengthscalingofthecouplingconstant.Weconcludewithsomeobservationsregardingimplicationsofourresultsforexperimentsontheputativesupersolidphaseof4He. Westartbyconsideringthecorrelationsbetweentwopointsalongasinglesuperuidtube,separatedbyadistanceL.Usingthenotationoftheprevioussection,'(0)='1and'(L)='2arethevaluesofthesuperuidorderparameterattwositesalongthetube.ThecorrelationsarecapturedbythepropagatorK('2,'1;L),whichcan Partsofthissectionarereproducedfromthepublishedarticle:D.Goswami,K.Dasbiswas,C.-D.YooandA.T.Dorsey,Phys.Rev.B84,054523(2011). 63

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Figure3-2. Atanglednetworkofdislocationlinesoneislikelytondinarealcrystal.Inourpicture,bulksuperuidorderdevelopsalongthisdefectstructureembeddedin3D.TheaveragespacingbetweenthenodesorlatticepointsinthisnetworkisL,andtheXYmodellikeexchangecouplingbetweenthesuperuidorderbetweentheithandjthnodesisJij. beobtainedfromthefunctionalintegral K('2,'1;L)=exp()]TJ /F4 11.955 Tf 9.3 0 Td[(He)=Z'(L)='2'(0)='1D(',')exp()]TJ /F4 11.955 Tf 9.3 0 Td[(ZL0dzc 2j@z'j2+a 2j'j2+b 4j'j4),(3)wherea,bandcaretheparametersoftheone-dimensionalLandautheoryderivedintheprevioussection,andHeistheeffectiveHamiltonianthatcharacterizesthecouplingbetweenthelatticesites.Castinthisform,weseethatthefunctionalintegralfortheclassicalone-dimensionalsystemisequivalenttotheFeynmanpathintegralforaquantumparticleinatwo-dimensionalquarticpotential(two-dimensionalbecausetheorderparameter'iscomplex),withzintheclassicalproblemreplacedbytheimaginarytimeforthequantumsystem[ 86 ].Indeed,previousauthorshaveusedthisanalogytostudytheeffectofthermaluctuationsontheresistivetransitioninone-dimensionalsuperconductors[ 87 88 ],obtainingessentiallyexactresultsforthecorrelationlengthandthermodynamicproperties.ConsistentwiththeMermin-Wagnertheorem[ 75 ],theseauthorsndnosingularitiesinthethermodynamicproperties,andaone-dimensionalcorrelationlengththatgrowsasthetemperatureisreduced,butneverdiverges[ 88 ].MostoftheseresultsareconvenientlycapturedusingasimpleHartreeapproximation[ 89 ],inwhichthequartictermisabsorbedintothequadratictermwiththequadratic 64

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coefcientredenedas a=a+1 2bhj'j2i,(3) wherea=a0(T)]TJ /F5 11.955 Tf 12.96 0 Td[(Tcond)=Tcondandhj'j2iastatisticalaveragewithrespecttotheeffectivequadratictheory.Carryingouttheaveraging,weobtain[ 89 ] hj'j2i=kBT c2?,(3) where=(c=a)1=2isthecorrelationlengthfortheone-dimensionalsystemalongthesuperuidtube,and?isthecross-sectionaldimensionofthe1Dsuperuidregion(thetransversecorrelationlength,asshowninFig. 2-1 inChapter2).InsertingthisresultintotheHartreeexpressionofEq.( 3 ),weobtainacubicequationforthecorrelationlengthas 1 2=1 20+kBTb 2c22?,(3) where0=(c=a)1=2istheGaussiancorrelationlength.Whilethemean-eldcorrelationlength0divergesatTcond,theHartreecorrelationlengthremainsniteatalltemperatures[ 88 ][seeFig. 3-3 ],reectingthelackoflong-rangeorderintheone-dimensionalsuperuidtube. ContinuingwiththeHartreeapproximation,wecanndtheexplicitformofthepropagatorbyexploitingananalogywiththepartitionfunctionforatwo-dimensionalquantumharmonicoscillator;theresultis[ 86 ] K('2,'1;L)=k(L)exp()]TJ /F4 11.955 Tf 37.14 8.08 Td[(c 2sinh(L=)(j'2j2+j'1j2)cosh(L=))]TJ /F3 11.955 Tf 11.96 0 Td[(2j'2jj'1jcos(1)]TJ /F4 11.955 Tf 11.96 0 Td[(2)), (3) wheretheprefactork(L)=c=2sinh(L=),and'1=j'1jei1and'2=j'2jei2.TheeffectiveHamiltonian(uptoanadditiveconstant)isgivenby He=c 2coth(L=)(j'2j2+j'1j2))]TJ /F5 11.955 Tf 11.95 0 Td[(J12(L)cos(1)]TJ /F4 11.955 Tf 11.96 0 Td[(2),(3) 65

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Figure3-3. DependenceofHartreeandmeaneldcorrelationlengthsontemperature:thedoesnotdivergeatTcunlikeMF.ThelatterisobtaineddirectlyfromtheLandautheorywhiletheformertakesGaussianuctuationsintoaccount. where J12(L)=cj'2jj'1j sinh(L=)=8><>:cj'2jj'1j=L,L=1;(2cj'2jj'1j=)e)]TJ /F11 7.97 Tf 6.58 0 Td[(L=,L=1. (3) ThelastterminEq.( 4 )istheoneofinterest,asitcouplesthephasesattheneighboringsitesthroughaneffectiveferromagneticcouplingconstantJ12.ThebehaviorofJ12asafunctionoftheinter-siteseparationLisoneofourimportantresultsforsmallL=,J121=L,reproducingtheresultofToner[ 78 ],whereasforlargeL=,J12e)]TJ /F11 7.97 Tf 6.58 0 Td[(L=.SinceisniteforallT,forasufcientlydilutenetworkofdislocationswewillalwayssatisfythelatterconditioni.e.,dilutenetworksofdislocationspossesscouplingconstantsexponentiallysmallinthedislocationdensity.Itmightbereemphasizedherethatsincetruelongrangeorderisnotpossibleinone-dimension, 66

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thephaseoftheorderparameterisalsonotwelldenedinone-dimension.Hence,aphaseonlyapproximationoftheproblemasshownbyToner[ 78 ]wouldbeinadequateincapturingthecompletebehaviorofthecouplingconstantandacompleteorderparameterdescriptionasshownhere,inthespiritoftheworkbyScalapinoetal.[ 88 ]isnecessary.Toner'sphase-onlydescriptionwouldbevalidwhenthecorrelationlengthislargerthanthedislocationnetworklengthscaleLasthentherewouldbephaseorderingbetweenneighboringlatticesitesofthedislocationnetwork,anditisinsuchalimit(L)thatwerecoveradependencesimilartoToner's[seeEq.( 3 )]. SofarwehavesystematicallyderivedaneffectiveHamiltonianthatdescribesthephasecouplingbetweentwopoints(latticesites)onasingledislocation.Tomaketheconceptualleaptothedislocationnetwork,weproposethattheappropriatemodelforthenetworkisarandombondXYmodeloftheform H=)]TJ /F16 11.955 Tf 11.29 11.35 Td[(XhijiJijcos(i)]TJ /F4 11.955 Tf 11.95 0 Td[(j),(3) wherehijirepresentsnearest-neighborlatticesitesandJijisa(positive)couplingbetweenthesitesthatscalesase)]TJ /F11 7.97 Tf 6.59 0 Td[(Lij=forasufcientlydilutenetworkofdislocations.AsnotedbyToner[ 78 ],therandomnessisirrelevantintherenormalization-groupsense[ 90 ],andinthreedimensionsweexpectthesuperuidityinthenetworktoorderwhenthetemperatureisoforderthetypicalcouplingstrength[Jij];i.e.,kBTc=O([Jij]).Again,foradilutenetworkofdislocations(witharealdislocationdensityn1=L2),wewouldndTce)]TJ /F6 7.97 Tf 6.58 0 Td[(1=(n1=2),aremarkablysensitivedependenceonthedislocationdensity. Itisnaturaltoaskherehowwellthesetheoreticalndingsrelatewiththerecenttorsionaloscillatorexperimentsonsolidhelium.Therecentconsensusfromexperimentalstudies,suchasbyRittnerandReppy[ 25 ],isthattheputativesupersolidresponsedependsonsamplequalityandpreparation.Accordingtotheseexperiments,theNCRIfractionshowsmuchmoresensitivedependenceonthesamplequality(varyingfromasmuchas20%forsamplespreparedbytheblockedcapillarymethodandthus 67

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havingmoredisorderto0.5%forthosepreparedunderconstantpressure[ 26 ])thantheonsettemperatureitself,whichshowsnocleardependenceondisorder.ThisseemstocontradicttheexponentialdependenceofTcondislocationdensitythatwepredictbutwewouldliketopointoutthatsofartherehavebeennosystematicstudiescharacterizingsamplesofsolid4Hefortheirdislocationdensity,andrelatingthistotheonsettemperatureofsupersolidsignalfoundinthesesamples.Ultrasoundattenuationexperimentsonsolid4Heinthepast[ 91 92 ]haverevealeddislocationdensitiesrangingfrom104-106cm)]TJ /F6 7.97 Tf 6.58 0 Td[(2.Thismeansthedensitycannotbedeterminedtoevenwithinanorderofmagnitude. WhatmakesrelatingourtheorytoexperimentevenmoredifcultisthefactthatwecannotmakequantitativepredictionsfortheTcfromourtheory.ThatwouldrequireknowledgeoftheLandautheoryparametersforsolid4Heinordertobeabletocalculatethecorrelationlength(inEq.( 3 )).Thesearenotknownforsolid4He,unlikeforexample,inconventionalsuperconductors,wheretheLandauparameterscanbeobtainedthroughexperimentbyrelatingtothemicroscopictheory.Thislackofcertainknowledgeofatleasttwovariablesinourtheory(Land)makeanycomparisonwithexperimentinconclusive.Itisalsoentirelypossible,thatwhiledislocationsuperuidityisarealeffect,theremightbeotherfactorsatworkbehindthehighNCRIfractionobservedindisorderedsampleswhichtendtoswampoutthesensitivedependenceofTcthatourmodelfordislocationsuperuiditypredicts.Alsoourtheoryatpresentdoesn'tincludedynamics(seeAppendixCforarudimentarydiscussionofdynamics)andcannotcommentonexperimentsthatstudytherateofsuperow,suchasbyRayandHallock[ 93 ].Howeverwebelievethatthemodelproposedaboveservesasausefulstartingpointtonumericallysimulateandunderstandsomeoftheseeffects. 68

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CHAPTER4VORTICESINTRAPPED,DILUTEBOSE-EINSTEINCONDENSATE Thegenerationofquantizedvorticesingaseoussampleshasbeenthesubjectofmuchstudy[ 39 ]sincetherstobservationofBose-Einsteincondensationinatomicgases.ThedegreeofcontrolandtheeasewithwhichvorticescanbeobservedinaBECinatrapmakeitanexcitingalternativeneutralsuperuidsystemtoliquidhelium,wherevortexcreationanddynamicscanbestudied.Inthischapter,wereviewsomeoftheknownresultsrelatedtothestabilityandmotionofasinglevortexinaBECinatrap,andsketchaderivationofthedynamicsofavortexinarotatingframeofreferencefromahydrodynamicsperspective.Theseresultsareostensiblydifferentfromthatofavortexinliquidhelium,becauseoftheeffectofatime-dependentpotential(thatrotatesthetrap),andtheinhomogeneitiesoftheconningpotential. 4.1Gross-PitaevskiiFormulation Thezerotemperaturepropertiesofadilute,interacting,BosegasofultracoldatomsinatrapisdescribedbytheGross-Pitaevskii(GP)equation[ 102 ].HerewesketchabriefderivationofthisGPformalism,followedbyadiscussionofsomeofitsbasicmathematicalfeatures.Acollectionofinteracting(bosonic)atomsinatrapischaracterizedbyitsmanybodywavefunction.WeemployaHartreeormeaneldapproachwherethismanybodywavefunctionisreplacedbyitsstatisticallyaveragedmeaneldwavefunction,whichcanbewrittenasasuitablysymmetrizedproductofsingle-particlewavefunctions.Afurthersimplicationisachievedinthefullycondensedstatewhenallatomsoccupythesamesingle-particlestate,(r).ThereforethewavefunctionoftheN-particlesystemcanbewrittenas (r1,r2,....rN)=NYi=1(ri),(4) wherethesingle-particlewavefunctionisnormalizedasRdrj(r)j2=1.Thiswavefunctiondoesnottakeintoaccountcorrelationsbetweensingle-particleparticle 69

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wavefunctionsthatareproducedbytheirinteractions.Forthedilutegasesofneutralalkaliatomswetypicallyconsider,theinteractingforcesareshortrangeandcanbetakenintoaccountbyaneffectiveinteractiong(r)]TJ /F17 11.955 Tf 12.37 0 Td[(r0)whichincludestheinuenceofshort-wavelengthdegreesoffreedomthathavebeenintegratedout.Infactforshortrangescattering,gisrelatedtothescatteringlengthasas[ 103 ] g=4~2as m.(4) TheeffectiveHamiltonianmaybewrittenas H=Xi=1Np2i 2m+Vtr(ri)+gXi
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TheGPenergyfunctionalofthediluteBECcanthenbewrittenintermsofthecondensatewavefunctionas E[ ]=Zdr~2 2mjr j2+Vtr(r)j j2+1 2gj j4.(4) Theexternaltrappingpotentialwilltypicallybeassumedtobethatofa3Dsphericalharmonicoscillator, V(r)=1 2m!2r2.(4) TonondimensionalizeEq.( 4 ),wechoosetheharmonicoscillatorlengthscaleobtainedfromthelinearSchrodingerequationpartoftheproblemastheappropriatelengthscaleintheproblemlahoq ~ m!,andwavefunctionamplitudescale,q ~! g.Usingtheabovetorescaleonlythelength,thefreeenergyfunctionalisnondimensionalizedas E=E0=l3Zd3xjr j2+r2j j2+j j4,(4) wherethescaleoffreeenergyisE01 2~!.Thisisthefundamentalenergyscaleinthisproblem,andisrelatedtothegroundstateenergyE0oftheharmonicoscillator.IntheunitsofE0,thegroundstateenergyis3.ThedimensionlessparameterisameasureofthestrengthofthenonlinearityintheGPfunctionalanditsformcanbederivedfromEq.( 4 )as g ~!=Nas aho.(4) Thisformisparticularlyusefulasittellsuswhetherweareinaregimeofstrongnonlinearity(largecondensatewavefunctionanddensity)orweaknonlinearity(smallcondensatewavefunctionanddensity)intermsofexperimentallyknownparameters:theparticlenumber,thescatteringlengthoftheinter-atomicinteractionandtheharmonictraplengthscale(relatedtoitsfrequency).WeapplyaccordinglyaThomas-Fermioraperturbativeapproximation.Boththesemethodsarediscussedinsubsequentsections.Forcompletenessofthediscussion,wenowpresentthefullynondimensionalizedGP 71

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functionalafterscalingoutthewavefunctionamplitudeaswellasthelength, E=E0=l32Zd3xjr j2+r2j j2+j j4,(4) Togettheequationofmotionforthecondensatewevarytheabovefreeenergywithrespectto undertheconstraintthattotalnumberofatomsinthetrapisxed.Thisgives E =0=r2 +[r2)]TJ /F4 11.955 Tf 11.96 0 Td[(] +j j2 ,(4) whereisthenondimensionalizedchemicalpotential(inunitsofE0)whichservesastheLagrangemultiplierforthisconstraint. 4.1.1WeaklynonlinearAnalysisforSmallCondensates TheGPEq.( 4 )ismathematicallysimilartotheLandautheoryforsecondorderphasetransitionswithchemicalpotentialplayingtheroleoftemperature.Whenthechemicalpotentialisclosetothegroundstateenergy,thereareveryfewatomsinthetrap,andweareinalowdensitylimit.ThecubictermintheGPequationisthensmallcomparedtothelineartermsandcanbetreatedperturbativelyfollowingtheprescriptionoftheweaklynonlinearanalysisintroducedinChapter2.Toseethisclearly,weintroduceacontrolparameter )]TJ /F4 11.955 Tf 11.96 0 Td[(0,(4) whereisthechemicalpotentialand0thegroundstateeigenvalueofthelinearpartoftheproblem,whichisbasicallytheSchrodingerequationforthetrappotential,inthiscasea3Disotropicharmonicoscillator.Nowrescalethecondensatewavefunctionas, =1=2.(4) TheGPequationcanthenberecastinaformconvenientforaperturbativeanalysis, ^L=)-222(jj2,(4) 72

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wheretheHermitianlinearoperator^Lisgivenby ^L=r2+r2)]TJ /F5 11.955 Tf 11.95 0 Td[(E0.(4) Next,weexpandinpowersof, =0+1+22+....(4) Collectingterms,weobtainthefollowinghierarchyofequations: O(1):^L0=0,O():^L1=0)-222(j0j20, (4) O(2):^L2=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(320)1. ThesolutionoftheO(1)equationisthenormalizedgroundstateeigenfunction,0(r)=1=3 4e)]TJ /F11 7.97 Tf 6.59 0 Td[(r2=2;thereisanoverallintegrationconstantA0,sothat 0=A00(r).(4) SubstitutethisintotherighthandsideoftheO()equation, ^L1=0A0)]TJ /F3 11.955 Tf 11.95 0 Td[(30jA0j2A0.(4) WecandetermineA0byleftmultiplyingthisequationby0,integratingond3r,andusingthefactthat^LisHermitian,tond A0)]TJ /F4 11.955 Tf 11.96 0 Td[(jA0j2A0=0,(4) wheretheconstantisgivenby =Zd3r40(r),(4) theintegralofthenormalizedgroundstatewavefunctionraisedtothefourthpower.Calculatingfortheharmonicoscillatorwavefunction,weget=1=(2)3 2.From 73

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Eq.( 4 ),A0=1=p ,andtherefore0=23=4e)]TJ /F11 7.97 Tf 6.59 0 Td[(r2=2.Theoverallwavefunctionistherefore (r)=1=223=4e)]TJ /F11 7.97 Tf 6.59 0 Td[(r2=2+O(3=2).(4) Thenexthigherordertermrequiresthesolutionofaninhomogeneousdifferentialequation.Thisresultisexpectedtoholdinthelowdensitylimitwhenisclosetothegroundstatevalueof3(inourunits).Inthislimit,thetotalnumberofparticlesuptothezeroethorder,isobtainedbyndingthenormalizationofthecondensatewavefunctioninEq.( 4 )andputtingthedimensionfullengthandamplitudescalesbackin, N=(2)3=2()]TJ /F3 11.955 Tf 11.96 0 Td[(3)2l3=r 2aho as.(4) Thusweidentifyanimportantexperimentalparameter=Nas ahoidenticaluptoamultiplicativeconstantwiththedenedearlierastheshiftinchemicalpotential,thathastobesmall(1)fortheproblemtobeweaknonlinear,andtheaboveperturbativeanalysistohold. 4.1.2ThomasFermiAnalysis Intheoppositelimitofhighdensity(quantitativelyexpressedasNas=aho1),thenonlineartermintheGPequationbecomessignicantandawayfromtheedgesofthetrapitisthegradienttermthatcanbeneglected.Thecondensatewavefunctioninthislimitis (r)= (0)p )]TJ /F5 11.955 Tf 11.96 0 Td[(r2,(4) whereristheradialdistancefromthecenterofthetrap.ThecondensatedensitythusgoestozeroataradialdistanceofRintheTFdescription,anditisthevicinityofrRthatthisapproximationbreaksdown.Wenowlistseveralquantitiesassociatedwiththehighdensitycondensateandtheirexpressionsintermsofotherquantities,intheTFlimit. 74

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chemicalpotential, =152=5 22 5~!; condensatesizeortrapradius, R2=2 m!2; densityatthecenterofthetrap, n0==g=152=5 22 5 asa2ho; totalnumberofparticles: N=8=155=2; healinglengthorsizeofthevortexcore. =a2ho R. 4.2VortexEnergeticsandStability Weconsideranaxiallysymmetrictrap,andcalculatetheenergycostofcreatingavortexinitscenterfromatime-independentmeaneldtheory.Theresultsobtainedinthissectionholdonlyforastationaryvortex.Thedynamicshavetobeincorporatedthroughatime-dependenttheory,whichwediscusslaterLettheaxisofthetrapbedirectedalongthez-axis,andweworkinacylindricalsystemofcoordinateswiththeradialdistanceintheplane,andtheazimuthalangle.Inthesubsequentpresentation,wewillrstaddressthesimplestcaseofauniformcondensate,andthenaddresstheissueofinhomogeneitiesintroducedbyatrap.Wediscussrstthecaseofthe2Dcondensate,andthengeneralizeourconclusionstothe3Dcase.A2Dsituationcanberealizedinaatorpancakecondensatewherethetrappingismuchstrongerinthez-direction,thanintheplanetransversetoit(!z!x=!y=!).Theaxialextentofsuchatrapisgoingtobemuchsmallerthanthetransverseextent.Wethen 75

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Figure4-1. Condensatewavefunction (r)versusr.correspondstoproximityofthechemicalpotentialtothegroundstateeigenvalueofthetrap.Thesolidbluelineisthenumericalsolution,thedottedredlineisthenaiveone-termperturbativeresult,andthedashedgreenlineistheThomasFermiapproximatedresult.Thepanelontheleftisinthelowdensitylimit(lowervalueof)wherethenumericresultisclosetotheperturbativeone.Thepanelontherightisforahighdensitylimit(highvalueof)andthenumericresultmatchestheTFresultclosely,asexpected.Theoscillationinthenumericresultisanartifactandcanberemovedbyprogressivelyreningthevalueoftheshootingparametertogreaternumberofsignicantgures. makenecessarymodicationsinourcalculationtodescribea3Dorsphericaltrap,wherethetrappingfrequenciesandcondensatesizearethesameinalldirections(!x=!y=!z=!).Whereverotherwisementioned,weassumethatthecondensatehasalargedensityofatoms,andtheThomasFermi(TF)limitapplies.Theappearanceofavortexchangesthedensityonlylocallyarounditselfwithoutaffectingthebulkcondensateprole,andthehealinglengthismuchsmallerthanthecondensatesize,R. Avortexwithwindingnumberllocatedinthecenterofthetrapcanbedescribedbythecondensatewavefunction (r)=f(,z)eil,(4) wherethephaseofthewavefunctionissimplyanintegralmultipleoftheazimuthalangle,=l.Itiseasytocheckthatthisphasesatisesthevortexquantization 76

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condition,Hr.dl=2l,andminimizestheelasticenergy,r2=0.Thesuperuidvelocityinducedbythevortexvs=(~=m)r=l~ m(1=)hastherequisitequantizedcirculation,andfallsoffslowlywithdistance.RecallfromEq.( 4 )thattheGrossPitaevskii(GP)energyofacondensate(strictlyvalidatzerotemperaturewhentherearenothermalexcitationsthatdepletethecondensate)isgivenasafunctionalofthecondensatewavefunction.UsingourAnsatzofEq.( 4 )forawavefunctionwithavortex,intheGPenergyfunctionalofEq.( 4 ),wend E=Zdr(~2 2m@f @2+@f @z)2+l2f2 2+Vtr(,z)f2+1 2gf4).(4) Theonlydifferencebetweenthisresultandtheoneforacondensatewithoutavortex,andnospatialdependenceofthecondensatephase,isthe1=2termwhichcomesfromthekineticenergydensityoftheazimuthalsuperowinducedbythevortex.TheenergyrequiredtocreateavortexatthecenterofthetrapisthedifferenceinGPenergycalculatedforacondensateofthesamedensity,withandwithoutthevortex,andsocorrespondstothisazimuthalkineticenergyterm.Simplepowercountingshowsthattheenergyislogarithmicallydivergentintheradialcoordinate.Inacondensateofnitesize,asonewouldexpectinaconnedtrap,thelargedistancecutoffisnaturallygivenbythecondensatesizeorradiusR,whereasanaturallowerlengthscaleisthehealinglength.Theleadingordertermofthevortexenergyisthusexpectedtobev'(~2l2n0=2m)ln(R=0)wheren0issomecharacteristiccondensatedensity1. 4.2.1EnergyofaVortexattheCenteroftheTrap WenowpresentaderivationofthisvortexenergyandcalculatehigherordercorrectionstoitusingtheGPenergyfunctionalpresentedinEq.( 4 ).Foralarge 1Thesizeofal-foldvortexisactuallyjlj(canbeveriedbycomparingtermsinGPfunctional)butthisfactoroflwillonlybealogarithmiccorrectionandhenceisdroppedintherestofthisanalysis. 77

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uniformcondensatewithavortex(inaTFlimit),thecondensatedensityhealstoitsbulkvalueofn0=N=Vwithinthecoherencelength,andtheradialintegraliseasilyperformed(thecondensatedensitycanbeassumedtobezerowithinthevortexcoreradius)togettheenergyperunitlengthofavortex v=ZR02d l2~2n m'n0~2l2 2mlnR 0.(4) ThemoreaccurateresultisfoundbyusingthenumericalsolutionoftheGPequationinthevortexenergyas[ 42 ] v=n0~2l2 mln1.464R 0.(4) Nowconsidertheaddedcomplicationofaharmonictrap.Thecondensatedensityinsteadofbeinguniformoutsidethevortexcore,nowfollowsaparabolicprole,n()=n0)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F11 7.97 Tf 14.88 4.71 Td[(r2 R21 2.Thedensityvariesslowlyonthelengthscaleofthecondensatesize.Soforlengthscalesmuchsmallerthanthatbutlargerthanthevortexsize,01R,thesystemisessentiallylikeauniformcondensatewithavortexanditsenergycanbefoundinthesamewayasshownaboveinEq.( 4 )orEq.( 4 ).Atdistanceslargerthanthis,thedensityprolehastheparabolicTFdependence.Puttingthetwocontributionstogether,wecalculatetheenergyofavortexperunitlengthina2Dtrapas, v=n0~2l2 mln1 0+ZR2d 21)]TJ /F4 11.955 Tf 14.63 8.09 Td[(2 R2=n0~2l2 mlnR 0)]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 2+O21 R2. (4) Thelogarithmictermisacontributionofthenear-uniformslowlyvaryingcondensateprole,butthereisaslightreductionfromthatbecauseofthesharplyreduceddensityneartheedgeofthecondensate. Ifthecondensatesizeisalsolargeinthez-direction(0Rz),aswouldbethecaseforaspherical3DtrapintheTFlimit,thetotalenergyofavortexiscalculatedby 78

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summingovereachhorizontalsliceofthecondensateandignoringgradientsinthez-direction.TheradiusofthecloudateachsuchslicedependsontheverticalpositionasR(z)=R)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F11 7.97 Tf 14.06 4.71 Td[(z2 R21 2,andthedensityisafunctionofboththeradialandaxialdistance.Thecoherencelengthdependsontheinversesquareofthecondensatedensityandontheaxis,goesas(z)=)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(n(0,0)=n(0,z)1 2.Usingtheexpressionforkineticenergycontributionin2DfoundinEq.( 4 ),wecalculatethetotalkineticenergyofa3Dcondensatewithavortexas E=~2l2 mZRz)]TJ /F11 7.97 Tf 6.58 0 Td[(Rzdzn(0,z)lnR(z) (z))]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2=~2l2n(0,0) mZRz)]TJ /F11 7.97 Tf 6.59 0 Td[(Rzdz1)]TJ /F5 11.955 Tf 14.36 8.09 Td[(z2 R2lnR 01)]TJ /F5 11.955 Tf 14.36 8.09 Td[(z2 R2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2=4~2l2n(0,0) 3mRzln0.458R 0. (4) 4.2.2EnergyofanOff-centerVortex Wenowdeterminethedependenceoftheenergyofavortexonitspositionwithinthecondensate.Sincethecondensatedensityissuppressedinaregionoftheorderofthehealinglengtharoundthevortex,itisexpectedthattheenergycostofcreatingavortexwouldbemorenearthecenterofthetrapwherethecondensatedensityishigher.Foranaxiallysymmetrictrap,theenergywilldependonlyontheradialdistanceofthevortexfromthecenter,v=p x2v+y2vwherethevortexcoordinatesarelabeled(xv,yv).Wewillgenerallyrefertothevortexposition,xv(v,v,0),reservinguseofvonlywhenwewanttoexplicitlyrefertotheradialdistanceofthevortexfromthecenterofthetrap/axisofrotation.Thedominantlogarithmiccontributiontotheenergyofanoff-centervortexiscalculatedusingthesameideasasforavortexatthecenter:Eq.( 4 )toEq.( 4 ).Theuniformcondensatedensityatthevortexpositionintheabsenceofavortexdeterminesthisenergycontribution.Thusfora2Dcondensate,the 79

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energyperunitlengthofthevortexisgivenby v'n(rv)~2l2 2mlnR 0=n0~2l2 2mlnR 01)]TJ /F17 11.955 Tf 15.24 8.09 Td[(r2v R2,(4) wherewehaveusedtheTFdensityprole.FollowingthesamestepsforavortexinthecenterofatrapthatleadfromEq.( 4 )toEq.( 4 ),whichisbasicallyintegratingalongtheaxialdirection,wendtheenergyofavortexina3Dtrap: E=~2l2 mZRz)]TJ /F11 7.97 Tf 6.59 0 Td[(Rzdzn(v,z)lnR(z) (v,z)'E01)]TJ /F17 11.955 Tf 15.23 8.09 Td[(r2v R23=2,(4) whereE0=4~2l2n(0,0) 3mRzlnR=0istheenergytoleadingorderofavortexatthecenterofthetrap,asfoundinEq.( 4 ).Intheaboverelations,theoff-centervortexenergyissmallerthanthecentralvortexbyafactorof)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F6 7.97 Tf 15.43 5.7 Td[(r2v R21=2,whichisfromthelowerdensityofthecondensateatthevortexposition.Anextrafactorof)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F6 7.97 Tf 15.45 5.7 Td[(r2v R21=2arisesinEq.( 4 )forthe3Dtrapbecausethez-heightofthecondensateissmalleratthevortexpositionthanatthecenterofthetrap. 4.2.3VortexStabilizationinaRotatingTrap ThestationaryGPdevelopmentsofarsuggeststhatthevortexenergyisalwayshighestatthecenterofthetrapanddecreasesmonotonicallywithoutwardradialdistance.Soifthereisanydissipativemechanismforexample,avortexcreatedatthecenterwouldtendtoloseenergyandmovetowardstheperipheryofthetrap.Wewillnowshowthatthistendencycanbecompensatedtosomeextentbytheeffectofrotationofthetrap,andthatforsomerangeofangularfrequencies,thevortexatthecentercanactuallybeinanenergeticallystabilizedormetastablestate. Theadditionalcomplicationwhenthetrapisbeingrotatedisthattheexternalpotentialfeltbythecondensateatomsbecomestime-dependentinthelaboratoryframeofreference.Thisdifcultycanbecircumventedbytransformingtoacomovingframethatrotateswiththetrap.Inthiscasethetrappotentialremainstime-independent,andthestaticdescriptionwehaveusedsofarstillholds.Quantitiesintherotating(primed) 80

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andlaboratory(unprimed)frameofreferencearerelatedthroughawell-knowntransporttheoremfrommechanics[ 76 ], dA0 dt=dA dt)]TJ /F17 11.955 Tf 11.96 0 Td[(A,(4) whereistheangularvelocitywithwhichthetrapisrotatingandAisanyvector.Theresultofinteresttous(whichcanbederivedfromthisbasicrelation)isthatenergiesintherotatingandlaboratoryframesofreferencearerelatedby E0=E)]TJ /F17 11.955 Tf 11.95 0 Td[(L,(4) whereListheangularmomentumofthecondensatecalculatedinthelaboratoryframe. Calculationofcondensateangularmomentum.Forthesakeofsimplicityweconsiderasituationinwhichthetrapisbeingrotatedaboutitsaxisofsymmetry,i.e.thez)]TJ /F1 11.955 Tf 12.62 0 Td[(direction.Eq.( 4 )suggeststhatweneedtocalculatetheexpectationvalueofthez)]TJ /F1 11.955 Tf 9.3 0 Td[(componentoftheangularmomentumabouttheaxisofthetrap,producedbythesuperowinducedbyavortexlocatedataradialdistancevfromtheaxisofthetrap.Thisisgivenbytheexpression D^LzE=Zdri~ ^z)]TJ /F17 11.955 Tf 5.48 -9.68 Td[(rr ,(4) wherethecondensatewavefunctionforanoff-centervortexcongurationisgivenbyamodicationoftheAnsatzinEq.( 4 ), (r)=f(,z)eil0, (4) 0=arctany)]TJ /F5 11.955 Tf 11.96 0 Td[(yv x)]TJ /F5 11.955 Tf 11.95 0 Td[(xv. (4) Thephaseisnowmeasuredinacoordinateframecenteredatthevortexposition.Thedominantcontributiontotheangularmomentum(likeforthekineticenergy)isgoingtocomefromtheazimuthalsuperuidvelocityinducedbythepresenceofthevortexaccordingtothedenitionofsuperuidvelocity,v=i~r.Anotherwaytoseethisis 81

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thatthegradientofthedensityisignoredintheTFlimit.Anitegradientofthephaseisimperativebecauseavortexisanexcitationinthephaseeldoftheorderparameter.ThisisplausiblebecausetheenergycostassociatedwithanamplitudeuctuationismorethaninphaseasseenfromtheGross-Pitaevskiifunctional.ThisdiscussionsuggeststhattheexpressionoftheangularmomentuminEq.( 4 )canbewrittenintheform, Lz'Zddmnv,(4) fora2Dcondensate,whichisalsoclassicallyintuitive.Here,vistheazimuthalcomponentofthesuperuidvelocity,andmisthemassofeachparticleinthecondensate.Thedensitynintheaboveexpressionisactuallyn(rv),i.e.thedensityevaluatedatthepositionofthevortexandthecorrectioncomingfromtheinhomogeneitiesindensityduetothetrappotentialatlargerlengthscalesisfoundtobesmall,aswasthecaseinthederivationofvortexenergyintheTFlimitEq.[ 4 toEq.( 4 )].TheintegralinEq.( 4 )canbeevaluatedsimplybyrealizingthat Zdv=Iv.dl=I(rv).da=2l~ mZda(2)()]TJ /F4 11.955 Tf 11.95 0 Td[(v)=8>><>>:2l~mif>v,0if
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Thisimpliesthatforavortexontheaxiseachatomcarriesaquantizedangularmomentumofl~.Thisisexpectedbecausetheatomsarecondensedtothesamegroundstatewavefunction,andthetotalenergyorangularmomentumisanextensivequantitythatisamultipleofthequantitycarriedbyeachatom.Inthispictureeachatomsrotatesaboutthecentralaxis,wherasanoff-axisvorteximpliesthatrotationisaboutanaxisdifferentfromthecentralaxisaboutwhichangularmomentumwascomputed.Wecannowworkouttheangularmomentumofa3Dcondensatebyintegratingalongtheaxial(z)direction, Lz'2~ZRz)]TJ /F11 7.97 Tf 6.59 0 Td[(Rzdzn(v,z)ZRvd=Nl~1)]TJ /F17 11.955 Tf 15.24 8.08 Td[(r2v R25 2.(4) Theangularmomentuminthe3Dtraphasanextrafactorof(1)]TJ /F17 11.955 Tf 12.21 0 Td[(r2v=R2)1=2overthe2Dcaseaswasseenintheexpressionsforenergy. Vortexenergyintherotatingframe.Nowthatwehavecalculatedbothenergyandangularmomentumforageneralvortexstateofthecondensateinthelaboratoryframeofreference,weareinapositiontocalculatetheenergycostofcreatingavortexintherotatingframethatiscomovingwiththetrap.UsingEq.( 4 ),wendtheenergyperunitlengthofavortexintherotatingframefora2Dcondensate 0v=01)]TJ /F17 11.955 Tf 15.24 8.09 Td[(r2v R2)]TJ /F3 11.955 Tf 19.07 8.09 Td[( 2Dc1)]TJ /F17 11.955 Tf 15.24 8.09 Td[(r2v R22,(4) wheretheprefactor0=n0~2l2 2mln)]TJ /F11 7.97 Tf 7.29 -4.97 Td[(R 0istheenergyperunitlengthofavortexontheaxisofa2DcondensatecalculatedinEq.( 4 ).2Dcisthecriticalangularvelocityatwhichthetrapneedstoberotatedabovewhichthevortexatthecenterofthetrapisstabilized.Thisisdeterminedbytheratioofenergyandangularmomentumofthecondensateforavortexatthecenterinthelaboratoryframe: 2Dc=~l2 2mR2lnR jlj0.(4) 83

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Thecorrespondingenergyofavortexintherotatingframefora3Dcondensateis Ev=E01)]TJ /F17 11.955 Tf 15.24 8.09 Td[(r2v R23=2)]TJ /F3 11.955 Tf 15.72 8.09 Td[( c1)]TJ /F17 11.955 Tf 15.24 8.09 Td[(r2v R25=2,(4) wheretheprefactorE0istheenergyofavortexontheaxisofa3DcondensatecalculatedinEq.( 4 ),andcisthecriticalvelocity(sameuptosomenumericalfactorin2Dand3Dcases),whichisdenedsimilarlyasinEq.( 4 )tobe c=5 2~l2 mR2lnR jlj0.(4) Theenergyprolesforavortexintherotatingframe,whenplottedagainstitsradialdistancefromthetrapcenterforvariousregimesofrotation,hasthesamebehaviorasshowninFig. 5-1 intheChapter5.Thisshowsthatthevortexatthecenterofthetrapcanbeinstable,metastableorunstablecongurationsdependingonhowfastthetrapisbeingrotated.Themetastablestateisofparticularimportancetousasthissuggeststhatthereissomenite,possiblymeasurable,timeatwhichavortexatthecentercantunnelthroughthispotentialbarriertowardstheperipheryofthetrap. 4.2.4VortexinaSmallCondensate SofarwehaveconsideredtheenergeticsofavortexinacondensateoflargedensityorparticlenumberwheretheTFanalysisisapplicable.Therethepresenceofavortexdoesnotalterthedensityofthecondensateexceptaroundasmalldistance,whichcorrespondstothehealinglength,thatismuchsmallerthanthecondensatesizeRinthislimit.Nowweconsidertheoppositelimit,wherethenonlineartermcanbetreatedperturbativelybytheweaklynonlinearmethodintroducedinSection4.1.Herethedensityissmalltobeginwithandthepresenceofthevortexradicallyaltersthecondensatedensity,alloverthetrap.Thehealinglengthisnowofthesameorderasthecondensatesize,whichmeansthatthevortexoccupiesalmostthewholeofthetrap.Thelonglengthscalehydrodynamicsvalidfarawayfromthevortexcorethusfailinthislimit,makingthisahardercasetodealwith. 84

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ThenondimensionalizedGPenergyfunctionalforavortexatthecenterofatrapis, E=Zd3x(~2 2m@f @2+@f @z2+l2f2 2+V(,z)f2+1 2gf4),(4) wherethe1=2termisthecontributionoftherotationtothekineticenergy(i.e.thecentrifugalbarrier).Thelengthandamplitudescalesl=ahoandarethesameasdenedearlierinEq.( 4 ). Forillustrativepurposes,weconsiderthecommonest(lowestenergy)situationofavortexwithasinglequantumofcirculation,i.e.l=1.Inthegroundstatethewavefunctionshouldnothaveanyz-dependence.Thetrappingpotentialisisotropicandharmonic.ThenondimensionalizedGPequationis, )]TJ /F3 11.955 Tf 13.34 8.09 Td[(1 xd dxxdf dx+f x2+(x2)]TJ /F4 11.955 Tf 11.95 0 Td[()f+f3=0,(4) wherex==listhenondimensionalizedradialcoordinateinacylindricalcoordinatesystem.ThelinearizedversionofthisproblemisthereforeaSchrodingerequationseparableintoanaxialpartalongthez)]TJ /F1 11.955 Tf 9.3 0 Td[(axiswhichcorrespondstothegroundstateofa1Dharmonicoscillator,andaradialpartwhichcorrespondstotherstexcitedstateofa2Dharmonicoscillatorproblem(l=0ingroundstate,l=1istherstexcitedstate).Theeigenvalueofthelinearprobleminourunits(1unitofenergy1 2~!)is1+4=5. Followingtheweaklynonlinearanalysisoftheprevioussection,theO(1)equationisthenormalizedproductofa1Dharmonicoscillator'sgroundstateeigenfunctionanda2Dharmonicoscillator's1stexcitedstateeigenfunction.ThereisanoverallintegrationconstantA0,whichisrecoveredfromthesolvabilityconditionoftheO()equation,asA0=2p .Theoverallamplitudeofthewavefunctionistherefore, f(x)=25=4p e)]TJ /F10 7.97 Tf 6.59 0 Td[(2=2)]TJ /F11 7.97 Tf 6.59 0 Td[(z2=2+O(3=2),(4) 85

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Figure4-2. Condensatewavefunction (r)versusrforthel=1vortexstateinasmallcondensate.Thesolidbluelineisthenumericalsolution,andthedashedgreenlineistheone-termperturbativeresult. wherethecoordinatesandarepolarcoordinatesnondimensionalizedbythelengthscaleaho.Theresultisexpectedtoholdwheniscloseto5andisthesmallparameterintheproblemdenedas)]TJ /F3 11.955 Tf 11.96 0 Td[(5. TheenergyneededtocreateavortexisthedifferenceoftheGPenergyofacondensatewithandwithoutavortex.ThiswasreadilydoneintheTFlimit,becausethekineticenergyfromtheazimuthalsuperowwastheonlyextratermintheGPenergyfunctionalofavortexstate.Weassumedthatthedensityofthecondensateremainedthesamewhileintegratingoverthiskineticenergydensity.Thisassumptionwouldnotberightinthecaseofthesmallcondensatebeingconsideredhere,becausethedensityofthecondensateisalteredoveralengthscalecomparabletothecondensatesizebythepresenceofavortex.Sowehavetobemorecarefulwithouranalysis. Werstcalculatethetotalenergyofacondensate(inourappropriatelynondimensionalizedunits)withasinglyquantizedvortexatitscenter,andwithachemicalpotentialdeterminedby.Innondimensionalunits,thisisobtainedbyinsertingthewavefunction 86

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ofEq.( 4 )intoasuitablynondimensionalizedGPenergyfunctionalas E=Z1dzZ102d"@f @2+@f @z2+1 2+2+z2f2+f4#=20p 23=2+O(2). (4) Thetotalnumberofparticlesinthisstateis N=Z1dzZ102djf(,z)j2=4p 23=2+O(2).(4) Nowtondtheactualextraenergycostofcreatingavortex,wehavetocalculatetheenergyofthecondensateinEq.( 4 )toavortex-freereferencesystemwhichhasthesamenumberofparticles.Thewavefunctionofsuchasystemwillbedescribedbythegroundstateofa3Dharmonicoscillator fR(,z)=23=2p Re)]TJ /F10 7.97 Tf 6.59 0 Td[(2=2)]TJ /F11 7.97 Tf 6.58 0 Td[(z2=2+O((R)3=2),(4) wherethequantitiesassociatedwiththisreferencesystemhavebeenlabeledwithasuperscriptR.Inparticularweexpectthechemicalpotentialoftheoriginalandreferencestatestobedifferent.Equatingthetotalnumberofparticlesforthevortex-freereferencestatetothatoftheoriginalstatewediscoverasimplerelationshipbetweentheirrespectivechemicalpotentials R=2.(4) TheenergyofthereferencesystemisfoundbyinsertingthewavefunctionofEq.( 4 )intotheGPenergyfunctional ER=Z1dzZ102d"@fR @2+@fR @z2+1 2+2+z2(fR)2+(fR)4#=6p 23=2R+O((R)2). (4) 87

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TheenergycostofcreatingavortexisthenthedifferenceofEq.( 4 )andEq.( 4 ): Ev=E)-222(ER=8p 23=2.(4) Similarlysincethisisacenteredvortex,theangularmomentumcarriedbyeachparticleis~andthetotalangularmomentumofthecondensatewithasinglyquantizedvortexatitscenteris L=4p (2)3=2+O(2),(4) wheretheunitofangularmomentumis~.Thecriticalvelocityoftraprotationatwhichvortexisstabilizedisthen(uptozeroethorderinE,andafterputtingbacktheunits) cEv=L!(4) Therewillbecorrectionstothisexpressionforcriticalvelocity.Howeverwecanalreadyseefromthezeroethorderresult,thatforasmallcondensate,thetraphastoberotatedatanangularvelocityofthesameorderofmagnitudeasthetrapfrequencytostabilizeavortexatthecenterofthetrap.Howeverthecentrifugalforcesatthisrotationfrequencyarelikelytodestroythetrapping.Thus,ndingstableormetastableregimesforthevortex,whilenotdestroyingthestabilityofthetrap,isruledoutforasmallcondensate.WeconcentrateonthedynamicsandtunnelingofalargeTFcondensateintheanalysestofollow. 4.3VortexDynamicsintheCo-rotatingFrame TheGross-PitaevskiiLagrangiandensitywrittenintherotatingframeofreferenceis L =i~ 2 @t )]TJ /F4 11.955 Tf 11.95 0 Td[( @t )]TJ /F4 11.955 Tf 11.96 0 Td[( )]TJ /F3 11.955 Tf 7.04 -7.03 Td[(^H0)]TJ /F3 11.955 Tf 11.96 0 Td[(^Lz )]TJ /F5 11.955 Tf 13.15 8.08 Td[(g 2j j4,(4) where ^H0=)]TJ /F7 11.955 Tf 12.92 8.08 Td[(~2 2mr2+Vtr(r)(4) 88

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istheHamiltonianforthenoninteractingcondensate,and ^Lz=i~^z)]TJ /F17 11.955 Tf 5.48 -9.69 Td[(rr(4) isthecomponentoftheangularmomentumoperatorinthez-direction. WewritetheaboveLagrangiandensityintermsofsuperuidphaseanddensityusingtheso-calledMadelungtransform, (r,t)=p n(r,t)ei(r,t)[ 107 ]: Ln,=)]TJ /F5 11.955 Tf 9.3 0 Td[(n(~@ @t+~2 2m(r)2+(rp n)2 n+Vtr+g 2n)]TJ /F7 11.955 Tf 11.96 0 Td[(~(^zr).r).(4) Theterminvolvingatimederivativeofdensitydoesnotcontributetothedynamicsandisneglected.Thetimederivativeofthephaseishoweverimportantbecausethephaseisnotananalyticeldinthepresenceofavortex. ByvaryingtheaboveLagrangianwegetthehydrodynamicequationsofmotioninarotatingframe: @n @t)]TJ /F3 11.955 Tf 11.95 0 Td[(@n @+~ mr(nr)=0;(4) )]TJ /F7 11.955 Tf 11.95 0 Td[(~@ @t=~2 2m(r)2)]TJ /F3 11.955 Tf 13.15 8.08 Td[((r2p n) p n+(Vtr)]TJ /F4 11.955 Tf 11.95 0 Td[()+gn)]TJ /F7 11.955 Tf 11.96 0 Td[(~@ @.(4) ThesuperuidvelocityintheinertiallaboratoryframeisrelatedtothephaseviatheJosephsonrelation,v=(~=m)r.Thephaseisdescribedintermsofthecoordinatesoftherotatingframe.Re-expressingEq.( 4 )andEq.( 4 )intermsofthesuperuidvelocity,v,werecoverthehydrodynamicequationsofsuperuidsinarotatingframeofreferenceas2 @n @t+r(n(v)]TJ /F17 11.955 Tf 11.96 0 Td[(r))=0,(4) and, @v @t+rv2 2+Vtr m+g mn)]TJ /F7 11.955 Tf 18.04 8.09 Td[(~2 2m2(r2p n) p n)]TJ /F17 11.955 Tf 11.96 0 Td[(v(r)=0.(4) 2Notethattheequationswhileintherotatingframeofreferenceareexpressedintermsofthesuperuidvelocityintheinertialorlaboratoryframe. 89

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Thesearethecontinuityandmomentumequationsrespectivelyforasuperuidinarotatingframe.Themomentumtransportequationcontainstheforcesexperiencedbyaninnitesimalelementofuidasagradientoftheexternaltrappotential,theinteractionforcesfromthecondensate,termsfromtherotatingframeofreference(whichcanbeshowntoberelatedtotheCoriolisandcentrifugaleffects)andaquantumpressureterm(socalledbecauseofitsdependenceonPlanck'sconstant)thatgoesasthegradientofthedensity.Thistermispurelyquantuminoriginanddistinguishesasuperuidfromaclassical,inviscidandirrotationaluid.IntheThomas-Fermi(TF)approximation,validforalarge,densecondensatewherethecondensatedensityvariesslowlyinspaceexceptneartheedgeofthetrap,thequantumpressuretermisdroppedintheaboveequation.Thisgivesthewell-knownEulerequationforaninviscidandirrotational,classicaluidinarotatingframeofreference. Nextweintegrateoutthedensityuctuationstoobtainthehydrodynamicactionsolelyintermsofthephase.ThiscanbeeasilydonebyexpressingthedensityintermsofthephaseusingEq.( 4 )andtheTFapproximation.TheTFcondensatedensityisthengivenby nTF(x,t)=)]TJ /F3 11.955 Tf 10.73 8.09 Td[(1 gh~@ @t+Vtr)]TJ /F4 11.955 Tf 11.95 0 Td[(+~2 2m(r)2)]TJ /F7 11.955 Tf 11.95 0 Td[(~@ @i(4) andtheactioncanthenbeexpressedas, S=Zdtg 2Zd3x)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(nTF2.(4) Nowtomakecontactwiththestationary(time-independent)GPequation,wegaugeawaythetimedependenceofinthestationarycase,i.e.dene~@ @t+asthenew~@ @t,whereistheconstantchemicalpotentialinthecorrespondingstationaryGPequation,gn(0).Alsodenes(r)=[)]TJ /F5 11.955 Tf 12.09 0 Td[(V(r)]=gtobethestationaryTFcondensatedensity.Usingtheabovenotation,thehydrodynamicdensitycanbedecomposedintostationary 90

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andnon-stationaryparts nTF(x,t)=s(r))]TJ /F3 11.955 Tf 13.4 8.09 Td[(1 gh~@ @t+~2 2m(r)2)]TJ /F7 11.955 Tf 11.95 0 Td[(~@ @i.(4) Afterintegratingouttheuctuationsindensity,wehaveanactionexpressedentirelyintermsofthephase.Thedetailedformoftheactioniswrittenasfollows: S=S1+S2+S3+S4,S1=)]TJ /F7 11.955 Tf 9.3 0 Td[(~2ZdtZdxs(r)@ @t,S2=~2 2gZdtZdx@ @t2,S3=)]TJ /F16 11.955 Tf 11.29 16.27 Td[(ZdtZdxs(r)~2 2m)]TJ /F2 11.955 Tf 5.47 -9.68 Td[(r2)]TJ /F7 11.955 Tf 11.95 0 Td[(~@ @,S4=)]TJ /F7 11.955 Tf 11.32 8.08 Td[(~2 2gZdtZdx2@ @@ @t+2@ @2. (4) Thereasonwemakea(purelyconceptual)distinctionbetweenthefourpiecesintheactionabove,willbecomeclearerinthediscussioninthesubsequentparagraphs. Letusrstconsideronlythevortexpartofthephase,andignorethephononmodesexcitedbythemotionofthevortex.OurchosenAnsatzforthevortexpartofthephaseis v(x,t)=~qv marctanhy)]TJ /F5 11.955 Tf 11.96 0 Td[(yv(t) x)]TJ /F5 11.955 Tf 11.95 0 Td[(xv(t)i,(4)xv(t)beingthevortexpositionasafunctionoftime,andforsimplicityweconsiderasinglevortexofchargeqv.Thesameideacouldbeusedtorepresentasystemofmultiplevortices.Thefollowingrelationshold: rrv=2qv(2))]TJ /F17 11.955 Tf 5.48 -9.69 Td[(x)]TJ /F17 11.955 Tf 11.95 0 Td[(xv^z,r2v=0. ByusingtheabovevortexAnsatzinthedetailedactionstatedinEq.( 4 )andthenintegratingoutthespatialcoordinates,wewouldbeleftwithanactionentirelyinterms 91

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ofthevortexcoordinates.Startingfrompurelyhydrodynamicconsiderations,wewouldhavederivedthesingle-particledynamicsofavortex.ThetermsS1toS4forwillbeshowntocorrespondtothesuperuidMagnusforce,thevortexmass,thestaticvortexpotential,andthenoninertialeffectsoftherotatingframe,respectively.Wenowdiscussinsomedetail,variousaspectsofthewell-establishedMagnusforcefeltbyamovingvortexinasuperuid,andthenot-so-well-establishedideaofthemassofavortex,withtheabovehydrodynamicactionstoillustratethepoint. 4.3.1TheMagnusForce ThevortexpartofS1inEq.( 4 )canbeshowntocorrespondtoatransverseforceakintotheLorentzforcefeltbyachargedparticleinamagneticeld[ 109 ],whereastheanalyticpartofS1isatotalderivativeandcanbedroppedfromtheaction.LetusnowtakeacloserlookatthevortexcontributiontoS1.Thetimederivativeofthevortexphasecanberelatedtothevortexvelocitythrough@tv(x)]TJ /F17 11.955 Tf 13.28 0 Td[(xv)=vv.rv(x)]TJ /F17 11.955 Tf 11.95 0 Td[(xv(t)),wherevvisthevelocityofthevortex,andrvdenotesthespatialderivativewithrespecttothevortexcoordinates.Thisidentity,whichwewillmakefrequentuseof,impliesthatthephaseofamovingvortexchangesintimebecauseofthechangeinitscoordinates. ThisletsuswritetheS1partoftheactionas )]TJ /F7 11.955 Tf 11.95 0 Td[(~ZdtZdxs(r)@v @t=ZdtqvvvA,(4) where qvA(xv)=)]TJ /F7 11.955 Tf 9.29 0 Td[(~Zdxs(r)rv(x)]TJ /F17 11.955 Tf 11.95 0 Td[(xv).(4) ThetermS1aswritteninEq.( 4 )resemblestheactionofachargedparticleinamagneticeld,whichisknowntobeRdt[(m=2)v2+qvA],theAdenedbyEq.( 4 )beingthemagneticvectorpotential.Theeffectivemagneticeldinwhichthevortex 92

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Figure4-3. Magnusforceonarotatingobjectinaowinguid:theobjectspinsclockwiseastheuidowspastitfromrighttoleft.Thevelocitystreamlinesareshown.Thetotaluidvelocityisasumoftheuniformowandtherotationalow(whichisanalogoustoavortexvelocityeld).Thevelocitiesarehigherinthebottomhalfplaneinthisgure,leadingtopressuredifferencesfromBernoulli'sprinciple.TheobjectthusfeelsadownwardMagnusforce. movesisthenidentiedas Bv=rvA=)]TJ /F7 11.955 Tf 9.3 0 Td[(~Zdxs(r)rvrv(x)]TJ /F17 11.955 Tf 11.95 0 Td[(xv)=)]TJ /F5 11.955 Tf 9.3 0 Td[(h2Ds(xv)^z.(4) TheLorentz-liketransverseforceFm=qv(vvBv)thatoriginatesfromthisactionisexactlyliketheMagnusforce(alsocalledtheKutta-Joukowskiforceinaerodynamicsliterature)familiarfromclassicaluiddynamics[ 106 ],withthedensityoftheclassicaluid,replacedbythesuperuiddensity.Thisisapurelyclassicaluid,andcanbeunderstoodintuitivelybyconsideringaspinningobjectplacedinaclassicaluid,asshowninFig. 4-3 .Astheobjecrotatesitentrainssomeofthesurroundinguidandswirlsitaround.Thenetuidvelocityisthereforedifferentononesideoftheobjectfrom 93

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thatonthediametricallyoppositeside.Bernoulli'sprincipleofenergyconservationinauidsuggeststhatthisleadstothebuildupofapressuredifference,andaresultingtransverseliftorMagnusforce.UsingtheGalileaninvarianceoftheGross-Pitaevskiiformalismandtheresultantclassicalhydrodynamicsdescribedabove,wecanimposeanoverallconstantvelocityeldvscorrespondingtotheoverallsuperuidowinsomeinertialframeofreference,andthemorecompleteexpressionoftheMagnusforcethenbecomes Fm=qv~2Ds(xv)^z(vv)]TJ /F17 11.955 Tf 11.96 0 Td[(vs).(4) Wenoteherethattheinhomogeneouspotentialfromthetrapisresponsibleforinhomogeneitiesinthedensity,andsoultimatelyinthemagneticeld,andthemagnitudeoftheMagnusforce.Forexampletheforceisexpectedtobestrongerwhenthevortexisnearthecenterofthetrap,becausethatiswherethecondensatedensityisthehighest.Alsonotethatthes(xv)usedintheabovetreatmentisactuallythe2Dsuperuiddensityatthevortexposition,whichisobtainedbyintegratingthe3Ddensityalongthez-direction. 4.3.2VortexMass Theideaofavortexmasshasbeenamatterofsomedebate.Yetthenotionofmassisveryimportantincalculatingtherateofvortextunneling,andothersuchexperimentallyobservablephysicalphenomenainvolvingvortices.Avortexinaclassical,invisciduidisusuallyassumedmasslessastheuiddensitygoestozerointhecoreofthevortex.Thispointofviewhasalsobeenadoptedinsomeapproachestostudyquantizedvortices[ 110 ].Othersassumeittobethemassofatomsinthenormalvortexcore[ 111 ],whichistheonlysourceofinertiainastrictlyincompressiblesuperuid.Severalauthors,suchasPopovinanolderwork[ 112 ],andDuanandLeggett[ 113 ]whotakethecompressiblenatureofthesuperuidandconsequentdensityuctuationsintoaccount,ndavortexmassthatislogarithmicallydivergentwithsystemsize.Wenoteatthispointamajordifferencebetweenvortex 94

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energyinsuperuidhelium,andaBECinatrap.Thelatteralwayshasanitesize,andthiscondensateradiusRprovidesanaturaluppercutofflengthscale.TheproblemoflogarithmicdivergencesisthereforeabsentinthetrappedBECsystemsweareconsidering.Inotherwordsln(R=)isnite,andoforder1formostexperimentalsituations,andtheabovetwoestimatesofmassarenotradicallydifferent.WewillnowpresentanestimateofthevortexmassbasedonthehydrodynamicactionwehavewrittendowninEq.( 4 )3. ThevortexpartofS2inEq.( 4 )afterperformingthespatialintegral,canbeshowntoberelatedtotheinertiaofthevortex.WebeginbynoticingthatthetermS2canbecastinaformsuggestiveofvortexkineticenergyas ~2 2gZdtZdx@ @t2=Zdt1 2v2vZdx~2 grvv(x)]TJ /F17 11.955 Tf 11.95 0 Td[(xv)2.(4) ThisisoftheformRdt(1=2)Mvv2vwherethevortexmassMvisobtainedbyperformingthespatialintegrationoverthephasetermabove.Weperformthisintegralbyrsttranslatingtheoriginofthespatialcoordinatestothepositionofthevortex,whichmeansthatthevortexphaseisnowsimplytheazimuthalangle.Thegradientofthephasenowhasthesimpleform rv=)]TJ /F3 11.955 Tf 11.8 10.74 Td[(^0 0,(4) where0and0aretheazimuthalandradialcoordinatesrespectivelyinthenewcoordinatesystemcenteredatthevortexposition.Thevortexmassisnowobtainedbyintegratingoverthenewspatialcoordinatesas Mv~2 gZdx)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rvv2=~2 gZdzZ2d0 0'2~2Rz glnR .(4) 3ThederivationissimilarinspirittothatpresentedinthebookInhomogeneousSu-perconductors(pp257)bySimanekinthecontextofchargedsuperuids[ 114 ]. 95

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Theideaofavortexmassisproblematic,andestimatesintheliteratureforitsvaluerangefrom0toinnite!Wehaveobtainedhereaworkingdenitionofmass,butwiththeobviouscaveatthatthevortexAnsatzusedinthispictureisnotthewholestorytothesuperuidphase.Amovingvortexislikelytointroducedensityuctuationsintheformofacousticwaves,andthesephononswouldcoupletothevortex,producingtemporallynonlocalmemoryeffectsinthevortexinertia.Thisresultsinafrequency-dependentvortexmassaswendlaterinthischapter,andaswasfoundbypreviousauthorssuchasThoulessandAnglin[ 115 ].TheexpressionforvortexmassweobtaininEq.( 4 )ispresumablyjustalocalapproximationofthismorecomplicatedinertialtermwhichwediscusslater.Fornow,wecanputtheresultofEq.( 4 )onarmerphysicalfootingbynoticingthatbarringthelogarithmicfactor,itissimplyrelatedtothemassoftheatomsinthenormalvortexcore.ThiscanbeeasilyseenbyusingtherelationsbetweenthecondensateandtrapparametersthatareexpectedtoholdintheTFregime,specicallygn0=,=~=(m!R)and=m!2R2.Usingthemwecanshowthat Mv'4mn02RzlnR ,(4) wheremisthemassofeachatominthecondensate,n0issomecharacteristiccondensatedensity,thevortexcoreradius,andRzthelengthofthevortexline.Thefactorofmn02Rzisthenjustthemassofthevortexcore,wheretheuidisinthenormalphase. Staticvortexpotential.ThevortexpartofS3inEq.( 4 )afterperformingthespatialintegral,correspondstothevortexenergyEvobtainedearlierfromstaticenergeticconsiderationsinEq.( 4 ).Thiscontributiontotheactionhastheformofaparticleinanexternalpotential,Vv(xv). Effectofrotation.ItisintuitivelyapparentfromthestructureofS4thatthetermscontainingtheangularfrequencyofrotationanditsquarecorrespondtotheCoriolisandcentrifugalforcesactingonthevortexintherotatingframe. 96

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Wenowdecomposethephaseintoanalyticandvortexparts,=a+v,rra=0.Wewishtointegrateouttheanalyticpartofthephaseandderiveanactionsolelyintermsofthevortexcoordinates.Withthisobjectiveinmind,letuscollectalltermsintheactionthatdependontheanalyticpartofthephaseandwriteitinthefollowingsuggestiveform, Sa=ZdtdzZd2x()]TJ /F7 11.955 Tf 11.31 8.09 Td[(~2 2g@a @t2+2@a @t@v @t+2@a @t@a @+@a @t@v @+@a @@v @t+2@a @2+2@a @@v @+~2 2ms(r)h)]TJ /F2 11.955 Tf 5.47 -9.68 Td[(ra2+2ra.rv). (4) Thisleavesoutsometermswhichjustdependonvwhichwewilltakeintoaccountlater.ForthetimebeingwevarytheactionwithrespecttoaandwritedownthecorrespondingEuler-Lagrangeequation, @2a @t2)]TJ /F5 11.955 Tf 14.76 8.09 Td[(g mr.)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(sra+@2a @t@+2@2a @2=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(@2v @t2+g mrs(r).rv)]TJ /F3 11.955 Tf 11.95 0 Td[(@2v @t@)]TJ /F3 11.955 Tf 11.96 0 Td[(2@2v @2,(4) TheanalyticpartofthephasecaninprinciplebesolvedfromtheGreen'sfunctionofthegeneralized,inhomogeneouswaveequationabove:a(x,t)=Rdt0d2x0G(x,x0;t,t0)f(x0,t0)wheref(x,t)isthesourcefunctionintermsofthevortexphase,ontherighthandsideofEq.( 4 ).Thispartoftheactionthereforehasthestructure, Sa=~2Rz 2gZdtdt0Zd2xd2x0G(x,x0;t,t0)f(x,t)f(x0,t0).(4) Nowreassemblingallthepiecesintheaction,wehavetheform, S=Sm+Sa)]TJ /F16 11.955 Tf 11.95 16.27 Td[(ZdtVv(xv)+Sv2+Sv4,(4) 97

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HereSmistheMagnusterm[calledS1inEq.( 4 )],Vvisthevortexpotentialdescribedintheprevioussection,andtheotherpiecesaredenedas Sv2=~2 2gZdtZd2x@v @t2,Sv4=)]TJ /F7 11.955 Tf 11.32 8.09 Td[(~2 2gZdtZdx2@v @@v @t+2@v @2. (4) HereSv2andSv4arethevortexpartoftheexpressionsS2andS4denedearlierinEq.( 4 )andcorrespondtothestaticvortexmassandthenoninertialtermsrespectively,asdiscussedinaperviousparagraph.Saistheonlynonlocalpieceintheactionofavortex,thedeterminationofwhichwouldrequireasolutionoftheGreen'sfunction. CalculationoftheGreen'sfunctionfortherotatingwaveequationintheneighbor-hoodofavortex.Wereplacethesuperuiddensitybyitsconstantvalueatthevortexpositioni.e.s(xv)inEq.( 4 ).Thewavespeedisthusdeterminedbythedensityofthecondensateatthevortexposition.Thishomogenizingassumptionisplausiblebecausemostofthecontributiontotheenergy/actionofthecondensatecomesfromnearthevortexasthesuperuidvelocityfallsawayas1=rfarfromthevortex. ThecomputationoftheGreen'sfunctionforthesolutionofEq.( 4 )isrenderednontrivialbythepresenceofthetermsinvolvingderivativewithrespecttotheazimuthalangle.Thesearisebecauseoftherotationofthetrap.Weproposetosolvetheproblemthroughaseparationofvariablesinacylindricalsystemofcoordinatesintherealspace.ByFouriertransformingtofrequencyspaceintimeandseparatingtheradialandazimuthalcoordinates,theGreen'sfunctioncanbewrittenas G(x)]TJ /F17 11.955 Tf 11.96 0 Td[(x0)=1 21Xm=gm(,0;!)eim()]TJ /F10 7.97 Tf 6.58 0 Td[(0),(4) wherethemtakesallintegervaluestopreservetheperiodicboundaryconditionintheazimuthaldirection.InsertingthisintotheequationfortheGreen'sfunctionresulting 98

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fromEq.( 4 ),andwhichiswrittenas )]TJ /F4 11.955 Tf 11.96 0 Td[(!2)]TJ /F5 11.955 Tf 11.95 0 Td[(c2r2+i!@ @+2@2 @2G(x)]TJ /F17 11.955 Tf 11.96 0 Td[(x0)=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(2 ()]TJ /F4 11.955 Tf 11.95 0 Td[(0)()]TJ /F4 11.955 Tf 11.95 0 Td[(0),(4) weobtainadifferentialequationfortheradialpartoftheGreen'sfunction: 1 @ @@gm @)]TJ /F16 11.955 Tf 11.96 16.85 Td[(!2m c2m2 2gm=)]TJ /F3 11.955 Tf 10.5 8.08 Td[(2 ()]TJ /F4 11.955 Tf 11.95 0 Td[(0).(4) Here!mp !2+m!+m22.When6=0,theabovecorrespondstoamodiedBessel'sequationwhosegeneralsolutionisalinearcombinationofthemodiedBessel'sfunctionsoftherstandsecondkind.Nowbyintegratingoverthejumpdiscontinuityinthederivativeat6=0andensuringtheappropriateconditionsforawell-behavedfunction(i.e.functiondecaystozerosufcientlyfarawayandremainsniteattheorigin),weobtainanexpressionfortheGreen'sfunctionas G(x)]TJ /F17 11.955 Tf 11.95 0 Td[(x0;!)=1Xm=Im!m< cKm!m> ceim()]TJ /F10 7.97 Tf 6.58 0 Td[(0).(4) Thisinprinciplewouldletuscalculatethecontributionofthephononparttothehydrodynamicaction,whichafterintegratingoutthephononsandthespatialcoordinates,wouldgiveanactionintermsofthevortexcoordinatesthatisnonlocalintime. Vortexdynamicsinanonrotatingframe.Theproblemofvortexdynamicsbecomestractableinthespecialcasewherethetrapisnotrotating,i.e.=0.Whilethisdoesnotallowforthemetastabilityofthevortexatthecenterofthetrap,andthereforenopossibilityofthemacroscopicquantumtunnelingofthevortex,itrevealsexplicitlythenonlocalnatureoftheactionofthevortex,andshowsthatthevortexmassisinfactfrequencydependent.Intheabsenceofrotation,theanalyticpartofthephasesatisesaninhomogeneouswaveequationobtainedbysetting=0inEq.( 4 ): @2a @t2)]TJ /F5 11.955 Tf 11.95 0 Td[(c2r2a=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(@2v @t2.(4) 99

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TheGreen'sfunctionforthiswaveoperatorhasthefamiliarform G(q,!)=1 )]TJ /F4 11.955 Tf 9.3 0 Td[(!2+c2q2,(4) infrequencyspace,andthecorrespondingpartoftheactionSa[denedinEq.( 4 )]canbeexpressedas Sa=Rz~2 2gZdtdt0Zdq)]TJ /F4 11.955 Tf 13.15 8.09 Td[(@2G(q;t)]TJ /F5 11.955 Tf 11.95 0 Td[(t0) @t2@v(q,t) @t@v()]TJ /F17 11.955 Tf 9.3 0 Td[(q,t0) @t0,(4) wherewehaveusedintegrationbypartstotransferthetwotimederivativestotheGreen'sfunctionfromthesourcefunctionsgivenintermsofthevortexphase.Wehavetobecarefulwhileintegratingbypartsasthevortexphaseisanonanlyticfunction.ThespatialFouriertransformofthesourcefunctionisgivenby @v @t(q,t)=evh mijqi q2(vv)je)]TJ /F11 7.97 Tf 6.58 0 Td[(iq.xv(t),(4) whereijisthecompletelyantisymmetrictensorin2D.InsertingtheaboveexpressionintoEq.( 4 )andusingtheGreen'sfunctionfromEq.( 4 ),weget Sa=Rz~2 2gZdtdt0vv(t)vv(t0)F(xv(t))]TJ /F17 11.955 Tf 11.95 0 Td[(xv(t0),t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0),(4) wherewehavedenedthekernelasanintegralinfrequencyspaceas F(x,t)=Zdq q2e)]TJ /F11 7.97 Tf 6.58 0 Td[(iq.xZd! 2!2 )]TJ /F4 11.955 Tf 9.29 0 Td[(!2+c2q2ei!t.(4) Thisintegralcanbeeasilyevaluatedafterseparatingoutalocalpieceusingpartialfractions F(x,t)=Zdq q2e)]TJ /F11 7.97 Tf 6.58 0 Td[(iq.xZd! 2)]TJ /F3 11.955 Tf 11.95 0 Td[(1+q2c2 )]TJ /F4 11.955 Tf 9.29 0 Td[(!2+c2q2=)]TJ /F5 11.955 Tf 9.29 0 Td[(F0(x)(t)+F1(x,t), (4) where F1(x,t)=c(ct)-222(jxj) p c2t2)-221(jxj2(4) 100

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istheusualretardedGreen'sfunctionforawaveequationin2D.ThelocalpieceoftheactionobtainedfromF0actuallycancelsthetermSv2correspondingtostaticvortexmass,anddenedinEq.( 4 ).Thetotalactionforavortexmovinginanonrotatingtrapcanthenbewrittenas S=Sm)]TJ /F16 11.955 Tf 11.96 16.27 Td[(ZdtVv(xv)+~2sRze2v 2m2cZdtdt0vv(t).vv(t0)(c(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0))-221(jx)]TJ /F17 11.955 Tf 11.95 0 Td[(x0j) p c2(t)]TJ /F5 11.955 Tf 11.95 0 Td[(t0)2)-222(jx)]TJ /F17 11.955 Tf 11.96 0 Td[(x0j2.(4) ItinvolvestermscorrespondingtotheMagnusforceorBerry'sphase,thepotentialfromthetrapfeltbythevortex,andinterestingly,anonlocalinertialtermobtainedbyintegratingoutthephononpartofthephase.Thisisintuitivelyunderstoodtobeacouplingofthevortexwithitselfatsomeearliertimemediatedbythephononsexcitedbythemotionofthevortex. Thevortexmasscanbecalculatedfromthisinertialterminthelimitwherethevortexmovesveryslowlyrelativetothesoundspeedsuchthatc(t)]TJ /F5 11.955 Tf 12.47 0 Td[(t0)jx)]TJ /F17 11.955 Tf 12.47 0 Td[(x0jisalwayssatised.Thisgivesthedynamicvortexmassas Mv~2sRze2v 2m2c2ln(!),(4) where!issomefrequencycharacteristicofthevortexdynamics,andissomecutofftimescaleintroducedtoregulatethedivergenceinthetimeintegral.Itcanbeidentiedwithasmalltimescaleinthesystem,R=c,whereRisthesizeofthecondensateandc,thespeedofsound.Thislogarithmicbehaviorcanalsobereproducedfromalocalexpansionofthegeneral,realspaceGreen'sfunctionobtainedinEq.( 4 ).Bysetting=0andexpandingforsmall!,weseethatthedominantcontributionisfromthem=0terminthesum.TheasymptoticbehaviorforthemodiedBessel'sfunctionsforsmallargumentisK0(x))]TJ /F3 11.955 Tf 26.37 0 Td[(ln(x)forsmallxandI0(0)=1.Onmakingalowfrequencyapproximation(whichcorrespondstothelongtimeintervalapproximationusedabove),theinertialtermisfoundtodependlogarithmicallyonfrequency. 101

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Thisfeatureofanonlocalinertialtermwhichcontributesamass'thatislogarithmicallydivergentinfrequencyisalsoexpectedtoholdinageneral,rotatingtrap.Sowehavederivedthedynamicsofthevortexfromtheunderlyinghydrodynamictheoryandhaveidentiedthethreemainfactorsthatdetermineitsdynamics:theMagnusforce,thevortexpotential,andthenonlocalinertialtermwhichcanbeidentiedwithamassonmakingalocalapproximation.Thisissensibleinthelimitthatthevortexvelocityissmallcomparedtothespeedofsoundinthesystem.Insuchasituation,therefore,itispossibletowritedownanequationofmotionofthevortexasifitwereapointparticleobeyingNewtonianmechanics.Wehavethereforestartedfromacontinuumtheory,integratedoutvariousmodes,andobtainedtheLagrangianofasingleparticle. 102

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CHAPTER5MACROSCOPICQUANTUMTUNNELINGOFAVORTEXINAROTATINGBOSEGAS Asuperuidvortexinalargecondensateisamacroscopicobjectexpectedtobehaveclassically.Thisisbecausethevortexcorecontainsalargenumberofatomswhenthecondensatedensityishigh,andsoisdescribedbyamany-bodywavefunctionwithmanydegreesoffreedom.InChapter4,wederivedthedynamicsofavortexusingpurelyclassicalconsiderations.ThisislegitimatebecauseBose-Einsteincondensationleadstoamacroscopicquantumstatethatcanbedescribedbyaclassicaleldtheoryfeaturingasinglescalarwavefunction,asisdoneintheGross-PitaevskiiformalismforBECatzerotemperature.TheclassicaldynamicsofasinglevortexinarotatingBEChavebeenworkedoutinRefs.[ 100 101 ]asalsoinChapter4.Hereweconsiderthepossibilityofquantumtunnelingofavortex. Thetunnelingofmicroscopicparticlesthroughsomeclassicallyforbiddenregionisawell-knownquantumphenomenon.Additionally,thesuperpositionofmacroscopicstatesiskeytothefoundationsofquantummechanics.Thedirectexperimentalevidenceofthiswouldbethetunnelingofamacroscopicobject[ 116 ],suchasavortex.Thesearchforsuchmacroscopicquantumtunneling(MQT)hashadsomesuccessinthecontextofstronglycorrelatedsystems,suchassuperconductingJosephsonjunctions[ 117 ],buthasnotbeenobservedinaBEC.PreviousauthorshavediscussedthepossibilityoftunnelingofasinglevortexbetweenpinningsitesinBECinatrap[ 118 ],oringeneral,two-dimensional,superuids[ 119 121 ].Inthischapter,wewillconsidertheyet-unexploredquestionoftunnelingofasinglevortexinatrappedBECthroughthepotentialbarriercreatedbytherotationofthetrap[ 100 103 ]. Intherstsection,weintroducethegeneraltheoryoftheclassicaldynamicsofavortextreatedasapointparticle,andestablishananalogywithachargedparticletravelinginamagneticeld.Wethendeterminetheprobabilityofquantumtunnelingofavortexfromsemiclassicalconsiderationsinthesecondsection.Thequantum 103

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mechanicsofaslowly-movingvortexcouldbedescribedbytheSchrodingerequationofachargedparticleinamagneticeld,andthetunnelingeventcanthenbedescribedinaWKBframework.However,ourderivationinChapter4ofthefullactionofavortex,showsthattherearenonlocalcontributionstothevortexinertiafromthecouplingswiththephononmodesitexcitesduringitsmotion.Inthiscase,simpleWKBmethodsfail,andwehavetoresorttomoresophisticatedtoolsliketheinstantonmethod.Weintroducethismethodandsketchoutapossibletunnelingcalculation.Finally,wediscussthefeasibilityofsettingupactualexperimentsinrotatingtrapsofultracoldatoms,withobservableratesoftunnelingforasinglevortex. 5.1ClassicalMechanicsofaVortex Avortexcanbethoughtofasapointmassinthe2Dplane,whosemotionisdescribedbyaNewton'slawoftheform Mvxv(t)=F?+Fjj+Fext(5) whereMvisthemassassociatedwiththevortex,xvisthepositioncoordinateofthevortexinthex)]TJ /F5 11.955 Tf 12.56 0 Td[(yplane,andthetermsontherighthandsidecorrespondtovariousforces,whoseexactformsandphysicaloriginswenowdiscussinturn.Theexactvalueofthevortexmassiscontroversial,andhasbeenfoundtotakevaluesfromzerotoinnitybydifferentauthors!Forthepurposesofthischapter,weassumeanitevalueforthevortexmass. ThetermF?consistsofforceswhichacttransversetothevortexvelocity,andhaveaformakintotheLorentzforceactingonanelectricchargemovinginamagneticeld.ThesuperuidMagnusforce(seederivationinSection4.3.1),anexactanalogoftheMagnusforcefeltbyrotatingobjectsmovinginclassicaluids,isonesuchtransverseforcehavingtheform FM=s(_xv)]TJ /F17 11.955 Tf 11.96 0 Td[(vs).(5) 104

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Thesaboveisthe2Dsuperuiddensity(theMagnusforceinclassicaluidsdependsexactlyinthesamewayonthetotaluiddensity).Wecallthequantizedcirculationofthesuperuidvelocityaroundavortex,,anditisgivenby =evh m^z,(5) whereevisthechargeorwindingnumberofthevortex,andmthemassofeachconstituentparticle,whichcouldbeatomsinatrappedBEC,orheliumatomsinstronglycorrelatedliquidHe.Anothertransverseforcesuggestedbysomeauthors(butdisputedbyothers)istheIordanskiiforce[ 122 ],whichhasthesameformastheMagnusforceabovebutarisesfromthemotionofthevortexrelativetothenormalfractionoftheuid: FI=n(_xv)]TJ /F17 11.955 Tf 11.95 0 Td[(vn).(5) Themotionofthevortexrelativetothenormaluidalsosubjectsittoalongitudinaldampingforcewrittenas Fjj=)]TJ /F4 11.955 Tf 9.3 0 Td[((_xv)]TJ /F17 11.955 Tf 11.96 0 Td[(vn).(5) Thedampingconstant(T)isstronglydependentontemperature.Thisarisesfromvortexinteractionwiththequasiparticlesinthenormalfraction.Weconnetherestofourdiscussiontothezerotemperaturelimit,wheretheGross-Pitaevskiidescriptionisapplicable.Heretheentiresystemisinthecondensedorsuperuidphase,andwesteerclearofthecontroversysurroundingtheIordanskiiforce.Thedampingforcedescribedaboveisalsoabsent.Otherforcesonthevortexcouldarisefromexternalsources,suchastheeffectivepotentialVv(r)feltbecauseofthetrap,andpinningforcesfromimpurities Fext=rVv+Fpinning.(5) Wearemostlyinterestedintheeffectofthetrapanditsrotation.Theexternalforceisthenjustthegradientofthepotentialexperiencedbythevortexinarotating,isotropictrap(foundinChapter4).SoatzeroT,afterignoringtheeffectsofdissipationand 105

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pinning,theclassicaldynamicsofthevortextreatedasasingleparticle,isdictatedbytheMagnusforceandthetrap-dependentpotential.ThisissummedupinasimpleNewton'slaw,whichessentiallydescribestheclassicaldynamicsofachargedparticleinaplanewithanon-uniformmagneticeld1intheMagnusforceactingperpendiculartoitandinthepresenceofacentralpotential, Mvxv(t)=s(_xv))-222(rVv.(5) Theenergycostofcreatingavortexatanarbitrarypositioninarotatingtrapwasfoundearliertobe Vv(r)=V01)]TJ /F5 11.955 Tf 14.9 8.08 Td[(r2 R23=2)]TJ /F3 11.955 Tf 15.72 8.08 Td[( c1)]TJ /F5 11.955 Tf 14.89 8.08 Td[(r2 R25=2,(5) wherethesymbolshavetheirpre-denedmeanings. Forthesakeofcompleteness,wenowreviewtheLagrangianmechanicsoftheanalogousproblemofachargetravelinginamagneticeldandsubjecttoacentralpotential.TheLagrangianforsuchaparticle(freetomoveintheplaneperpendiculartothemagneticeld)incylindricalcoordinatesis L(r,;_r,_)=1 2M(_r2+r2_2))]TJ /F5 11.955 Tf 11.95 0 Td[(V(r)+qAv,(5) whereABr^=2isthemagneticvectorpotentialinthesymmetricgauge,V(r)isageneralcentralpotentialin2D,andtherestofthesymbolshavetheirusualmeanings.Thesymmetricgaugeisanaturalchoiceinacentralpotentialwherethexandy-directionsareonequalfooting. 1Wewillusethetermmagneticeldinterchangeablywiththeeldsh=mcorrespondingtotheMagnusforce,throughoutthischapter 106

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Figure5-1. Energycost(nondimensionalized)associatedwithasinglyquantizedstraightvortexinarotatingasymmetrictrapintheTFlimitasafunctionofafractionalvortexdisplacement0fromthesymmetryaxis.Differentcurvesrepresentdifferentxedvaluesoftheexternalangularvelocity(a)=0(unstable);(b)=m(theonsetofmetastabilityattheorigin);(c)=c(theonsetofstabilityattheorigin);(d)=3 2c.Inthemetastableregime,m<
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Wecanthenusethisexpressionfortheangularvelocitytondanotherconstantfromtheequationsofmotion,intermsoftheradialcoordinatealone,as 1 2M_r2+V(r)+1 8M!2cr2+L2 2Mr2=constant,(5) whichisbasicallytheenergyconservationequationforaparticletravelingin1Dinaneffectivepotential,Ve(r)=V(r)+M!2cr2=2+L2=2Mr2,thathascontributionsfromthemagneticeldandthecentrifugalbarrier.Theexacttrajectoryoftheparticlecouldthenbesolvedinprinciplebyquadrature.Thisderivationfurthersuggeststhattheprobleminacentralpotentialisstillseparableinradialandazimuthalcoordinates,andcanbereducedtoaneffectively1Dproblem,eveninthepresenceofamagneticeld.Thishasimportantimplicationsforthetunnelingcalculation. Whenthemagneticeldisverylow,theparticletrajectorywouldbeslightlyperturbedbythesmallmagneticeldfromtheusualellipticalorbitexpectedinacentralpotential.Whenthemagneticeldisveryhigh,theinertialtermmaybeneglectedintheequationofmotion,andthentheparticlesimplymovesalongtheequipotentiallines.Thesearejustcirclesinthecaseofacentralpotential.ForthevortexinatrappedBEC,thiswouldmeanaprecessionaroundthecenterofthetrapatafrequencywhichisdeterminedbybalancingthegradientofthepotentialagainsttheMagnusforce, rV'evvB.(5) Ifthemagneticeldinthecasebeingconsideredisperpendiculartotheplaneoftheparticle'smotion,theaboveequationimpliesthatthemotionisperpendiculartothegradientofthepotential,i.e.,alongequipotentiallines.Letusnowusethisequation,andtheexpressionforthemagneticeldcorrespondingtotheMagnusforceonavortexasfoundinEq.( 4 ),topredictthevelocityofavortexinthepotentialofarotatingharmonictrapgivenbyEq.( 5 ).Thevortexisthenfoundtomoveincirclesaroundthecenterofthetrap.Itsspeedhasasomewhatcomplicateddependenceontheradial 108

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positionofthevortexinthetrap: vv=4evh MR2lnR jevjs n(0) n(rv))]TJ /F3 11.955 Tf 16.92 8.09 Td[( ms n(rv) n(0)(^z^r),(5) wheren(r)istheThomas-Fermi(TF)densityproleofthecondensate,isthetraprotationfrequency,mthemetastablefrequencyfoundearlier,andtherestofthesymbolshavingtheirpredenedmeanings.ThisresulthasbeenfoundinadifferentformbyamoredetailedanalysisinvolvingmatchedasymptoticsolutionsofthedynamicGPequationnearandfarawayfromthevortex,byRubinsteinandPismenforgeneralsuperuidsin1994[ 123 ],andbyFetterandSvidzinskyforsimilartrappedBECsin2000[ 101 ].Thelatterauthorsalsolookatthegeneralcaseofasymmetrictrapsandcurvedvortexlines,andreachessentiallysimilarconclusions.Thevortexisthusseentoprecessaroundthecenterofthetrap,withaprecessionfrequencythatdependsontheradiusofthevortexorbit.Thisfrequencynearthecenterofthetrapisfoundtobe !p'4evh MR2lnR jevj1)]TJ /F3 11.955 Tf 16.91 8.09 Td[( m.(5) Thus,apositivevortexprecessesinacounterclockwisedirectionwhenthetrapisrotatedatafrequencyhigherthanthemetastability(>m),andclockwisewhenlowerthanit.Thereisnoprecessionwhenrotatedatmetastability,andthevortexenergyproleattens.Wehavetorememberthatthisiswithrespecttotherotatingframeofthetrap. Further,itisusefultocomparetheprecessionfrequencywithothertimescalesinthesystem.Weseethattheprecessionfrequencyisapproximatelyafactorof(1)]TJ /F3 11.955 Tf 13.06 0 Td[(=m)timesthemetastableangularvelocity.Therefore,ifthetrapisrotatednearmetastability,theprecessionisveryslow(aspointedoutearlier).Theprecessionfrequencydependsdirectlyonthepotentialgradientbutinverselyontheeffectivemagneticeld.Thisisinsomesensethereverseofthemagneticcyclotronfrequency!c=qB=M,whichisdirectlyproportionaltothemagneticeld.Asmallvalueof 109

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precessionfrequencyrelativetothecyclotronfrequency(!p!c)thensuggestsahighmagneticeldlimit.Thecorrespondingtrajectoryisaslowcircularprecessionaroundthecenterofthetrap,andrapidcyclotronoscillationsintightcirclessuperimposedonit,whichgoawayonaveraging-leadingtotheequipotentialcircularorbitsexpectedinthehigheldcase.Theratiooftheprecessionandthecyclotronfrequenciesisthereforeausefulquantitythatcanbeestimatedfromexperimentalparameters. Alsotheprecessionfrequencyorvelocityofthevortexseemstoincreasewithincreasingdistancefromthecenterofthetrapasthegradientofthepotentialissharper,andtheMagnusforcebecomesweakerfurtherout.Ofcourse,beyondacertaindistancethehighmagneticeldapproximationabovefails,andthevortexinertiabeginstocontributetoitstrajectory.Inourpresentanalysis,weareinterestedinoutwardtunnelingofthevortexfromthemetastableminimumatthecenterofthetrap...soweexpecttoremaininthehighmagneticeldlimitdiscussedaboveaslongasthereisahighenoughcondensatedensityatthecenterofthetrap. 5.2SemiclassicalEstimatesofVortexTunneling Avortex,thoughmacroscopic,isaquantumobject,especiallyatlowT.Itsquantumdynamicsisrigorouslydescribedthroughtheevolutionofitsdensitymatrix[ 124 ]whichintheclassicallimityieldsaNewton'slawoftheformgiveninEq.( 5 ).Whentreatedasasingleparticleasdonesofarinthischapter,weoughttobeabledescribeitusingaSchrodinger'sequation,whichthenyieldsthequantummechanicsofachargedparticleinamagneticeld. 5.2.1SchrodingerEquationforaChargeinaMagneticField Thetime-independentSchrodingerequationin2DforachargedparticleinamagneticeldB^zandpotentialV(r)is 1 2Mi~r)]TJ /F5 11.955 Tf 24.57 0 Td[(qA2+V(r) =E .(5) 110

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Inthesymmetricgaugeofthemagneticvectorpotential,A=1 2Br^,andtheSchrodingerequationtakestheform, )]TJ /F7 11.955 Tf 16.34 8.09 Td[(~2 2Mr2+!c 2r)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(i~r+1 8M!2cr2+V(r) =E ,(5) where!c=qB=Misthecyclotronfrequencyforparticleinthemagneticeld.Asthepotentialiscentral,thewavefunctioncanbeseparatedintoradialandangularpartsas (r,)=f(r)eil,(5) wherelistheangularmomentumquantumnumber.TheSchrodingerequationfortheradialpartlookslike ~2 2Md2f dr2+1 rdf dr=E0)]TJ /F5 11.955 Tf 11.95 0 Td[(Ve(r)f(r),(5) whereE0=E+1 2l~!c,andthetrappotentialismodiedbythecentrifugalbarrierfromangularmomentumaswellasthemagneticeldas Ve(r)=V(r)+1 8M!2cr2+l2~2 2Mr2.(5) ThiscorrespondsexactlytotheLagrangianmechanicsofachargedparticleinamagneticeldasderivedintheprevioussection,asseenfromEq.( 5 ). 5.2.2WKBAnalysisofTunneling TheWKB[ 125 ]methodiswidelyusedtondsemiclassicalapproximationsofthequantumwavefunctioninboththeclassicallyallowedandtheforbiddenregionsofquantummechanicsproblemsin1D2.Notethatalthoughourproblemisin2D,thepotentialiscentralwhichallowsaseparationofvariables,andweendupwithaneffectively1Dproblemintheradialdirection.Wecarryoutanexpansionoftheradial 2Generalizationstomultidimensionalproblemsispossiblebutisusuallymoreinvolved. 111

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wavefunctionintermsofasmallparameterrelatedtothePlanck'sconstantas[ 126 ] 2=~2 2M,f(r)eS0(r)=+S1(r)+O(),df drS00 +S01f(r),d2f dr2S000 +S001+)]TJ /F5 11.955 Tf 6.67 -1.59 Td[(S00 +S012. (5) SubstitutingtheperturbativeexpansioninEq.( 5 )intotheSchrodingerequationstatedinEq.( 5 ),weget O(1):)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(S002=Ve(r))]TJ /F5 11.955 Tf 11.96 0 Td[(E0,O():S000+2S00S01+S01 r=0. (5) Theprobabilityoftunnelingintotheclassicallyforbiddenregionofthepotentialisthengivenby )-278(=Ae)]TJ /F11 7.97 Tf 6.59 0 Td[(B=~,(5) wherethetunnelingexponentBisidentiedwiththeleadingordertermS0intheWKBexpansionabove,andAisaprefactortobedeterminedseparately.Therefore, B=Zrfrip 2M(Ve(r))]TJ /F5 11.955 Tf 11.96 0 Td[(E0),(5) whereriandrfaretheclassicalturningpoints.ThemostfamoushistoricalapplicationofthisexpressionwastotheproblemofalphadecaybytunnelinginanucleusworkedoutbyGamow. Theexactevaluationoftheturningpointsandalsotheintegralitselfhastobecarriedoutnumericallybecauseofthecomplicatedformofthevortexpotential(andtheaddedcentrifugalbarrierterm),butitispossibletoanalyticallyestimatethisusinganapproximateformofthevortexpotential.Webeginbynoticingthathigherratesoftunnelingareexpectedwhenthetrapisbeingrotatedclosetometastability,becausethe 112

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potentialbarrierheightislower(andalsonarrower)atsuchfrequencies.Letusfocusonthisregimebydeningarelevantsmallparameter,==m)]TJ /F3 11.955 Tf 12.31 0 Td[(1.Thisimpliesthattheturningpoints,andthetunnelingphenomenonitself,aresituatednearthecenterofthetrap,andsoitisinstructivetoexpandthevortexpotentialinEq.( 5 )intheradialcoordinateas Vv(r)=V01)]TJ /F5 11.955 Tf 14.89 8.09 Td[(r2 R23=2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(3 5(1+)1)]TJ /F5 11.955 Tf 14.89 8.09 Td[(r2 R25=2'V01+3 5(1+)+3 2r2 R2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(3 4r4 R4+Or6 R6. (5) ThemagneticeldcontributesatermquadraticinrtotheeffectivepotentialinEq.( 5 )andsoforthezeroethangularmomentumstate,theeffectivepotentialisstillaquartic,writtenas(ar2)]TJ /F5 11.955 Tf 12.4 0 Td[(br4),whereaandbarepositiveconstantsthatdependonvarioustrapparameters.Thecoefcientofthequadraticterm,a,isrenormalizedbythecyclotronfrequencyormagneticeld,anddominatesthecontributionforhighermagneticelds.Thisformofthepotentialisvalidforsmallradialdistances,hasthestructureofabarrierthroughwhichtunnelingcantakeplace,andletsuscalculateanapproximateexpressionforthetunnelingprobabilityinaclosedformquitesimply,atleastforthezeroangularmomentum(l=0)channel. Inthissimpletunnelingcalculation,weassumetheinitialenergyofthevortextobethesameasitspotentialenergyattheorigin.Theclassicalturningpointsarethentheoriginitself,ri=0,andthepointdeterminedbythetrapparameters,rf=p a=b.ThetunnelingexponentinEq.( 5 )isthengivenby B'Zp a b0drp 2M(ar2)]TJ /F5 11.955 Tf 11.95 0 Td[(br4)=p M(2a)3=2 3b.(5) Thedependenceofthisestimateforthetunnelingprobabilityexponentonthemagneticeldisofparticularinteresttous.RecallfromtheformoftheeffectivepotentialinEq.( 5 ),andtheshortdistanceexpansionofthepotentialinEq.( 5 ),thatthe 113

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exactformsofthecoefcientsaandbinourmodelpotential,aregivenas a=3 2V0 R2+1 8M!2c,b=3 4V0 R4.(5) Usingtheseexpressions,wendtheapproximateexpressionsforthetunnelingprobabilityexponentinvariouslimitingcasesfromEq.( 5 )as B'8 9p 2MV0R3 21+O!2c !2t,(5) forlowmagneticelds,and B'1 12p 2MV0Rm!2cR2 V03 21+O!2t !2c,(5) forthehigheldlimit.Here,!t'p (V0=MR2)issomecharacteristicfrequencythatdependsontrapparametersandisinfactsimilartotheprecessionfrequency!mfoundinEq.( 5 ). Inthiscalculation,theeldissaidtobelow(high)ifthecyclotronfrequency!cismuchsmaller(bigger)thanthecharacteristicfrequency!t.Theseresultssuggestthatthetunnelingcanbeappreciableinlowermagneticelds,andisgivenbyathree-halveslawin,theshiftinangularfrequencyoftraprotationfromtheonsetofmetastability;whereasthetunnelingforhighmagneticeldsisexponentiallysuppressedasapowerofthemagneticeld.Thehigheldresultisconsistentwithourexpectationthattheparticlemovesinequipotentiallinesathigheldandthereisnegligibletunnelingintheradialdirection.Theloweld,particularlythezeroeld,resultisusefulinthatitestablishesanupperboundonthetunnelingprobability.Weshouldremarkherethatveryhighmagneticelds,wouldactuallychangethecharacterofthepotentialbarriersothatthesmalldistanceexpansionasaquarticpotentialisnolongervalid,andtheabovecalculationbreaksdown. Sofarwehaveaddressedtunnelinginthel=0angularmomentumchannel.Thisiswhereweexpectmaximumtunnelingprobability,asthecentrifugalbarriersuppresses 114

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tunnelingforhigherangularmomentumstates.Itisinfactpossibletoestimateacriticalvalueoflatwhichtheeffectivepotentialchangesitsshapeandnolongerpossessesametastableminimumtotunnelfrom.Thisisobtainedusingthemodelquarticeffectivepotentialandacriterionforthenumberofrootsofacorrespondingcubicequation.Itisexpressedintermsofthepotentialcoefcientsas lcritical=r 2 271 ~r ma3 b,(5) wherethesquarebracketsaboveimplythe`greatestintegerlessthanorequaltothequantitywithinthebrackets.Fortypicalvaluesofthetrapparameters,thisactuallydoesworkouttol=0furtherconrmingthatthisistheonlychannelwhichseesanytunnelingofinterest. TheWKBestimateshelptoestablishanupperboundonthetunnelingratewithazero-orlow-magneticeldestimate,andalsoconrmourexpectationthattunnelingissuppressedforhighermagneticelds.Thisisallthereistoaproblemoftunnelingofachargedparticleinamagneticeldandinthepresenceofanisotropicpotential.However,sofarwehavenottakenintoaccountthenonlocalormemoryeffectsintheactionofthevortex,whichresultfromthecouplingofitsmotiontophonons.TheproblemoftunnelingwhentheactionhasnonlocaltermsinitwillrequiretoolsbeyondtheWKBdescription,whichwethendescribeinthenextsubsection. 5.2.3TheMethodofInstantonsorBounceTrajectories Anoft-usedapproachtocalculatetheprobabilityofaquantumtunnelingeventistocomputetheactionofthesystemalongitsclassicalpathbetweentheturningpoints.Quantumuctuationsaroundthisclassicalpath(foundbyaminimizationoftheaction),computedinasaddlepointapproximation,thenhelpdeterminetheprefactorAintheexpressionfortunnelingprobabilitystatedearlierinEq.( 5 ).ThefactorBintheexponenthereisthenjusttheactioncorrespondingtotheclassicalpath.ThismethodhasbeendescribedbyColeman[ 127 ]andotherworkersinaeldtheoreticcontext, 115

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whohaveapplieditelegantlytosolvethetunnelingproblemindoublewellpotentials.Thetechniquebeencalledthemethodofinstantons,aftertheclassicalsolutionsofeldtheories,whicharelocalizedinspace-timelikesolitonsinspace.Theclassicalpathbetweentheturningpointsinthetunnelingproblemisalsosometimescalledabouncetrajectory,astheparticleisimaginedtobouncebackandforthbetweenthesepoints. Weintroducethismethodforasimpleactionforaparticleina1Dpotential,wheretheLagrangiancanbewrittenasL=m_x2=2)]TJ /F5 11.955 Tf 12.47 0 Td[(V(x).Letusnowconsiderthisactionandtheconsequentequationsofmotioninimaginaryoreuclideantime.Thiseffectivelymapstheproblemtoonewithanegativeorinvertedpotential,andthepotentialbarrier(classicallyforbiddenregion)betweentheclassicalturningpointsbecomesapotentialwell(classicallyaccessibleregion).TheEuclideanactionbetweenthesetwoturningpoints(chosentobexiandxf)isthencomputedas SE=Z0dh1 2Mdx d2+V(x)i,(5) withtheboundaryconditionschosenasx()=xi,andx(0)=xf.TheEuler-Lagrangeequationsyieldthetotalenergyasanintegralofmotion 1 2Mdx d2)]TJ /F5 11.955 Tf 11.96 0 Td[(V(x)=E,(5) whereE=V(xi)=V(xf).Thisreferenceenergylevelcanbesettozerowithoutanylossofgenerality,E=0.Thisenergyconservationrelationcanbeusedtoderiveaquadratureconditionas dx d=p 2MV(x),(5) whichcanthenbeusedtore-expresstheactioninEq.( 5 )asanintegraloverspaceinsteadoftime,as SE=Zxfxidxp 2MV(x).(5) Thusforanactionwiththissimpleform,weobtainthesameexpressionforthetunnelingprobabilityBasbytheWKBmethodseeninEq.( 5 )withenergysettozero. 116

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Theroleofthemagneticeldintunnelingbecomesintuitivelyapparentfromthisinstantonpicture.Themagneticeldtendstocurveanyclassicalpaththatmighttrytoproceedinaradiallyoutwarddirection,thussuppressingtunnelingunderanisotropicbarrier.Thiseffectbecomesverypronouncedinthehighmagneticeldlimit,wheretheparticlejustmovesalongcircularequipotentiallinesinacentralpotential.Thissuggests,discouraginglyforus,thatthesemiclassicalprobabilityoftunnelingtendstozerointhehighmagneticeldlimit. Thefullactionforavortexcanhoweverhavenonlocal(intime)inertialtermsasseeninChapter4.Thisactioncanthenbewrittenas S=Z10d1 2M_r2+1 2Mr2_2+V(r)+Z10dZ10d0()]TJ /F4 11.955 Tf 11.96 0 Td[(0)_x()_x(0), (5) whereMisthemassobtainedbyalocalexpansionoftheinertia,andtheresidualnonlocaleffectsaretakenintoaccountinthekernel()]TJ /F4 11.955 Tf 13.28 0 Td[(0).Thisactioncouldinprinciplebecalculatednumerically,butitwouldbeacomputationallyintensiveprocedure.OnewouldrstneedtonumericallysolvethenonlocalEuler-Lagrangeequationin2D(theproblemmaynotbeseparableoncethenonlocaltermsareincluded),andthenintegratethemtondtheaction.Ourhopeisthateveninahighmagneticeld,thenonlocaltermscouldservetodistorttheperfectlycircularclassicalpathstoenhancetunneling. Itispossibletoconsidersimplerlimitingscenariosoftheabove,e.g.asufcientlyslowlymovingvortexforwhichthenonlocaltermsaresmallandcanbetreatedperturbatively.Onethenjustconsiderstheclassicalpathforaparticleinamagneticeld,whichisaseparableproblem.Theangularvariableiscyclicandcanbeintegratedoutasshownin?.Thispathdeterminedwithoutthenonlocaleffectsisthenusedto 117

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calculatethefullactionas Sc=Z10d1 2M_rc2+V(rc)+1 8M!2cr2c+L2 2Mr2c+Z10dZ10d0()]TJ /F4 11.955 Tf 11.96 0 Td[(0)_xc()_xc(0). (5) 5.3ExperimentalDiscussion Asmentionedearlier,therearetwobasicexperimentalapproachesthathavebeenusedtocreateandstudyvorticesinaBEC.Therst,carriedoutinJILAin1999[ 36 ],usesamixtureoftwohypernecomponentsof87Rb,spinningoneuprespecttotheotherbyapplyinganexternalelectromagneticbeam.ThesecondwasperformedinENS,Parisin2000[ 37 ].Wewillusethelatterasourprototypeforourdiscussiononvortexdynamicsandtunnelinginthissection.Theytrappedaroundamillionatoms(N106),thetrappingfrequencyinthedirectiontransversetotherotationisoftheorderof100Hz(whichimpliesaharmonicoscillatorquantumlengthofmicronsaho10)]TJ /F6 7.97 Tf 6.58 0 Td[(6m,andthescatteringlengthisusuallytunedtoafewnanometers,as10)]TJ /F6 7.97 Tf 6.58 0 Td[(9m.ThissuggeststhattheparameterNas=aho103isquitehigh,andassuresusthatthetypicalcondensateinwhichvorticesarestudiedinthelabisindeeddescribedbytheThomas-Fermi(TF)approximation.Someotherquantitiesofinterestthatcanbeestimatedbasedontheseparametersare:sizeorradiusofthecondensate,R10aho10)]TJ /F6 7.97 Tf 6.59 0 Td[(5m;healinglengthorcoresizeofavortex,100nm;themassofavortexasthenumberofatomsinthecoreofthevortex,M103m10)]TJ /F6 7.97 Tf 6.58 0 Td[(22kg;thecriticalvelocityoftraprotationatwhichavortexisrstnucleated,c100Hz;thecyclotronfrequencycorrespondingtotheMagnusforceatthecenterofthetrap!c106Hz. Thesequantitiescanbeusedtoreachseveralimportantconclusions.Mostimportantly,wecancalculateanupperboundonthetunnelingprobability.Weexpectthehighestratesoftunnelingwhenthereislittleornomagneticeld,whenthetunneling 118

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exponentBisgivenbyp 2mV0R3 2,V0beingthevortexpotentialatthecenterofthetrap,andthedetuningofthetraprotationfrequencyfromthemetastablefrequency.Itturnsoutthatforsmallenoughdetuning(10)]TJ /F6 7.97 Tf 6.59 0 Td[(4),thetunnelingexponentcanbeoftheorderofPlanck'sconstant,B~,whichisnecessaryforappreciabletunneling(toohighamagnitudeoftheexponentwouldmeansuppressedtunneling).Soifourprototypicaltrapisrotatedveryclosetothemetastablefrequencyofaround100Hz,andextremelyclosetoit,itispossibletoattainestimatedtunnelingtimesoftheorderofseconds! Howeverthisloweldestimateisnotrealistic,becauseweareinfactalwaysinahigheldlimitinthekindofcondensatesintheTFregimethatweareconsidering,andalsowhicharerelevanttomostexperimentsonBECinthelaboratory.Intuitively,thisisbecausetheMagnusforceonavortexdependsonthecondensatedensity,whichisalwayshighintheTFregime.Itisalsopossibletoseethisfromquantitativeestimates,bycomparingthetwotimescalesinvolvedinthemotionofthevortex.Theprecessionfrequencyisroughlygivenby!pmj(1)]TJ /F3 11.955 Tf 12.87 0 Td[(=m)j,andweareinterestedintraprotationfrequenciesinthemetastableregimem<
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AnotherinterestingpossibilityispinningcentersforvorticesinaBEC.Arecentworktreatsthevortexpotentialresultingfromtwopinningcentersasamodeldoublewell[ 118 ].Fromamoregeneraltheoreticalperspective,wehaverealizedthattheproblemofmacroscopicquantumtunnelingforasinglevortexinaBECintheTFregime,isanalogoustothetunnelingofanelectroninahighmagneticeld.Thishasbeenelegantlysolvedforsomemodelpotentials[ 128 ],butnogeneralprescriptionexistsforsuchacalculation.Alsotheverynotionsofvortexmassandtheexactnatureofitsdynamicsaredebatedtopics.Wehopetohaveprovidedsometheoreticalinsightsintothislongstandingproblem. 120

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CHAPTER6CONCLUSION Inthiswork,wehavelookedattwodistinctrealizationsofthesuperuidphase-insolidhelium,andinadiluteBose-Einsteincondensateinatrap.Inparticular,wehavetriedtoexplaintheapparentlysuperuidresponse(NCRI,asobservedinatorsionaloscillatorexperiment[ 23 ])insolidheliumasaneffectofdefectslikedislocationsandgrainboundaries.InBECs,welookatthestabilityanddynamicsofvorticesinarotatingtrap,andestimatetheirtunnelingprobability. Wehaveconstructedamodelforasupersolidbasedonsuperuidityinducedalonganetworkofdislocations.StartingfromaLandautheoryforthebulksolid,weareabletosystematicallyderiveaone-dimensionalequationdescribingsuperuidityalongasingledislocation.Wethenconsideranetworkofdislocationsandtheeffectofoverlapofthesestrandsof1Dsuperuid,andshowthatitispossibletorealizeabulksuperuidorderalongthisnetworkstructurewithanonsettemperaturethatdependsonthesuperuidcouplingbetweenneighboringnodesinthisnetwork.ThisisanalogoustoanarrayofJosephsonjunctionsoranXYmodelwithrandombonds.Thiseffectshouldbeobservableinsuperconductors,whereitshouldbepossibletocalculatethevariousconstantsintheLandautheoryintermsofmicroscopicquantities(somethingwecannotdoforsolid4He).Oneofourmorestrikingresultsisthesensitivedependenceofthetransitiontemperatureforthedislocationnetworkonthedislocationdensity(i.e.,thesamplequality).Grainboundariesascollectionsofdislocationsarealsosimilarlyexpectedtoenhancesuperuiditylocally.WederiveadependenceofthecriticaltemperatureTcofagrainboundarysuperuidontheangleofthegrainboundaryintwooppositelimits.WealsoanalyzetheKosterlitz-Thoulesspropertiesofagrainboundarysuperuid,andshowthattheinherentanisotropyinthegrainboundaryisirrelevantasfarasthesuperuidityisconcerned. 121

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Theprocedureforestimatingtheshiftinsuperuidcriticaltemperaturecausedbyanedgedislocationledustothelongstandingquantumproblemofatwo-dimensionaldipolepotential.ThewavefunctionsandthespectrumareobtainedbynumericallysolvingtheSchrodingerequationwiththe2Ddipolepotential,andalsovariationallyinthecaseofthegroundstate.Wendthattheresultsobtainedfromthedifferentmethodsareconsistentandcomparefavorablywithpreviousestimatesintheliterature.Wealsodiscoverasimplepatterninthespectrum,wherethequantumnumberofthestatevariesinverselyasitsenergy(n)]TJ /F6 7.97 Tf 6.58 0 Td[(1),whichcanbejustiedfromsemiclassicalconsiderations.Certainfeaturesofthespectrumandwavefunctionsareyettobeexplainedandmightprovidescopeforfutureinvestigation.Forexample,thestatisticsofthelevelspacingscouldpossiblybeasignatureofquantumchaos.Thegroundstateofthequantumdipoleproblemisrelatedtothebindingenergyofelasticimpuritiestoedgedislocationlines.Thisisimportantinthecontextofheliumforexample,asthebindingof3Heimpuritiespresentinsolid4Hecanhaveaneffectonthestiffnessandheatcapacityofsolidhelium.Thisisforexamplethebasisofthedislocation-pinningpicturecommonlysuggestedtoexplaintheshearstiffeningatlowtemperaturesseenbyBeamishetal.[ 28 ],andalsoforaSchottkyanomalyinthespecicheat.Inthedislocation-basedsuperuidmodelweintroduce,thelinearizedLandauequationisisomorphictotheSchrodingerequation[Eq.( 2 )].Thegroundstateenergyanditswavefunctiondeterminedinthiscalculationprovideaninputtothecalculationoftheshiftinthesuperuidtransitiontemperature,andtherenormalizedcoefcientg[asdenedinEq. 2 ]ofthenonlinearterminthe1DLandaumodelforsuperuidalongasingledislocationline,respectively. WedevelopaweaklynonlinearanalysistoobtainanapproximatesolutionoftheLandauequationinthepresenceofsomedefectpotential.Thisisintermsofthesolutionofthelinearizedproblem,whichisnothingbutaSchrodingerequation.Thisisexpectedtoholdnearthethresholdofatransitionwhentheorderparameteris 122

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small,andtheextentofproximitytothethresholdprovidesthesmallparameterneededforsuchaperturbativeexpansion.WedemonstratetheefcacyofthismethodbycalculatinganalyticcorrectionstotheorderparameterinthepresenceofaCoulombpotential.Wealsoshowhowthisperturbationcanbedoneinamorecontrolledmannerbyastretchedasymptoticmethodthateliminatestheseculartermsfromanaiveperturbationtheory.Inessence,thisformalismcanbeappliedtoanynonlinearSchrodingerequationwhereasmallparameterthatgoestozeroatthebifurcationpointcanbeidentied.Forexample,thisletsuscalculateperturbativelythesolutionoftheGrossPitaevskiiequationwhenthechemicalpotentialisclosetothegroundstateenergyofthetrappingpotential.Thiscorrespondstothethresholdvaluerequiredtoloadatomsintothetrap.WeshowthatthismethodpredictsthecondensatewavefunctioninaBECquitewellforsmallcondensates. WeconsidertheproblemofafewvorticesinatrappedBEC.WereviewthewellknownresultsforvortexenergyandstabilityinarotatingtrapintheThomas-Fermilimit.MostextanttheoreticalworkonvorticesinBECinfactusestheTFapproximation,whichisvalidfortypicalexperimentalsituationsinvolvinglargecondensates.Weconsidertheproblemofavortexinasmallcondensatewithinouraforementionedweaklynonlinearapproximation,andderivethecriticalangularvelocityforvortexformationinsuchasystem.ThishasbeenobtainedpreviouslybyLinnandFetter[ 105 ]throughaformallydifferentperturbationtechnqiue.Weconcludethatthisrotationfrequencyislikelytomakethetrapunstable.Asmallcondensateisthusnotanexperimentallyfeasiblesystemforstudyingvortices,especiallyintheirmetastableregime. WethenderivetheexactdynamicsofavortexinaBECintheTFlimit,byusingthefullhydrodynamicalactioninarotatingframeofreference,andintegratingoutthephonon(orspinwave)modesinthesuperuid.Someinterestingconclusionsweestablishare: 123

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Thevortexactionhasatermthatisnonlocalintime.Thiscanbephysicallyinterpretedasaconsequenceofvortex-phononcoupling,whichcausesavortextoeffectivelyinteractwithitselfatsomeretarded/advancedtime. Thisnonlocalinertialtermcanbeapproximatedtoalocalformwhenthevortexmovesslowlycomparedtothesoundspeedinthemedium.Thiscorrespondstothemassofavortexwhichdependslogarithmicallyonsomefrequencyassociatedwiththevortexdynamics Fortypicalexperimentalparameters,thevortexmotionisdominatedbytheMagnusdynamics,whichisanalogoustothemotionofanelectroninhighmagneticeld.Thiscausesthevortextoprecessalongequipotentiallinesofthetrap. Wenextconsidertheproblemofvortextunnelingfromametastablestateatthecenteroftherotatingtrap.OursemiclassicalestimatesbasedonalocalversionoftheactionshowthatahighMagnusforcetendstosuppressthetunnelingintypicalcondensates.WeobtainanupperboundontherateoftunnelingbyconsideringasituationwithouttheMagnusforce,andndthatitwouldbebarelyobservablefortypicaltrapparameters.Itispossiblethatthenonlocaleffectsfromcouplingtophonons,ordissipationfromothersources,wouldenhancetunneling.Thisisworkinprogress.Iffoundplausible,vorticesinaBECcouldbeacandidatesforobservingmacroscopicquantumtunneling(MQT)inadditiontoprototypicalsuperconductingsystemssuchasSQUIDs. 124

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APPENDIXASTRAINFIELDFORANEDGEDISLOCATION Letusconsiderastraightedge-dislocationinanisotropicsolid,withinlinearelasticitytheory.Letthedislocationbealongthepositivez-axisandtheBurger'svectoralongthey-axis.IfbistheBurger'svectorandisthePoisson'sratiothenthecomponentsofthedisplacementeld(inpolarcoordinates)duetothedislocationaregivenby(see,forexample,Refs.[ 43 50 ]) ur=b 2"(=2)]TJ /F4 11.955 Tf 11.96 0 Td[()sin)]TJ /F3 11.955 Tf 18.03 8.09 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2 2(1)]TJ /F4 11.955 Tf 11.96 0 Td[()cosln(r=r0)#,u=)]TJ /F5 11.955 Tf 13.93 8.09 Td[(b 2"(=2)]TJ /F4 11.955 Tf 11.96 0 Td[()cos+sin+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)sinln(r=r0) 2(1)]TJ /F4 11.955 Tf 11.95 0 Td[()#, (A) wherer0isthesizeofthedislocationcore. Thecomponentsofthestraintensorareobtainedfromthedisplacementeldusing, urr=@ur @r,u=1 r@u @+ur r,2ur=@u @r)]TJ /F5 11.955 Tf 13.15 8.09 Td[(u r+1 r@ur @. (A) ThistraceofthestraintensorresultsinEq.( 2 ). 125

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APPENDIXBANALYSISOFALANDAUMODELWITHA1=RPOTENTIAL InthisAppendixwesolveasimpliedversionofthedipolepotential,replacingcos=rbytheattractivetwo-dimensionalCoulombpotentialV(r)=)]TJ /F3 11.955 Tf 9.3 0 Td[(1=r.Thesetwopotentialssharethesamelengthscaling;however,theCoulombpotentialisrotationallysymmetricandthelinearproblemcanbesolvedexactly.ThedetailsoftheperturbationcalculationfollowthegeneralschemedevelopedinSection2.3.Wecomparetheresultsoftheperturbationtheorywithnumericalsolutionsofthenonlineareldequation,andndcloseagreementforawiderangeof. TheenergyeigenvaluesfortheHamiltonian^H=r2?)]TJ /F3 11.955 Tf 11.95 0 Td[(1=raregivenby[ 95 96 ] En=)]TJ /F3 11.955 Tf 31.53 8.09 Td[(1 (2n+1)2,n=0,1,2,...,(B) soE0=)]TJ /F3 11.955 Tf 9.3 0 Td[(1,withagroundstateeigenfunction0(r)=p 2=e)]TJ /F11 7.97 Tf 6.59 0 Td[(r.Therelatedlinearoperator[seeEq.( 4 )]is ^L=r2?)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 r+1.(B) Expandingasbefore, =0+1+22+...,(B) wehave ^L0=0,(B) withthesolution 0(r)=A02 1=2e)]TJ /F11 7.97 Tf 6.59 0 Td[(r(B) (notethatwewillignorethez-dependenceinthisAppendix).SubstitutingintothelefthandsideoftheO()equation, ^L1=0)-222(j0j20=A02 1=2e)]TJ /F11 7.97 Tf 6.59 0 Td[(r)]TJ /F5 11.955 Tf 11.95 0 Td[(A302 3=2e)]TJ /F6 7.97 Tf 6.58 0 Td[(3r. (B) 126

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Weleftmultiplythisequationbyp 2=e)]TJ /F11 7.97 Tf 6.59 0 Td[(randintegrateond2rtoobtain0=A0)]TJ /F3 11.955 Tf -425.72 -23.9 Td[((1=2)A30,sothatA0=p 2.SubstitutingbackintoEq. B ,andassuming1hascylindricalsymmetry,weobtainaninhomogeneousequationfor1: )]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 rd drrd1 dr)]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 r1+1=2e)]TJ /F11 7.97 Tf 6.59 0 Td[(r)]TJ /F3 11.955 Tf 11.96 0 Td[(8e)]TJ /F6 7.97 Tf 6.59 0 Td[(3r.(B) Theexplicitsolutionofthisequation(thatdecaysto0forlarger)is 1(r)=ce)]TJ /F11 7.97 Tf 6.59 0 Td[(r+e)]TJ /F6 7.97 Tf 6.59 0 Td[(3r+1 2e)]TJ /F11 7.97 Tf 6.59 0 Td[(r2r+ln(2r)+Z12re)]TJ /F11 7.97 Tf 6.59 0 Td[(t t,(B) wherecisanintegrationconstant.Wesubstitute1(r)intotherighthandsideoftheO(2)equationtoobtain ^L2=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(320)1.(B) Weagainleftmultiplybyp 2=e)]TJ /F11 7.97 Tf 6.59 0 Td[(r,integrateond2randusethefactthat^LisHermitiantoobtainthesolvabilityconditionfortheconstantc,withtheresult c=)]TJ /F3 11.955 Tf 10.5 8.08 Td[(11 12+ 2+ln(2))]TJ /F3 11.955 Tf 13.16 8.08 Td[(3 4ln(3)=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.75887,(B) where=0.577210istheEuler-Mascheroniconstant.Wehavenowexplicitlycalculatedtwotermsintheperturbationexpansion;bothtermsarenonzeroattheorigin,with0(0)=2and1(0)=1 12+ln(2))]TJ /F6 7.97 Tf 13.15 4.71 Td[(3 4ln(3)=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.047478. Wehavesolvedthenonlineareldequationforthe)]TJ /F3 11.955 Tf 9.3 0 Td[(1=rpotentialnumericallyforawiderangeofvaluesof=E)]TJ /F5 11.955 Tf 12.6 0 Td[(E0=E+1,usingtheshootingmethod[ 97 ].TheresultsfortheorderparameterarepresentedinFig. B-1 fortwodifferentvaluesof.Thetwo-termperturbationtheorygivesanexcellentapproximationevenafairlylargevalueof=0.43.Togetasenseoftheefcacyoftheperturbationtheory,wecancalculatetheamplitudeoftheorderparameterattheorigin: (0)=1=20(0)+3=21(0)+...=21=2)]TJ /F3 11.955 Tf 11.96 0 Td[(0.0474783=2+.... (B) 127

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ThisresultisplottedinFig. B-2 ,alongwithournumericalresults.Again,weseetheexcellentagreementbetweenthetwo-termperturbationtheoryandthenumericalresults,evenforrelativelylargevaluesof. FigureB-1. Orderparameter (r)versusr.Thesolidbluelineisthenumericalsolution,thedashedgreenlineistheone-termperturbativeresult,andthedottedredlineisthetwo-termperturbativeresult.Theupperpanelcomparestheresultsfor=0.09,andthelowerpanelfor=0.43.Thetwotermexpansionprovidesanexcellentapproximationtothenumericalresult,evenfor=0.43.ReprintedwithpermissionfromRef.[ 48 ][D.Goswami,K.Dasbiswas,C.-D.YooandA.T.Dorsey,Phys.Rev.B84,054523(2011)],cAPS,2011. 128

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FigureB-2. Orderparameteramplitude (0)asafunctionof.Thecrossesarethenumericalresults,thedashedgreenlineistheone-termperturbativeresult,andthedottedredlineisthetwo-termperturbativeresult.Theupperplotcomparesthenumericalresultswiththeperturbationtheoryforawiderangeof;thetwo-termperturbativeresultprovidesanexcellentapproximationevenforvaluesofaslargeas0.8.Thelowerpanelisaplotof (0)=21=2asafunctionof,whichhighlightstheroleofthesecondordertermintheexpansion.ReprintedwithpermissionfromRef.[ 48 ][D.Goswami,K.Dasbiswas,C.-D.YooandA.T.Dorsey,Phys.Rev.B84,054523(2011)],cAPS,2011. 129

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APPENDIXCANALYSISOFATIME-DEPENDENTMODEL InthisAppendixwegeneralizetheresultsofSection2.3toderiveaone-dimensionaldynamicalmodelforthesuperuid.Forsimplicity,wewillassumethattherearenoconserveddensities,sothedynamicsaredescribedbymodelA(oftenreferredtoastime-dependentGinzburg-Landautheory)intheHohenberg-Halperinclassication[ 98 99 ].Therelaxationalequationofmotionis @ @t=)]TJ /F3 11.955 Tf 9.3 0 Td[()]TJ /F6 7.97 Tf 6.77 -1.8 Td[(0F +(x,t),(C) where)]TJ /F6 7.97 Tf 6.77 -1.79 Td[(0isarelaxationrateandistheuctuatingnoisetermwithGaussianwhitenoisecorrelations,i.e.,h(x,t)i=0andh(x,t)(x0,t0)i=2kBT)]TJ /F6 7.97 Tf 6.77 -1.8 Td[(0(x)]TJ /F17 11.955 Tf 12.77 0 Td[(x0)(t)]TJ /F5 11.955 Tf -429.9 -23.91 Td[(t0).Asbefore,tofacilitatethereductiontoaone-dimensionalmodelweintroducedimensionlessquantities,withatimescale~t=2c=()]TJ /F5 11.955 Tf 11.66 0 Td[(a20B2)=2l2=()]TJ /F5 11.955 Tf 11.66 0 Td[(c).Intermsofthedimensionlessvariables, @ @t=r2 )]TJ /F3 11.955 Tf 11.95 0 Td[([V(r))]TJ /F5 11.955 Tf 11.96 0 Td[(E] )-222(j j2 +(x,t),(C) wherethenoisecorrelationsaregivenby h(x,t)(x0,t0)i=2(kBT=F0)(x)]TJ /F17 11.955 Tf 11.95 0 Td[(x0)(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0).(C) Asbefore,weintroducethesmallparameter=E)]TJ /F5 11.955 Tf 12.79 0 Td[(E0,andintroduce =1=2,z=)]TJ /F6 7.97 Tf 6.58 0 Td[(1=2,t=)]TJ /F6 7.97 Tf 6.58 0 Td[(1,and=3=2~,toobtain ^L=)]TJ /F4 11.955 Tf 9.3 0 Td[(@+@2+)-222(jj2+~.(C) Weagainexpandinpowersof,andatO(1)wehave^L0=0,thesolutionofwhichis 0=A0(,)0(r).(C) 130

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SubstitutingthisintotherighthandsideoftheO()equation,leftmultiplyingby0,andthenintegratingond2r,thesolvabilityconditionyields @A0=@2A0+A0)]TJ /F5 11.955 Tf 11.96 0 Td[(gjA0j2A0+,(C) whereistheone-dimensionaluctuatingnoiseterm(thethree-dimensionaltermwiththetransversedimensionsprojectedout), (,)=Zd2r0(r)~(r,,).(C) Usingthefactthat0isnormalizedtoone,itisstraightforwardtoshowthathasGaussianwhitenoisecorrelations.Undoingthescalings,weobtainournalone-dimensionaltime-dependentGinzburg-Landautheory, @t'=@2z'+')]TJ /F5 11.955 Tf 11.96 0 Td[(gj'j2'+.(C) 131

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APPENDIXDVORTICESINATWO-DIMENSIONALXYMODEL Ingeneral,thephaseinanXYmodelcanbedecomposedintoananalytic,spinwavepartandasingular,vortexpart. (x)=a(x)+v(x).(D) Thecorrespondingphasegradientsorvelocitiesarelongitudinalandtransverserespectively,satisfying,vjjs=ra,rvjjs=0,v?s=rv,r.v?s=0. (D) Thecurlofthegradientofafunctionwilldisappeareverywhereexceptwhereitisnotanalytic,i.e.atthepreciselocationsofthevortices.Foradistributionofseverallinevorticesatpositionsxvallpointinginthez-direction,wecandeneavortexsourcefunction,suchthat rv?s=m(x)X2ev2(x)]TJ /F17 11.955 Tf 11.96 0 Td[(xv)^z. (D) Asimpleanalogycanbeestablishedbetweenthevelocityeldduetoavortexv?s,andamagneticeld,andsothesourcefunctionm(x)islikethecurrentdensitythatproducesthismagneticeld.Aftersomefurthervectorcalculus, r(rv?s)=r2v?s=rm,(D) thetransversevelocityeldfromavortexcanbederivedas v?s(x)=rZd2x0G(x)]TJ /F17 11.955 Tf 11.95 0 Td[(x0)m(x0),(D) whereG(x)]TJ /F17 11.955 Tf 12.17 0 Td[(x0)=)]TJ /F3 11.955 Tf 9.3 0 Td[(2ln)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jx)]TJ /F17 11.955 Tf 12.17 0 Td[(x0j=RistheGreen'sfunctionoftheLaplaceoperatorin2D. 132

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Nowwearereadytousetheexplicitexpressionforthevortex-inducedvelocityobtainedabovetocalculatetheenergyofasystemofvortices.ForatranslationallyinvariantsystemitisusefultoFouriertransforminspace.Wethenobtain Eel=1 2sZd2xZd2x0m(x0)G(x)]TJ /F17 11.955 Tf 11.95 0 Td[(x0)m(x)=1 2sZd2q (2)2m(q).m()]TJ /F17 11.955 Tf 9.3 0 Td[(q) q2. (D) Firstwederivetheenergyfortheparticular(andsimplest)caseofasinglevortex.Foravortexlocatedatxv,theFouriertransformofthevortexsourcefunctionis m(q)=Zd2xeiq.x2ev^z2(x)]TJ /F17 11.955 Tf 11.95 0 Td[(xv)=2ev^zeiq.xv. (D) InsertingthisintoEq.( D ),weobtain, Eel=1 2s)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(ev2Zd2q q2=(ev)2sln(R=a), (D) wheretheultravioletandinfraredcutoffsinfrequencywerechosentocorrespondtothelatticespacingaandsystemsizeRrespectively.NotethatthisexpressionforvortexenergywasderivedbyamoredirectmethodinEq.( 3 )inChapter3.Thetotalenergyforasystemofvorticeswillinvolvetheirenergyofinteractioninadditiontothesumoftheirindividualenergiesofcreation. m(q)=Zd2xeiq.xX2ev^z2(x)]TJ /F17 11.955 Tf 11.96 0 Td[(xv)=2^zXeveiq.xv,Eel=1 2sX,evevZd2q q2eiq.(xv)]TJ /F6 7.97 Tf 6.59 0 Td[(xv)=XE+X,2sevevln(R=r). (D) Theintegralinfrequencyspaceabovehasbeencarriedoutbyrstintegratingovertheangle(whichgivesaBesselfunctionofzeroethorder)andsubsequentlyovertheradialcoordinates,whichgivesalogarithmtoleadingorder.Thecutoffsimposedontheradialintegrationtotakecareofthelogarithmicdivergence,areinversesofthesystemsize(IRorlowfrequencycutoff)andoftheminimumdistancer(UVorhighfrequencycutoff) 133

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betweenneighboringvortices,i.e.theminimumvaluethatthedistancerbetweenapairofvorticeslabeledbyandcanhave.Thisisneededbecausetwovorticesatthesameplacewillinvolveaninniteenergycost: Zdq qZ20deiqrcos()=Z1=r1=RdqJ0(qr) q'lnR r.(D) Theenergyofinteractionbetweentwovorticesgoesasalogarithmofthedistancebetweenthem,andthesignisdeterminedbytheproductoftheircharges.TheexpressionforvortexenergyinEq.( D )canberecastintheform Eel=s)]TJ 7.47 1.67 Td[(Xev2ln(R=a)+2)]TJ 7.47 1.67 Td[(X,evevln(a=r).(D) Thisexpressionisparticularlysuggestiveasitshowsthatthelongwavelengthenergyofanon-neutralsystemoffreevortices,i.eoneforwhichtotalchargeisnotzero,divergeswithsystemsize.Thisiswhyitisalwaysenergeticallymorefavorabletoexcitevorticesinpairsofequalandoppositecharge.Thisalsosuggeststhevortex-antivortexpairbindingpictureoftheBKTtransition.Basedontheabove,wecannowwritedownthetotalHamiltonianforasystemofvorticesbyincludingthecorecontribution,as Hv=T=)]TJ /F4 11.955 Tf 9.29 0 Td[(KX,evevln(r=a)+Ec=T)]TJ 7.47 1.67 Td[(Xev2.(D) WecancomputevariousstatisticalaveragesbasedonthisHamiltonian.Freeenergycalculationsshowthattherenormalizedphasestiffnessinthepresenceofvorticesisdeterminedbythecorrelationfunctionoftransversevelocities.ThiscaninturnberelatedtothevortexsourcefunctionsthroughEq.( D )as hv?s(x)v?s(0)i=Zd2x1d2x2hm(x1)m(x2)irG(x)]TJ /F17 11.955 Tf 11.96 0 Td[(x1)rG(x)]TJ /F17 11.955 Tf 11.95 0 Td[(x2)jx=0.(D) 134

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ThetwoGreen'sfunctionsabovecanbetracedoverandcombinedintoasingleGreen'sfunctionthroughtheircompletenessproperty,as Zd2xrG(x)]TJ /F17 11.955 Tf 11.95 0 Td[(x1)rG(x)]TJ /F17 11.955 Tf 11.96 0 Td[(x2)jx=0=)]TJ /F5 11.955 Tf 9.3 0 Td[(G(x1)]TJ /F17 11.955 Tf 11.96 0 Td[(x2)jx1=0,(D) wherewehaveintegratedbyparts,andusedthepropertyoftheGreen'sfunction.NowweFouriertransformtofrequencyspace,andusethisidentityofGreen'sfunctiontoshowthat Zd2xhv?s(x).v?s(0)i=hm(q).m()]TJ /F17 11.955 Tf 9.3 0 Td[(q)i q2.(D) Letusnowevaluatethecorrelationofsourcefunctionsusingtheirdenitioninfrequencyspace,hm(q).m()]TJ /F17 11.955 Tf 9.29 0 Td[(q)i=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(22P6=heveveiq.(xv)]TJ /F6 7.97 Tf 6.58 0 Td[(xv)i.Inthelongwavelengthlimit,wecanexpandinsmallqandignorehigherordertermsinq.Alsoifweconsideravortexgaswithlowfugacity,thecongurationthathasthehighestordercontributiontothestatisticalaverageandsatisesthechargeneutralitycondition,istheonecorrespondingtoasinglepairofvorticeswithunitcharge.So hm(q).m()]TJ /F17 11.955 Tf 9.3 0 Td[(q)i=)]TJ /F16 11.955 Tf 9.29 9.69 Td[()]TJ /F3 11.955 Tf 5.48 -9.69 Td[(22q2X6=hevev(xv)]TJ /F17 11.955 Tf 11.95 0 Td[(xv)2i+O(q4)')]TJ /F16 11.955 Tf 28.56 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(22q2X6=(xv)]TJ /F17 11.955 Tf 11.96 0 Td[(xv)2e)]TJ /F6 7.97 Tf 6.59 0 Td[(2Ec=T)]TJ /F10 7.97 Tf 6.59 0 Td[(Kln(jxv)]TJ /F6 7.97 Tf 6.59 0 Td[(xvj=a). (D) Nowwegotothecontinuumlimitandsumoverallpossiblepositionsofthetwovorticestondthat hm(q).m()]TJ /F17 11.955 Tf 9.3 0 Td[(q)i q2=43e)]TJ /F6 7.97 Tf 6.59 0 Td[(2Ec=TZdr ar a3)]TJ /F6 7.97 Tf 6.59 0 Td[(2K.(D) Thisresultdescribesthepaircorrelationforvortexsourcefunctionsinthelimitoflongwavelengthandlowfugacity(validforlowtemperatureswherethevortexdensityisdilute),andisusedforobtainingtherenormalizedstiffnessinEq.( 3 ). 135

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BIOGRAPHICALSKETCH KinjalDasbiswaswasborninKolkata(formerlyCalcutta)ineasternIndiaandwaseducatedattheIndianInstituteofTechnologyinKanpur.Hemanagedtosurviveaveyearintegratedmasters'programinphysicsthereandjoinedtheDepartmentofPhysicsattheUniversityofFloridainthefallof2007.ThesunnyclimateandwarmpeopleofFloridahelpedhimsettledowntolifeherewhilehewasponderingonthemysteriesofmatteratverylowtemperatures.HeworkedintheareaoftheoreticalcondensedmatterphysicswithProf.AlanT.Dorsey,havingjoinedhisgroupin2008.HegratefullyreceivedhisPhDdegreefromthisinstituteinthesummerof2012. 143