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Phenomenology of Supersolids

Permanent Link: http://ufdc.ufl.edu/UFE0024203/00001

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Title: Phenomenology of Supersolids
Physical Description: 1 online resource (128 p.)
Language: english
Creator: Yoo, Chi-Deuk
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dislocations, hydrodynamics, superfluid, supersolid, viscoelasticity, vortices
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract: We investigate the phenomenological properties of supersolids - materials that simultaneously display both crystalline order and superfluidity. To explain the recent observation in the torsional oscillator experiments on 4He solid by Kim and Chan we adopt a viscoelastic solid model which is characterized by a frequency-dependent complex shear modulus. In this model, we found that a characteristic time scale which accounts for dissipation in solids grows rapidly as the temperature is reduced, and results in a decrease in the resonant period and a peak in the inverse of Q-factor. We also briefly discuss the possible relation between the torsional oscillator results and the anomalous increase of shear modulus obtained by Day and Beamish. In a related study, we employ a variational principle together with Galilean covariance and thermodynamic relations to obtain the non-dissipative hydrodynamics and an effective Lagrangian density for supersolids. We study the mode structure of supersolids by calculating the second and fourth sound speeds due to defect propagation. We also calculate the density-density correlation function of a model supersolid using the hydrodynamics of Andreev and Lifshitz, and propose a light scattering experiment to measure the density-density correlation function (which is related to the intensity of scattered light). We find that the central Rayleigh peak of the defect diffusion mode of a normal solid in the density-density correlation function splits into an additional Brillouin doublet due to the longitudinal second sound modes in supersolid phase. Finally, we study the dynamics of vortices and dislocations in supersolids by using the derived Lagrangian for supersolids. We obtain the effective actions for vortices and dislocations in two-dimensional isotropic supersolids emphasizing the differences from the dynamics in superfluids and solids. As a result we obtain the frequency-dependent inertial masses for slowly moving vortices and dislocations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chi-Deuk Yoo.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dorsey, Alan T.

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Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024203:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024203/00001

Material Information

Title: Phenomenology of Supersolids
Physical Description: 1 online resource (128 p.)
Language: english
Creator: Yoo, Chi-Deuk
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dislocations, hydrodynamics, superfluid, supersolid, viscoelasticity, vortices
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We investigate the phenomenological properties of supersolids - materials that simultaneously display both crystalline order and superfluidity. To explain the recent observation in the torsional oscillator experiments on 4He solid by Kim and Chan we adopt a viscoelastic solid model which is characterized by a frequency-dependent complex shear modulus. In this model, we found that a characteristic time scale which accounts for dissipation in solids grows rapidly as the temperature is reduced, and results in a decrease in the resonant period and a peak in the inverse of Q-factor. We also briefly discuss the possible relation between the torsional oscillator results and the anomalous increase of shear modulus obtained by Day and Beamish. In a related study, we employ a variational principle together with Galilean covariance and thermodynamic relations to obtain the non-dissipative hydrodynamics and an effective Lagrangian density for supersolids. We study the mode structure of supersolids by calculating the second and fourth sound speeds due to defect propagation. We also calculate the density-density correlation function of a model supersolid using the hydrodynamics of Andreev and Lifshitz, and propose a light scattering experiment to measure the density-density correlation function (which is related to the intensity of scattered light). We find that the central Rayleigh peak of the defect diffusion mode of a normal solid in the density-density correlation function splits into an additional Brillouin doublet due to the longitudinal second sound modes in supersolid phase. Finally, we study the dynamics of vortices and dislocations in supersolids by using the derived Lagrangian for supersolids. We obtain the effective actions for vortices and dislocations in two-dimensional isotropic supersolids emphasizing the differences from the dynamics in superfluids and solids. As a result we obtain the frequency-dependent inertial masses for slowly moving vortices and dislocations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chi-Deuk Yoo.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dorsey, Alan T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024203:00001


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Iamgreatlyindebtedtomyadvisor,ProfessorAlanT.Dorsey,forhisguidance,encouragement,andpatiencehedemonstratedthroughoutmywork.Withouthissupportthisworkwouldnothavebeenpossible.IwouldliketothankProfessorP.J.Hirschfeld,ProfessorY.-S.Lee,ProfessorK.Matchev,ProfessorM.W.Meisel,ProfessorS.R.Phillpot,andProfessorA.Roitbergforvaluablediscussionandsupport.IalsothankProfessorM.H.W.ChanofthePennsylvaniaStateUniversityandProfessorH.KojimaoftheRutgersUniversityforsharingtheirvaluableexperimentaldata.FinallyIthankmyparents,Hae-SeunYooandHee-SookYoo,fortheirunagginginterest,support,andencouragement.MostofallIwanttothankmydearestwife,Sae-il,andchildren,DanandSeul,forstandingbesidemewithendlessloveandtrust. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTIONTOSUPERSOLIDS ....................... 12 1.1HistoryofSupersolids ............................. 12 1.2TorsionalOscillatorExperimentson4HeSolidbyKimandChan ...... 13 1.3RecentTheoreticalandExperimentalWorks ................. 15 2VISCOELASTICSOLIDS:ALTERNATIVEEXPLANATIONOFNCRI .... 28 2.1EquationofMotionfortheTorsionalOscillator ............... 28 2.2PropertiesofViscoelasticSolidsunderOscillatoryMotion ......... 30 2.2.1InniteCylinder ............................. 32 2.2.2FiniteCylinder ............................. 33 2.2.3InniteAnnulus ............................. 34 2.3TorsionalOscillatorwithViscoelasticSolids ................. 34 2.4PossibleConnectionbetweenAnomaliesinShearModulusandNCRI ... 37 3NON-DISSIPATIVEHYDRODYNAMICSOFAMODELSUPERSOLID .... 49 3.1VariationalPrincipleinSupersolids ...................... 49 3.1.1IntroductiontotheVariationalPrincipleinContinuumMechanics 49 3.1.2IsotropicSupersolids .......................... 52 3.1.3AnisotropicSupersolids ......................... 58 3.1.4QuadraticLagrangianDensityofSupersolids ............. 62 3.2CollectiveModesandtheDensity-DensityCorrelationFunction ...... 66 4DISSIPATIVEHYDRODYNAMICSOFAMODELSUPERSOLID ....... 72 4.1AndreevandLifshitzHydrodynamicsofSupersolids ............. 73 4.2Density-DensityCorrelationFunctionanditsDetection ........... 77 4.2.1NormalFluidsandSuperuids ..................... 78 4.2.2NormalSolidsandSupersolids ..................... 79 5DYNAMICSOFTOPOLOGICALDEFECTSINSUPERSOLIDS ........ 85 5.1VortexDynamics ................................ 86 5.2DislocationDynamics .............................. 93 5

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.................................... 101 APPENDIX ACALCULATIONOFBACKACTIONTERMS .................. 105 A.1InniteCylinder ................................. 105 A.2FiniteCylinder ................................. 106 A.3InniteAnnulus ................................. 109 BVARIATIONALPRINCIPLEINSUPERSOLIDSWITHTHEROTATIONALVELOCITYOFSUPERCOMPONENTS ..................... 111 CSTATICCORRELATIONFUNCTIONSOFISOTROPICSUPERSOLIDS ... 114 DKUBOFUNCTIONSANDCORRELATIONFUNCTIONS ........... 115 ECALCULATIONOFTHEDENSITY-DENSITYCORRELATIONFUNCTION 117 FDERIVATIONOFANEFFECTIVEACTIONFOREDGEDISLOCATIONS 119 REFERENCES ....................................... 124 BIOGRAPHICALSKETCH ................................ 128 6

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Table page 2-1Fittingparameters .................................. 48 7

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Figure page 1-1Resonantperiodchangeintemperature ....................... 20 1-2Resonantperiodchangeintemperatureforvariousconcentrationof3Heimpurities ....................................... 21 1-3Resonantperiodandtheamplitudeofoscillationofannularcell(panelA)andofblockedannularcell(panelB)asafunctionoftemperature .......... 22 1-4Annealingeectintheresonantperiod ....................... 23 1-5AnnealingeectintheinverseofQ-factor ..................... 24 1-6Specicheatpeaksof4Hesolidwithdierentconcentrationsof3Heimpurities 25 1-7Shearmodulusof4Hesolidasafunctionoftemperature ............. 26 1-8Shearmoduluschangeforvariousconcentrationsof3Heimpurities ........ 27 2-1SchematicillustrationofTOandgeometryofatorsioncell ............ 39 2-2Eectivemomentofinertiaofaninnitecylinderofviscoelasticsolidasafunctionofthedrivingfrequency! 40 2-3Eectivedampingcoecientofaninnitecylinderofviscoelasticsolidasafunctionofthedrivingfrequency! 40 2-4Displacementvectorinahalfcyclefor!1=E. ................. 41 2-5Displacementvectorinahalfcyclefor!=3=E. ................. 41 2-6Displacementvectorinahalfcyclefor!=4=E. ................. 42 2-7Eectivemomentofinertiaofanitecylinderofviscoelasticsolidasafunctionofthedrivingfrequency!with=E=1=100 ................... 42 2-8Eectivedampingcoecientofanitecylinderofviscoelasticsolidasafunctionofthedrivingfrequency!with=E=1=100 .............. 43 2-9Eectivemomentofinertiaofaninniteannulusofviscoelasticsolidasafunctionofthedrivingfrequency!with=E=1=1000 .............. 43 2-10Eectivedampingcoecientofaninniteannulusofviscoelasticsolidasafunctionofthedrivingfrequency!with=E=1=1000 .............. 44 2-11F(x)ofnitecylinder ................................ 44 2-12F(x)ofinniteannulus ............................... 45 8

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........... 45 2-14InverseofQ-factoroftheblockedcapillarysampleofBeCuTO ......... 46 2-15ResonantperiodoftheannealedblockedcapillarysampleofBeCuTO ..... 46 2-16InverseofQ-factoroftheannealedblockedcapillarysampleofBeCuTO .... 47 2-17ResonantperiodoftheconstanttemperaturesampleofBeCuTO ........ 47 2-18InverseofQ-factoroftheconstanttemperaturesampleofBeCuTO ....... 48 4-1Brillouinspectrumofliquidargon .......................... 82 4-2Brillouinspectraof4Hesuperuid .......................... 82 4-3Density-densitycorrelationfunctionsofisothermalandisotropicnormalsolidsandsupersolids) .................................... 83 4-4SplittingoftheRayleighpeakduetothedefectdiusionmodeofanormalsolidintotheBrillouindoubletofthesecondsoundmodes ............ 84 5-1Cutforanedgedislocation ............................. 100 9

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Weinvestigatethephenomenologicalpropertiesofsupersolids-materialsthatsimultaneouslydisplaybothcrystallineorderandsuperuidity.Toexplaintherecentobservationinthetorsionaloscillatorexperimentson4HesolidbyKimandChanweadoptaviscoelasticsolidmodelwhichischaracterizedbyafrequency-dependentcomplexshearmodulus.Inthismodel,wefoundthatacharacteristictimescalewhichaccountsfordissipationinsolidsgrowsrapidlyasthetemperatureisreduced,andresultsinadecreaseintheresonantperiodandapeakintheinverseofQ-factor.WealsobrieydiscussthepossiblerelationbetweenthetorsionaloscillatorresultsandtheanomalousincreaseofshearmodulusobtainedbyDayandBeamish. Inarelatedstudy,weemployavariationalprincipletogetherwithGalileancovarianceandthermodynamicrelationstoobtainthenon-dissipativehydrodynamicsandaneectiveLagrangiandensityforsupersolids.Westudythemodestructureofsupersolidsbycalculatingthesecondandfourthsoundspeedsduetodefectpropagation.Wealsocalculatethedensity-densitycorrelationfunctionofamodelsupersolidusingthehydrodynamicsofAndreevandLifshitz,andproposealightscatteringexperimenttomeasurethedensity-densitycorrelationfunction(whichisrelatedtotheintensityofscatteredlight).WendthatthecentralRayleighpeakofthedefectdiusionmodeofanormalsolidinthedensity-densitycorrelationfunctionsplitsintoanadditionalBrillouindoubletduetothelongitudinalsecondsoundmodesinsupersolidphase. 10

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1 ],andAllenandMisener[ 2 ]simultaneouslydiscoveredthesuperuidityofHeIIatatemperaturearound2.17Kin1938,thereweretheoreticalspeculationsaboutthecoexistenceofcrystallineorderandsuperuidityinmatter.In1956PenroseandOnsagerstudiedthepossibilityofsuchasupersolidphaseofmatterusingthedensitymatrixformalism,andconcludedthatsupersolidscouldnotexist[ 3 ].However,in1969severalnoveltheoreticalproposalsforasupersolidphaseweremade.AndreevandLifshitzproposedthepossibilityofacondensationofzero-pointdefectsina4Hesolid.Everysolidcontainsdefects:vacancies,interstitials,andsoon.Classicallythesedefectsareconsideredtobeobjectslocalizedatthelatticesites.However,atlowtemperatures,duetoquantumuctuations,defectsin4Hesolidvibratefromthelatticesitesandbecomemobile.Theycalledthesequantumexcitations\zero-pointdefectons".Theygeneralizedthetwo-uidmodeldevelopedbyLandauforsuperuidstosolidswithdefects,andobtainedanewcollectivemode(thepropagationofdefects)atzerotemperature[ 4 ]. Oneyearlater,ChestersuggestedthatasystemofinteractingbosonscanexhibitbothcrystallineorderandBose-Einsteincondensationatthesametime[ 5 ].Also,Leggettsuggestedthatthe\non-classicalrotationalinertia"(NCRI)ofliquidHeIImaybeobservedinthesolidphase.NCRIofHeIIcanbeexplainedbythetwouidmodel.WhenavesselcontainingHeIIisrotated,theabsenceofviscosityofthesuperuidpartcausesonlythenormalparttobedraggedbythewallofcontainer.Thus,theeectivemomentofinertiaofHeIIislessthanthatofnormalheliumliquid.Leggettpredictedansupersolidfractions=of3104,andasimpleanddirectexperimentofrotatingasolidwassuggestedtodetectit[ 6 ]. Thereafter,Saslow[ 7 ]andLiu[ 8 ]improvedAndreevandLifshitz'scalculationofhydrodynamicmodes.InRef.[ 9 ],thesupersolidfractionforafcclatticeasafunction 12

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10 ].Otherfeasiblesystemsthatcanexhibitsupersoliditywerealsoproposedtheoretically:vortexcrystalsintype-IIsuperconductors[ 11 ],Wignercrystalsformedbyexcitonsinelectron-holebilayers[ 12 ],andcoldatomsinopticallattices[ 13 ]. However,experimentalsearchesforpossiblesignaturesofsupersolidityofsolid4Hewerenotsuccessfulpriorto2004.Theseincludesoundspeedexperiments[ 14 ],massowexperiments[ 15 ],andtorsionaloscillator(TO)experiments[ 16 17 ].TheearlyexperimentssearchingforthesupersolidphasearesummarizedinRef.[ 18 ]. 19 20 ].Bothexperimentsobserveddropsofresonantperiodinthesolidphaseof4He,whichmightbeanindicationoftheNCRIproposedbyLeggett. TheTOusedbyKimandChanconsistofaBe-CutorsionbobandaBe-Cutorsionalrodwhichalsowasusedtointroduce4Heintothetorsionbob.InRef.[ 19 ],thetorsionbobcontainedaporousmedium(Vycorglass),whileinRef.[ 20 ]theexperimentwasperformedwithbulk4Heconnedinanannularchannel.Theyusedpressuresof62barand51barfor4Heinporousmediaandbulk4He,respectively.Theythenelectricallydrovetheoscillatorandmeasuredtheresonantperiodataxedtemperature.ThecharacteristicdependenceofresonantperiodonthetemperatureisshowninFigs. 1-1 and 1-2 A.ThedropofresonantperiodoccurredbelowthecriticaltemperaturearoundTc=175mKforinporousmediaandTc=250mKforbulkhelium. TheresonantperiodofaTOwithoutdissipationisgivenby whereIisthemomentofinertiaofthetorsionbobwith4He,andkisthetorsionalspringconstant.UsingEq. 1{1 KimandChaninterpretedtheirresultsofthedecreaseinthe 13

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21 ], whereT0istheonsettemperature,andIemptythemomentofinertiaoftheemptyTO.ThelargestobservedNCRIFsareabout0.5%and1.7%for4HeinVycorglassandbulk4He,respectively. KimandChanalsoperformedseveralcontrolexperimentstosupporttheirinterpretation.First,theyinvestigatedontheeectofthecriticalvelocity,andfoundthat,inbothexperiments,thedropinperioddecreaseswithincreasingrimvelocity.KimandChanestimatedthecriticalvelocity,atwhichtheNCRIdisappears,tobe300m/sfor4HeintheVycorglassand420m/sforbulk4He(Figs. 1-1 and 1-3 A).Second,theyrepeatedthesameexperimentwithsolid3He,whichisafermionicsolid;consequently,noBosecondensationispossible.Withsolid3Hetheydidnotobservedanychangeintheresonantperiod.However,animportantandinterestingfeatureisfoundthatincreasingtheconcentrationof3Heimpuritiesinsolid4Heincreasestheonsettemperatureandbroadensthechangeintheresonantperiod(Fig. 1-2 ).Third,takingadvantageofthecellgeometryofthebulk4Hesampletheyinsertedabarrierintotheannuluschannelaroundwhichsuperowisblocked,andfoundthattheresonantperiodissignicantlyreduced(Fig. 1-3 B).KimandChanconcludedthatthisisduetothedisruptionofsuperowaroundtheannulus.Finally,theymeasuredtheamplitude,whichisrelatedtodissipation,andobservedabroadminimumovertherangeoftemperatureswheretheresonantperioddropped.Theminimumintheamplitudehasthesametrendastherimvelocity,Fig. 1-3 A. 14

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22 { 26 ].RittnerandReppyreportedrsttheannealingeectonTOresults[ 22 ]andthesamplepreparationeect[ 24 ].BothsignaturesofTOexperiments-adropintheresonantperiodandapeakintheinverseofQ-factor-disappearedbyannealingthe4Hesolid(Figs. 1-4 and 1-5 ).InRef.[ 24 ]theyreportedtheeectsofquenchingthesamplebycoolingitrapidly,theeectofwhichistomakeasolidofpoorqualitywithalargenumberofdefects.TheyreportedNCRIFsashighas20%.Basedontheirresults,RittnerandReppysuggestedthatextendeddefectssuchasdislocationsandgrainboundariesplayanimportantroleinunderstandingtheTOresults. Ontheotherhand,Aokietal.studiedthefrequencydependenceoftheNCRIusingadoubletorsionaloscillator.Theyaddedanotherdummycellconcentricallyabovethetorsionbobwithsolid4He.ThisallowedthemtoinvestigatetheNCRIofthesamesamplewithtwodierentfrequencies:theresonantfrequencyoftheout-of-phasemodewasalittlemorethanatwicethatofthein-phasemode.TheyfoundnofrequencydependencetotheonsettemperatureofNCRI.Inadditiontothis,Aokietal.foundahysteresisthatdependsontherimvelocity:atT=19mKtheNCRIFincreaseduponloweringtherimvelocity,butsaturatedatthemaximumvalueastherimvelocitywasagainincreased. 27 ].ReviewsonbothrecenttheoreticalandexperimentalworkscanbefoundinRefs.[ 28 ]and[ 21 ].Saslowhassuggestedthatoneshoulduseathree-uidmodel,insteadofthetwo-uidmodelofLandau,tocorrectlydescribethesupersolid.Inhismodelthemassdensityandthemasscurrentcontainanadditionaltermconsistingofthelatticevelocityandthelatticemassdensity[ 29 ].CeperleyandBernucalculatedexchangefrequenciesinperfectbulkhcp4HeusingthePathIntegralMonteCarlo(PIMC)methodandconcludedthatsuperuiditywouldnotbeobservedinaperfectcrystal[ 30 ].Prokof'evandSvistunovhavefoundasimilarconclusionthatzero-pointdefectswerenecessary 15

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31 ].However,Boninsegnietal.obtainedalargeactivationenergyof13Kforvacanciesandof23Kforinterstitials,andsuggestedthatpointdefectsareunlikelytobepresentinthelowtemperaturerangeofexperimentalconditions[ 32 ]. Ontheexperimentalside,severalresultsthatareunfavorabletothesupersolidinterpretationarereported.Dayetal.performedexperimentsofmassowthroughsmallcapillarieswithsolid4HeinVycorglass[ 33 ]andinbulk[ 34 ].Oneexpectsapersistentmassowinasupersolid;however,theydetectednomassowineitherexperiments.Onthecontrary,amassowinsolid4HewasdetectedbySasakietal.[ 35 ]andbyRayandHallock[ 36 ].Sasakietal.observedaowin4Hesolidonthemeltingcurvewithgrainboundaries(apoorqualitycrystal).Forasinglecrystal(goodqualitycrystal)noowwasdetected.Initiallytheysuggestedthattheowtookplacethroughgrainboundaries,butfurtherexperimentsshowedthatmasscouldowalongthechannelbetweenagrainboundaryandawall[ 37 ].RayandHallockinjectedsuperuidthroughonelineintoacelllledwithsolid4He,anddetectedachangeontheotherline.Theyhavefoundamassowinsolid4Heatpressureothemeltingcurve[ 36 ].Similarly,superuidityingrainboundaries[ 38 ]andinscrewdislocations[ 39 ]wasstudiedusingPIMCsimulations.Polletetal.foundthatsuperuidisformedwithingrainboundariesinsolid4Heattemperaturearound0.5K.Boninsegnietal.foundsuperudityalongthecoreofascrewdislocationina4Hesolidatzerotemperature.Ontheotherhand,therearetwoneutronscatteringexperimentstomeasurethecondensatefraction[ 40 ]andtodetectchangesintheDebye-Wallerfactor[ 41 ].Neitherexperimentshowedanyevidencefortheexistenceofasupersolidphaseinsolid4He. Itiswellknownthatthetransitionof4Hefromthenormaluidtosuperuidisasecondorderphasetransitionaccompaniedwitha-anomalyinthespecicheatatthetransitiontemperature,e.g.seeRef.[ 42 ].Dorseyetal.suggestedthatthereshouldbea-anomalyinthespecicheatifthesupersolidtransitionisofsecondorder[ 43 ]. 16

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44 ].Theymeasureddowntoatemperatureofabout80mK,anddidnotobservedanysignature.Incontrasttothisnullresult,Linetal.reportedapeakinthespecicheatataboutT=75mK[ 45 ].Inthissecondexperimentasiliconsamplecellwasusedinsteadofanaluminumcell;siliconhasasmallerheatcapacityandhigherthermalconductivityatlowtemperaturesthanaluminum.Thus,theycouldmeasurethespecicheatdowntoatemperatureabout30mK,andtheyobserveddeviationsfromtheT3Debyespecicheat.Figure 1-6 showstheobservedpeaksinspecicheatofsolid4Hewithvariousconcentrationsof3Heimpuritiesaftersubtractingthecontributionsoftheemptycell,phonons,and3Heimpurities.Inaddition,theconstantspecicheatterm,whichmightbeduetothemobilityof3Heimpurities,wasalsofoundinthe10p.p.m.and30p.p.msamples(theinsetofFig. 1-6 ).Itisfoundthattheheightofpeaksisabout20Jmol1K1,anddoesnotdependonconcentrationof3Heimpurities.Linetal.estimatedtheNCRIFtobeabout0.06%whichiscomparabletooneoftheirTOresults[ 45 ].Finallytheyconcludedthattheobservedpeaksinspecicheatmeasurementsareanotherpossiblesignatureofthesupersolidphasetransition,inadditiontotheirTOresults. Anotherinterestingandimportantexperimentonsolid4HewasperformedbyDayandBeamish[ 46 ].Theymeasureddirectlytheshearmodulusatlowfrequenciesandamplitudesusingtwopiezoelectrictransducerslledwithsolid4He.Onetransducerwasusedtoapplyashearstresswhiletheotherdetectstheinducedsheardeformation.Theyfoundthattheshearmodulusofsolid4Heincreasedbyabout10%uponloweringthetemperature(Fig. 1-7 ).DayandBeamishexplainedtheirobservationusingtheadsorption(desorption)of3Heimpuritiesinto(from)adislocationnetwork.Theadsorbed3Heimpuritiespindislocationsatlowtemperatures,increasingtheshearmodulus.If3He 17

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1-8 thereducedchangesoftheshearmodulusfordierentvaluesof3Heimpurityconcentrationareshownasafunctionoftemperature,andincomparisonwiththereducedNCRI.Moreover,DayandBeamishalsostudiedtheresonanceinthecavity.TheymonitoredtheresonantfrequencyandtheQ-factor,andfoundsimilarbehaviortotheTOexperiments:theresonantfrequencyincreasedasthetemperaturewaslowered,accompaniedwithapeakintheinverseQ-factor.Thetwomeasurementsareverysimilar,suggestingthattheyarecloselyrelated.ThiswouldmeanthatdislocationsandgrainboundariespresentinsolidsmightberesponsibleforboththeshearanomalyandtheNCRI[ 47 ]. Ontheotherhand,severalalternativeexplanationsfortheobservedNCRIof4Hesolidhavebeenproposedaswell.DashandWettlaufergaveanargumentthatthereexistsathinlayerofliquidheliumbetweenthewallandtheheliumsolid,andtheyshowedthattheslippagebetweenthemcouldberesponsiblefortheNCRI[ 48 ].Nussinovatel.proposedaglassmodelforthe4Hesolid[ 49 ].Intheirmodel,solid4Hewasassumedtobeinglassyphaseatlowtemperatures,andtheystudieditseectontheTOexperiments.RemarkablytheycouldndthereasonableagreementwiththeexperimentofRef.[ 22 ].Finally,HuseandKhandkerproposedaphenomenologicaltwo-uidmodelforasupersolidwithatemperaturedependentcouplingconstant[ 50 ]toexplaintheTOresults. Tosummarize,theexistenceofasupersolidphaseinsolid4Hehasremainedcontroversial,boththeoreticallyandexperimentally.TheexplanationfortheobservedTOexperimentsisnotcomplete,openingpossibilitiesrangingfromasupersolidtransitiontoapossiblyalreadyknownmechanicaleectsuchasdislocationunbinding.Inthisworkwewillfocuson 18

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19

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Resonantperiodchangeintemperature.Theemptycelldataandlmdataareshiftedupby4,260nsand3290ns,respectively.P=971;000ns.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[E.KimandM.H.W.Chan,Nature427,225(2004)],copyright(2004). 20

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Resonantperiodchangeintemperatureforvariousconcentrationof3Heimpurities.P=971;000ns.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[E.KimandM.H.W.Chan,Nature427,225(2004)],copyright(2004). 21

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Resonantperiodandtheamplitudeofoscillationofannularcell(panelA)andofblockedannularcell(panelB)asafunctionoftemperature.istheresonantperiodatT=300mK.FromE.KimandM.H.W.Chan,Science305,1941(2004).ReprintedwithpermissionfromAAAS.Copyright(2004)byAAAS. 22

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Annealingeectintheresonantperiod.P=5:428053ms.Reprintedgure3withpermissionfromA.S.C.RittnerandJ.D.Reppy,Phys.Rev.Lett.97,165301(2006).Copyright(2006)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRevLett.97.165301). 23

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AnnealingeectintheinverseofQ-factor.Reprintedgure4withpermissionfromA.S.C.RittnerandJ.D.Reppy,Phys.Rev.Lett.97,165301(2006).Copyright(2006)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRevLett.97.165301). 24

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Specicheatpeaksof4Hesolidwithdierentconcentrationsof3Heimpurities.Opensquaresareof10p.p.m.,bluetriangles0.3p.p.m.,andredcircles1p.p.b.Theinsetshowsthedataofthe10p.p.m.samplebeforesubtractingaconstantterm(dottedline)of59Jmol1K1.ReprintedbypermissionfromMacmillanPublisherLtd:Nature[X.Lin,A.C.Clark,andM.H.W.Chan,Nature449,1025(2007)],copyright(2007). 25

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Shearmodulusof4Hesolidasafunctionoftemperature.Dataareshiftedforbetterclarity.ReprintedbypermissionfromMacmillanPublisherLtd:Nature[J.DayandJ.Beamish,Nature450,853(2007)],copyright(2007). 26

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Shearmoduluschangeforvariousconcentrationsof3Heimpurities.ReprintedbypermissionfromMacmillanPublisherLtd:Nature[J.DayandJ.Beamish,Nature450,853(2007)],copyright(2007). 27

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46 ],andannealingeectsofNCRIrelatedtosamplepreparation[ 22 24 ].Theseeectssuggestthattheelasticpropertiesofsolid4HemustbewellunderstoodbecausetheymightberesponsiblefortheNCRI. Attherststepofthischapter,westartdescribingthedynamicsofanemptyTO.Letusassumethatthetorsionbobiscompletelyrigid.TherigidbodymotionofthetorsionbobresultsinaconstantmomentofinertiaIosc.Thetorsionrodisassumedtobemasslesswithaspringconstantkthatprovidesarestoringforceproportionaltotheangulardisplacement.Thenthemodefrequencyoftheundampedharmonicoscillatoris!empty=p dt+k(t)=Mext(t);(2{1) whereoscisthedissipationcoecient.Withdampingtheresonantfrequenciesbecome 28

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Itot;(2{3) whereItot=Iosc+IHe.Therefore,theprincipaleectofloadingsolid4HeinthetorsioncellistosimplyincreasethetotalmomentofinertiabyaconstantIHe.However,ifweassumethatsolid4Heisnotcompletelyrigid,weneedtobecarefulindescribingthedynamicsoftheTOwithsolid4He.Infact,angulardisplacementsofthetorsionbobinduceashearstressinthesolid4He,anddragitalongintomotion.Thegeneratedstresscauseselasticsheardeformationstopropagatewithanitevelocitythroughoutthesolid4He.Thedeviationfromtherigidbodymotionresultsinaneectivemomentofinertiawhichdependsuponthedrivingfrequency.Moreover,dampingprocessespresentineverysolidproduceaneectivefrequency-dependentdampingcoecient. AsdiscussedinChapter1,themodicationofEq. 2{1 foraTOwithsolid4HewasmaderstbyNussinovetal.[ 49 ]byaddingthebackactiontorqueM(t)duetosolid4He: dt+k(t)=Mext(t)M(t):(2{4) Themotionofsolid4HeaectsthedynamicsoftheTObyexertingbackagainantorqueM(t)onthetorsionbob.FollowingNussinovetal.[ 49 ]thetorqueexertedbythesolid4HeontheTOistakentoberelatedtotheangulardisplacementthroughalinearresponsefunctionas Theresponsefunctiong(t)isreferredtoasthe\backactionterm"byNussinovetal.[ 49 ].Letusassumetimetranslationalsymmetrysothatg(t;t0)dependsontt0.Fourier 29

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2{5 ,weobtain Additionally,weassumethattheresponseof(t)totheexternaltorqueMext(t)islinearandrelatedbythesusceptibility(t);inFourierspace,wehave wherewehaveusedEq. 2{6 .Thezerosofthesusceptibility,1(~!)=0,givetheresonantperiod andthequalityfactoroftheTO Therefore,ifthebackactiong(!),andthereforethetorque,dependonthetemperatureT,thebackactiontermcontainsalltheinformationofthedynamicsoftheTOlledwithsolid4He.Inthefollowingsectionwecalculatethebackactiontermsbymodelingsolid4Heasaviscoelasticsolid. InTOexperiments,theshapesoftorsioncellsvaryfromasimplecylindricalcelltocomplicatedonessuchasablockedannulus.Inthisworkweconsider,forsimplicity,torsioncellswithcylindricalsymmetry.InFig. 2-1 weshowthegeometryofanitecylindertorsioncellofradiusRandheighth.Whenashearstressisappliedtoaviscoelasticsolid,duetothecylindricalsymmetry,theonlydisplacementisinthe 30

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Forlargedrivingfrequencies,viscoelasticsolidsrespondasviscousuidstotheshearstress,andtheNavier-Stokesequationforthevelocityeldv(t)issuitabletodescribeitsdynamics whereisthemassdensityandistheshearviscosity.Bycontrast,theelastodynamicequationforthedisplacementu(t)describesthesolid-likedynamicsforsmallfrequencies whereistheshearmodulus.Equation 2{11 predictsthetransversesoundspeedofcT=p 51 52 ] FromEq. 2{12 wealsoidentifyarelaxationtimedenedas==.When!1,aviscoelasticsolidrespondselasticallywhereasfor!1itrespondsviscously.Equation 2{12 canbegeneralizedtoaviscoelasticsolidmodelofaformlikeEq. 2{11 withafrequencydependentcomplexshearmodulus(!)withthefollowingproperties:forsmall! lim!!0=[(!)] andforlarge! lim!!1(!) TheshearmodulusoftheVoigtmodelofviscoelasticsolidsis(!)=i!. 31

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2{12 forthedisplacementeld,givenanoscillatoryboundarycondition.Fromtheobtaineddisplacementeldwecalculatetheshearstresses(rand/orz)ontheboundary,andthetorqueM(t).Thentheeectivedescriptionofasysteminoscillatorymotioncanbeinvestigatedbyanalyzingtheeectivemomentofinertia andtheeectivedampingcoecient where0istheinitialangulardisplacement.Theviscoelasticsolidmodelpredictsthat,forsmall!,e(!)vanishesandIe(!)becomesthemomentofinertiaofrigidbodyIRB.Inthefollowingwepresenttheresultsofcalculationforthreedierentgeometries:innitecylinder,nitecylinderandinniteannulus(AppendixAfordetails). whereIRB=R4h=2, andacharacteristictimeE=R=cT.InFigs. 2-2 and 2-3 weshowtheeectivemomentofinertiaandtheeectivedampingcoecients,respectively.Thereareelasticresonancesthatappearaspeaksinthedampingcoecientsandaneectivemomentofinertiathatincreasesandthendecreasestoanegativevalue.Tounderstandthenegativeeective 32

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TheresonantfrequenciesaregivenbythezerosofJ1(!E)inEq. 2{22 .InFig. 2-4 weshowhowthenormalizeddisplacementvectorevolvesduringahalfcyclewhen!1=E.Insuchalimit,themotionbecomespurelylikearigidbodymotion(straightlinesinFig. 2-4 ).Increasing!from0,thedeviationoftheamplitudeofdisplacementeldfromtherigidbodymotionbecomesapparent,developinganeectivelylargerout-of-phasemotionwiththeappliedshearstress(Fig. 2-5 ).Asaresult,elasticsolidshaveaneectivelylargermomentofinertiathantherigidbody.When!passesthroughtherstresonantfrequency,thedirectionofthedisplacementeldchanges.Figure 2-5 isaschematicshowingthein-phasemotion,thustheapparentmomentofinertiabecomesnegative.Weestimatetherstresonancefrequency!1'3:83cT=R=0:2MHzforcT300m/sandR0:5cm.Inexperimentalconditions,drivingfrequencies(1KHz)arewellbelowtherstresonancefrequencytoobserveelasticresonances.Theeectofviscosity(forlarge)istosmearouttheresonances. (2m1)22R2 wherem=p (2m1)22R2 Figures 2-7 and 2-8 illustratetheeectivemomentofinertiaandtheeectivedampingcoecientofnitecylinder,respectively.Thepresenceofthetopandbottomofthenite 33

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andtheeectivedamptingcoecientis qEAJ2(qER)+BN2(qER))=(2hR2i! qEAJ2(qERi)+BN2(qERi)); whereIRB=h(R4R4i)=2and InFigs. 2-9 and 2-10 weshowtheeectivemomentofinertiaandtheeectivedampingcoecientsoftheinniteannulusofviscoelasticsolid,respectively.Weobservethesametrendofresonantfrequenciesasinthepreviouscasesthatresonantfrequenciesdecreaseasthegapoftheannulusisincreased. 34

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InSection2.2,weinvestigatedthedynamicalresponseofanisotropicviscoelasticsolidtooscillatorydisturbancesbystudyingitseectivemomentofinertiaandeectivedampingcoecient.Ourmainconclusionisthatbothquantitieschangewiththedrivingfrequency!andarelaxationtime.Hence,theresonantfrequencyandthequalityfactoroftheTOdependontherelaxationtimeaswell. AsexplainedinSection2.1,togettheresonantperiodandthequalityfactorforeachcase,weneedtondthepolesofEq. 2{7 employingthebackactiontermofaviscoelasticsolidwithagivengeometry[AppendixAforthecalculationofg(!)].Butunderallexperimentalconditions,thewavelengthofthetransversesoundmodeismuchlargerthanthetypicaldimensionsoftorsioncells(jqERj;jqEhj1).Inthisregime,thebackactionterms,Eqs. A{8 A{29 and A{38 ,canbecastintoasingleform 24(1i!);(2{29) whereafunctionF(x)isdenedsuchthatforaninnitecylinderFinfcyl=1,foraninniteannulus andforanitecylinder (2m1)4~H(2m1) xi;(2{31) where~H(x)H(x)1.InFigs. 2-11 and 2-11 weshowFinfann(R=Ri)andFncyl(h=R),respectively.Notethat0Finfann;Fncyl1. NowweareinapositiontocalculatetheresonantperiodandtheinverseofthequalityfactoroftheTO.Withthebackactionterm,Eq. 2{29 ,thesusceptibilityof 35

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2{7 reducesto 24(1i!):(2{32) Forsimplicityweassumethatthereisnodampingfromtheoscillator(osc=0).Infact,thisassumptionisacceptablebecauseforahighqualityTO(Q106)thedampingoftheoscillator(osc)canbeneglected.SinceIHe=Iosc103underexperimentalconditions,thecontributionduetoviscoelasticitycanbetreatedasaperturbationfromtherigidbodymotion,whoseresonantfrequencyis!0giveninEq. 2{3 .Weexpandthepolesabout!0suchthat1(!=!0+!1)=0with!1!0.Thenweobtain 48Itot!20 Intheviscoelasticsolidmodel,theresonantperiodandtheinverseofqualityfactorofTOare,usingEqs. 2{8 and 2{9 PPP0'R2!0IHeF(h=R) 24Itot1 1+2!20;(2{34) and Q1'R2!20IHeF(h=R) 24Itot!0 whereP0=2=!0.Equations 2{34 and 2{35 areourcentralresultsinthischapter. Tointerprettheseresults,rstnoticethatwhenpassesthrough1=!0,therewouldbeapeakinQ1andadropintheresonantperiodwhosesizesaregivenby Q1max=P=P0=R2!20IHeF(h=R) 48Itot:(2{36) Therefore,theviscoelasticmodelpredictsthatQ1max=(P=P0)=1.However,theTOexperimentsshowedthatthemaximumdissipationQ1maxislowerthan(P=P0).Thetypicalexperimentalvaluevariesaround0.1[ 24 53 ]reachingupto0.65[ 22 ]anddownto0.01[ 54 ].ThisimpliesthatwecanonlyteitherthepeakinQ1maxorthedecreaseinPtakingtheshearmodulusasattingparameter.Otherquantitiessuchasthe 36

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2-11 and 2-12 ),thereductionofdimensionofthesystemfromtheinnitecylindergeometryresultsindecreasedperiodshiftsandsizesofpeakinQ1.Consequently,inttingtheexperimentaldata,theshearmodulus(ttingparameter)ofthenitecylinderturnsouttobegreaterthanthatoftheinnitecylinderbyafactorof1=F.Third,followingNussinovetal.[ 49 ],wetaketherelaxationtimeas anduse0andE0asttingparameters.AsthetemperatureisloweredbelowE0=kB,therelaxationtimebecomeslargerthan1=!0,andsolid4Hebehaveslikeaviscousuid.InFigs. 2-13 2-18 weshowchangeintheresonantperiodandintheinverseofqualityfactormeasuredinClarketal.[ 53 ]andtheirttingusingEqs. 2{34 and 2{35 ,respectively.WehavechosentotQ1maxusingtheinnitecylindermodel.InTable 2-1 welisttheshearmodulus,theactivationenergyE0,and0usedintting.Asmentionedearlier,thechangeintheresonantperiodoftheviscoelasticmodelonlyaccountsforabout10%ofwhatisactuallyobserved.Thismightimplythattheunexplainedpartiscausedbyasupersolidtransition.Moreover,itisusefultostudythedynamicaleectofdislocationsontheshearmodulusofnormalsolidsbecausethemotionofdislocationscanchangetheelasticpropertiesofregularcrystals,anddeterminethecharacteristictimeoftheviscoelasticsolidmodel(e.g.,Ref.[ 55 ]). 37

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46 ].Wefocusonlyonelasticitybecauseintherangeoftheemployedfrequencytomeasuretheshearmodulustheviscosityofsolid4Hecouldnotbeprobedsimultaneously.AswediscussedinChapter1,DayandBeamishalsoinvestigatedtheresonanceeectinthecavitywheretheapparatustomeasuretheshearmoduluswasembedded.TheyfoundbehaviorsthatresembletheNCRI.However,therelationbetweenthedissipationpeakobservedinthecavityresonanceandtheviscositypresentinthegapbetweentwotransducersisnotyetclear. Letusexaminersttheeectivemomentofinertiaofaninnitecylinder.Forsmall!,wehaveIinfcyl()'IRB[1+(!2R2=24)]withoutviscosity.Thesecondtermisthecorrectiontotherigidbodyvalueduetoaniteshearmodulus.SincethecorrectionterminIinfcyl()isinverselyproportionalto,ataxedfrequencyIinfcyl()decreasesas(!)increases.Thereforeanincreaseofshearmoduluswillenhancethechangeinresonantperiod.ThiscouldbeaconnectionbetweentheincreasingshearmodulusobservedbyDayandBeamishandNCRI. Wecalculatetheactualchangeintheresonantperiodinducedbyanincreaseofshearmodulus.TheresonantperiodofTOforanelasticsolidofaniteshearmoduluscanbeeasilyobtainedbysettingtozeroinEq. 2{34 .Theresultis 48Itot:(2{38) OnecanverifythisresultndingthezerosofEq. 2{32 withoutand.Let'sconsiderasmallchangeinshearmodulusfrom0.ThenthefractionalchangeinP()duetobecomes P P(0)'R2!20IHeF(h=R) 48Itot 0:(2{39) ForRef.[ 24 ],Eq. 2{39 predictsonly0.8105%decreasechangeintheresonantperiodwhereasthemeasuredchangeisabout2.6103%.Therefore,wendthatanincreasingshearmodulusaccompaniesadecreaseintheresonantperiodofaTO;however,theelasticsolidmodeldoesnotaccountforallthechangeobservedinexperiments. 38

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SchematicillustrationofTOandgeometryofatorsioncell. 39

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Eectivemomentofinertiaofaninnitecylinderofviscoelasticsolidasafunctionofthedrivingfrequency!. Figure2-3. Eectivedampingcoecientofaninnitecylinderofviscoelasticsolidasafunctionofthedrivingfrequency!. 40

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Displacementvectorinahalfcyclefor!1=E. Figure2-5. Displacementvectorinahalfcyclefor!=3=E. 41

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Displacementvectorinahalfcyclefor!=4=E. Figure2-7. Eectivemomentofinertiaofanitecylinderofviscoelasticsolidasafunctionofthedrivingfrequency!with=E=1=100. 42

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Eectivedampingcoecientofanitecylinderofviscoelasticsolidasafunctionofthedrivingfrequency!with=E=1=100. Figure2-9. Eectivemomentofinertiaofaninniteannulusofviscoelasticsolidasafunctionofthedrivingfrequency!with=E=1=1000. 43

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Eectivedampingcoecientofaninniteannulusofviscoelasticsolidasafunctionofthedrivingfrequency!with=E=1=1000. Figure2-11. 44

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Figure2-13. ResonantperiodoftheblockedcapillarysampleofBeCuTO.ExperimentaldatawereadaptedwithpermissionfromA.C.Clark,J.T.West,andM.H.W.Chan,Phys.Rev.Lett.99,135302(2007).Copyright(2007)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302). 45

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InverseofQ-factoroftheblockedcapillarysampleofBeCuTO.ExperimentaldatawereadaptedwithpermissionfromA.C.Clark,J.T.West,andM.H.W.Chan,Phys.Rev.Lett.99,135302(2007).Copyright(2007)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302). Figure2-15. ResonantperiodoftheannealedblockedcapillarysampleofBeCuTO.ExperimentaldatawereadaptedwithpermissionfromA.C.Clark,J.T.West,andM.H.W.Chan,Phys.Rev.Lett.99,135302(2007).Copyright(2007)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302). 46

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InverseofQ-factoroftheannealedblockedcapillarysampleofBeCuTO.ExperimentaldatawereadaptedwithpermissionfromA.C.Clark,J.T.West,andM.H.W.Chan,Phys.Rev.Lett.99,135302(2007).Copyright(2007)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302). Figure2-17. ResonantperiodoftheconstanttemperaturesampleofBeCuTO.ExperimentaldatawereadaptedwithpermissionfromA.C.Clark,J.T.West,andM.H.W.Chan,Phys.Rev.Lett.99,135302(2007).Copyright(2007)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302). 47

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InverseofQ-factoroftheconstanttemperaturesampleofBeCuTO.ExperimentaldatawereadaptedwithpermissionfromA.C.Clark,J.T.West,andM.H.W.Chan,Phys.Rev.Lett.99,135302(2007).Copyright(2007)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302). Table2-1. Fittingparameters. sample[g/cms2]0[s]E0=kB[mK] blockedcapillary0:341082.86158annealedblockedcapillary0:751080.21395constanttemperature1:531083.72166 48

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4 ].Intheirmodelpointdefects,suchasinterstitialsandvacancies,inbosonicsolidsundergoesaBose-Einsteincondensationandacrystalbecomesasupersolid.ThehydrodynamicsofsupersolidswasstudiedfurtherbySaslow[ 7 ]andLiu[ 8 ]. WeusethevariationalprincipletoderiveaLagrangianforthesupersolidandthenon-dissipativehydrodynamicequationsofmotion.Thevariationalprinciplewasoftenusedintheliteraturetoobtainthehydrodynamicsofvariouscontinuumsystems:normaluids[ 56 { 58 ],superuids[ 56 59 { 63 ],normalsolids[ 57 64 ],liquidcrystals[ 65 ],andsoon.Letusstartthissectionbygivingaplainexampleofvariationalprincipleappliedtoidealuidstoshowthesimplicityofthemethod. 2v2UIF();(3{1) whereisthemassdensity,vthevelocityeld,andUIFtheinternalenergydensity.Theinternalenergydensitysatisesthethermodynamicrelation 49

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ThevariationalprinciplestatesthattheequationsofmotioncanbederivedbyminimizingtheactionSofaLagrangiandensityL withrespecttoallthedynamicalvariablesi.e.,andvinthisexample.However,thevariationoftheactionwiththeLagrangiandensityinEq. 3{1 withrespecttothedynamicalvariablesandvdoesnotprovideuswiththerightequationsofmotion-theyarenotindependentandrestrictedbysideconditionssuchasconservationlaws.TheeasywaytoseethisistotakethevariationoftheactionofEq. 3{1 withrespecttov:theresultingequationofmotionisatrivialandirrelevantone(v=0).Inordertoovercomethisproblem,thesideconstraintsmustbeincludedintotheLagrangiandensity.Asoneknows,foruidsthetotalmassisconserved,andthisconservationlawisexpressedinthecontinuityequation Sinceweareconsideringisentropicidealuids,themassconservationistheonlysideconditiontobetakenintoaccount.Infactthemomentumisconservedaswell.Nonetheless,themomentumconservationlawisnotasidecondition,butthebyproductofthevariationalprinciple.ThecontinuityequationisincorporatedintotheLagrangiandensityEq. 3{1 byusingaLagrangemultiplier: 2v2UIF()+@t+@i(vi):(3{5) Thentheequationsofmotionareobtainedbytakingvariationsoftheactionwithrespecttoandv.WithEq. 3{5 weobtain 50

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2v2@UIF Dt=0;(3{7) whereD=Dt@t+vi@i.TheothertrivialequationofmotionisthecontinuityequationwhichoneobtainsbytakingthevariationwithrespecttotheLagrangemultiplier.Equation 3{6 impliesthat andthisconrmsthatthereisnovorticity(rvs=0)foridealuids,asexpected.WecanexpressEq. 3{7 inmorefamiliarformbytakingitsgradient.SinceinthermalequilibriumthechangeinthechemicalpotentialperunitmassisrelatedtothechangeinthepressurePbytheGibbs-Duhemrelationasfollows weobtain Equation 3{10 istheEulerequationforidealuidswithoutexternalforces.Consequently,wederivedthehydrodynamicsdescribingidealuidsusingthevariationalprinciple:thecontinuityequationandtheEulerequation. Asweshowedintheanalysisabove,thevariationalprincipleprovidesuswiththeLagrangiandensityforacontinuumsystem;Eq. 3{5 istheLagrangiandensityforisentropicidealuids.Moreover,aslongastheboundarycontributionsarenegligibleintheaction,theanalysisaboveisequivalenttothatwithaLagrangiandensity 2(@i)2UIF():(3{11) WehaveintegratedbypartstheactionofEq. 3{5 ,andreplacedthevelocitywithusingEq. 3{8 .ThisLagrangiandensityalsocanbederivedfromthetime-dependentGross-PitaevskiiLagrangiandensityforthesuperuidwhichis 2m(i~@j)(i~@j)g 51

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3{12 canbewrittenasfollows: Sincethersttermisatotalderivativeofananalyticaleld,itdoesnotcontributetothedynamicsandcanbeneglected.Hencewecanidentifythat=mnand=(~=m);therefore,wehaveshownthatthetime-dependentGross-PitaevskiiLagrangiandensityisthesameastheLagrangiandensityforisentropicidealuidsEq. 3{11 withaparticularformoftheinternalpotentialenergydensity. 2svs2+1 2(s)vn2USS(;s;s;Rij);(3{14) wheresisthedensityofsuper-components,thetotaldensity,vsthevelocityofsuper-components,vnthevelocityofnormalcomponents,stheentropydensity,and thedeformationtensorwithRandxbeingLagrangianandEuleriancoordinates,respectively.ThersttwotermsintheLagrangiandensityarethekineticenergydensitiesofthesuper-componentandnormalcomponents,andthelasttermistheinternalpotentialenergydensity.Becauseoftheisotropy,theLagrangiandensityofanisotropicsupersolidisverysimilartotheLagrangiandensityofsuperuid,e.g.theLagrangiandensityofasuperuidusedbyZisel(Eq.2.1inRef.[ 59 ]).TheonlyexceptionisthatUSSdependsonthedeformationtensor.Incontrasttouids,theelastic 52

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GiventheLagrangiandensity,Eq. 3{14 ,thetotalenergydensityforsupersolidsisdenedasthesumofthekineticenergydensitiesandthepotentialenergydensity 2svs2+1 2(s)vn2+USS(;s;s;Rij):(3{16) Thistotalenergydensitycanberelatedtotheenergydensitymeasuredintheframewherethesuper-componentisatrestasfollows: 2vs2+(s)(vnivsi)vsi+:(3{17) AsisaGalileaninvariant,itmustdependonGalileaninvariantquantities[ 66 ].Similartothetwo-uidmodelforsuperuid[ 66 ],hasathermodynamicrelation[ 4 ] withikthestresstensor.WenowcanobtainthethermodynamicrelationforESSbydierentiatingEq. 3{17 andreplacingEq. 3{18 ford.Theresultis 2(2vn22vnivsi+vs2)d+svsidvsi+(s)vnidvni: ThisthermodynamicrelationwasusedbySaslow[ 7 ]andbyLiu[ 8 ]inderivingthehydrodynamicsofasupersolidprovidedthatthechemicalpotentialperunitmassisgivenby FinallyfromEq. 3{16 andEq. 3{19 ,wegetthethermodynamicrelationforUSS: 2(vnivsi)2d1 2(vnivsi)2dsikdRik:(3{21) 53

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AsillustratedinSection3.1.1,thedynamicalvariablesofasupersolidintheLagrangiandensity,Eq. 3{14 ,arenotindependenteachother.ThesideconditionsrelatingthedynamicalvariablesmustbeincludedintheLagrangiandensityinordertoderivethecorrectequationsofmotion.Therearetwoimportantconservationlawsforasupersolid:themassconservationlawandtheentropyconservationlaw.Onceagainthemomentumconservationlawisnotasideconditiontobeimposed,butaconsequenceofthevariationalprinciple.Theconservationofmassisexpressedbythecontinuityequation withthetotalcurrent Fortheentropyconservationwehave Oneshouldnotethatintheequationfortheconservationofentropy,Eq. 3{24 ,onlythevelocityofthenormalcomponentareinvolvedbecausetheentropyiscarriedsolelybythenormalcomponent.Inadditiontothetwoconservationlawsabovethereisonemoresidecondition,calledLin'sconstraint,tobeincluded[ 56 ]: whereDn=Dt@t+vni@i.Lin'sconstraintforsolidsisanexpressionofthefactthattheLagrangiancoordinates,(i.e.,theinitialpositionsofparticles)donotchangealongthepathsofthenormalcomponent.Thesameconditionisalsousedforanisentropicnormaluidtogeneratevorticitywhereastheentropyconservationequationproducesvorticityforthenormaluid[ 56 ].Let'sincorporatetheseconstraintsintotheLagrangiandensityusing 54

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2svs2+1 2(s)vn2U(;s;s;Rij)+(@t+@isvsi+(s)vni)+@ts+@i(svni)+i@t(sRi)+@j(sRivnj); wherewehaveusedtheLin'sconstraintcombinedwiththeconservationequationofentropymakingitinaformofcontinuityequation.Itisalsopossibletousethecontinuityequation,insteadoftheconservationequationofentropy,combiningtheLin'sconstraint. Wearenowinapositiontogettheequationsofmotion.Wetakethevariationsoftheactionwithrespecttoallthedynamicalvariables.Weobtain 2vn2@USS Dt=0;(3{27) 2vs21 2vn2@USS Dn Dt+RiDni ObviouslythevariationswithrespecttotheLagrangemultipliersrecovertheimposedconstraints,Eqs. 3{22 through 3{25 .InthefollowingweshowthatthederivedequationsofmotioncanberearrangedtorecoverthehydrodynamicsofasupersoliddevelopedbyAndreevandLifshitz[ 4 ],bySaslow[ 7 ],andbyLiu[ 8 ]. 55

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57 ]forthevelocityofthesuper-component,fromEq. 3{30 andthevelocityofthenormalcomponent,fromEq. 3{31 s@i+Rj@ij:(3{34) Withtheserepresentationswendrstthat Consequentlythevelocityofthesuper-componentisirrotationalasoneexpectsforasuperuidwithoutvortices:vsisonlylongitudinal.Ontheotherhand,wealsondthatvorticitycanbegeneratedforvn: sr+rsRi InfactitisalsopossibletoincludesystematicallythetransversepartofvsbyintroducingasecondLin'sconstraint(AppendixB). TheuseofEq. 3{33 inEq. 3{28 corroboratesoneofthethermodynamicrelationsgiveninEq. 3{21 2(vnivsi)2:(3{37) TakingthegradientofEq. 3{27 andusingtheClebshrepresentationofvs,Eq. 3{33 ,wegettheJosephsonequationforvs whereDs=Dt@t+vsi@i.InderivingEq. 3{38 ,wehaveusedEq. 3{37 andvsj@ivsj=vsj@jvsibecausevsisirrotational. 56

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3{31 ,andusethederivedequationsofmotiontoeliminatetheClebschpotentialsoftherelativevelocity.TheresultingequationistheEulerequationforvn: (s)Dnvni 2(s)@i(vnjvsj)2@ts+@j(svsj)vsi(@t(s)+@j(s)vnj)vni=(s)@i@jjkRiks@iT1 2(s)@i(vnjvsj)2(@t(s)+@j(s)vnj)(vnivsi); wherewehaveusedanusefulidentityoftheconvectivederivative, Dt+a@iDb Dta@jb@ivj:(3{40) FinallytheJosephsonequationandtheEulerequationcanbeputtogetherintothemomentumconservationequation, becausejk@iRjk+@jjk@iRk=@j(jkRik).InderivingEq. 3{41 wehaveusedthemassconservationequationandthethermodynamicequationfor. Asshownintheexampleofidealuids,theLagrangiandensityfortheisotropicsupersolidcanalsobereducedinacompactformbyusingtheClebshrepresentationofvs.Neglectingtheboundarycontributions,theLagrangiandensityEq. 3{26 isequivalentto 57

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2svs2+1 2(s)vn2USS(;s;s;Rij)@tsvsi+(s)vni@isDn DtsRiDni ThenextstepistouseEqs. 3{29 through 3{31 toreplacevs,and.ThenweobtaintheLagrangiandensityforanisotropicsupersolid 2(@i)2+1 2(s)(vni@i)2f(;s;T;Rij);(3{43) wheref=USSsT.ThisLagrangiandensityusedinthederivationofthehydrodynamicsofasupersolidhasmanyotherapplications,includingthecalculationofcorrelationfunctionsandcollectivemodes,andthestudyofthedynamicsofdislocationsandvortices. 2sijvsivsj+1 2(ijsij)vnivnjUSS(;sij;s;Rij):(3{44) Theinternalpotentialenergysatisesthethermodynamicrelation 2(vnivsi)2dikdRik1 2(vnivsi)(vnjvsj)dsij:(3{45) ThederivationofthisthermodynamicrelationisidenticaltothecalculationofEq. 3{21 forisotropicsupersolids. Theconstraintstobeimposedfortheanisotropicsupersolidare 58

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WeincorporatetheseconstraintsintotheLagrangiandensityusingLagrangemultipliers,and.Thenwehave 2sijvsivsj+1 2(ijsij)vnivnjUSS(;sij;s;Rij)+@ts+@i(svni)+(@t+@isijvsj+(ijsij)vnj)+i@t(sRi)+@j(sRivnj): IntheaboveLagrangiandensityLin'sconstraintwasintroducedaftercombiningitwiththeequationoftheentropyconservation.Wecalculatetheequationsofmotionasfollow 2vn2@USS 2vsivsj1 2vnivnj@USS 2(vsjvnj)@i1 2(vsivni)@j=0;(3{51) Dn Dt+RiDni Additionally,thevariationswithrespecttotheLagrangemultipliersjustreproducetheimposedsideconditions,Eqs. 3{46 through 3{48 .First,thevariationwithrespecttovsileadsagaintoEq. 3{33 .Next,thederivationoftheJosephsonequation,theEuler 59

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2@ivs2;(3{56) theEulerequation andtheequationoftheconservationofmomentum where Notethatthismomentumconservationequationagreeswiththenon-dissipativemomentumcurrentderivedbyAndreevandLifshitz(Eq.12inRef.[ 4 ])exceptforthenonlinearstraintermthattheyneglected.Moreover,thederivedmomentumconservationequationalsocanbemappedintoEq.4.16ofSaslow[ 7 ]bytakingvsasaGalileanvelocity,andEq.3.40ofLiu[ 8 ]withthevanishingsuperthermalcurrentwiththechemicalpotentialgivenbyEq. 3{20 Similartotheisotropicsupersolidcase,theequivalentLagrangiandensityofanisotropicsupersolidstoEq. 3{49 is 2sij@i@j+1 2(ijsij)vnivnj(ijsij)vnj@if(;sij;T;Rij); 60

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67 ].First,wecanalsomaketheconnectiontotheLagrangiandensityderivedbySon[ 67 ]byreplacingvnwiththedisplacementvectorRandthestraintensorRijusingtheinvertedLin'sconstraint, whereR1ji@xi=@RjandRijR1jk=ik.Josserandetal.usedthehomogenizationmethodstartingfromthetime-dependentGross-Pitaevskiiequation[ 68 ].Ontheotherhand,YeproposedaLagrangiandensityforasupersolidintroducinganarbitraryphenomenologicalcouplingconstantbetweenelasticityandsuperuidityintheGross-Pitaevskiiequation[ 69 ]. AtthispointitisworthwhiletospeculateaboutthepossibleconnectionbetweenthederivedLagrangiandensityandthetime-dependentGross-PitaevskiiLagrangiandensitycoupledtoanelasticeld.InextendingthesuperuidGross-PitaevskiiequationEq. 3{12 ,toasupersolidonemustrequireGalileaninvariance.Thecovariantformofthegradientoftheorderparameterwavefunctionis wheremisthemassofthebosonicparticle.However,theuseofthecovariantgradientresultsinthecouplingconstantsbetweenvsandvnwhilefromtheinteractiontermintheLagrangiandensity,Eq. 3{60 ,thecouplingconstantis(ijsij)whichissetbyconservationlawsandGalileaninvariance.UnfortunatelytheconnectionbetweenthevariationalprincipleandtheGross-PitaevskiiLagrangiandensityisstillunclear. 61

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and Forthevelocityofthesuper-component,wetakeavariationintheClebschpotentialsothatwehaveitsnonvanishinggradientandthetimederivative:@iand@t.Inaddition,weassumethattheLagrangiancoordinatesdierfromtheEulercoordinatesbyasmalldisplacementvectoreldu: Thedeformationtensorthenbecomes wherewij@iuj.FromtheinvertedLin'sconstraint,Eq. 3{61 ,weobtainthelinearrelationbetweenthevelocityofthenormalcomponentandthedisplacementvector, Therefore,wefoundthatinlinearelasticitythevelocityofthenormalcomponentisgivenbythetimederivativeofthedisplacementvector.ItbecomesclearthatLin'sconstraintisthehydrodynamicequationfortheelasticvariablesthatarisesfromthetranslationalbrokensymmetriesofsolidsratherthantheconservationoftheinitialpositions. 62

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20(@i)2@ @wijwij1 2@ @wij()2+1 2n0ij(@tui@i)(@tuj@j)1 2@ij wheren0ij0ijs0ij,andwehaveused @=+1 2(vnivsi)2';(3{69) and @wij=ij:(3{70) Intheexpansionwehavedroppedconstantswhichdonotcontributetotheequationsofmotion.Additionallythetermsproportionalto@f=@sijareneglectedbecausetheyareofhigherorder: @sij=@USS 2(vnivsi)2'0:(3{71) Asaresult,thedependenceonsijisabsentinthequadraticexpansionoftheLagrangiandensity,Eq. 3{68 .OftenthersttwotermsinEq. 3{68 areneglectedbecausetheyaretotalderivatives,anddonotcontributetotheequationsofmotionaslongasboundarycontributionsarenotimportant.Thisistrueaslongastopologicaldefects,suchasvorticesordislocations,arenotpresentinthesupersolid.InChapter5wewillshowthatthersttermisresponsiblefortheMagnusforceactingonvortices[ 70 ]whilethesecondtermgivesthePeach-Koehlerforceonadislocation[ 71 72 ].Sincewearenotconsideringanytopologicaldefectsinthischapter,weneglectthesetermsfornow. ThelinearizedequationsofmotionareobtainedtakingthevariationsoftheactionofEq. 3{68 withrespectto,,andui: @wijwij@ @wij(@t+0)(3{72) 63

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@t+n0ij@i(@tuj)+s0ij@i@j=0(3{73) @wji@jn0ij@t@j=0(3{74) wherewehaveusedtheidentity @yz=@z @yx@x @zy:(3{75) NotethatEq. 3{73 isthelinearizedcontinuityequation.TheothertwoequationsofmotionareequivalenttoEq.19ofAndreevandLifshitz[ 4 ]whentheJosephsonequation,Eq. 3{56 ,(@t=0)isused. Densityuctuationsinsolidsarecausedbyuctuationsineitherthelatticedisplacementorthenetdefectdensitydenedas[ 73 74 ] whereiisthedensityofinterstitialsandvthedensityofvacancies.Thedefectdensityisconservedaslongassurfaceeectsareignored;vacanciesandinterstitialarecreatedanddestroyedbypairsinthebulk,butcanbecreatedordestroyedindividuallybymigratingtothesurface.Nowwecantakeastheindependentvariableinsteadof.Since @wijwij+@ @wij;(3{77) Eq. 3{72 reducesto @wijwij+@ @wij;(3{78) wherewehaveusedEq. 3{75 and @y0=@x @yz+@x @zy@z @y0:(3{79) InthehigherorderexpansionoftheLagrangiandensity,thetermsproportionaltothesuperuiddensityuctuationmustbeconsideredinEq. 3{78 .FollowingZippeliusetal.

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73 ],wecanidentify @wij=0ij;(3{80) and @wij=1:(3{81) ThesamerelationwasusedbyOstlundetal.whentheystudiedthehydrodynamicsofananisotropicnormalsolid[ 75 ].TakingthetimederivativeofEq. 3{78 andusingthecontinuityequationEq. 3{73 ,wegetthelinearizedconservationequationofthedefectdensity withthedefectcurrentproportionaltothedensityofthesuper-component Wefoundaveryimportantfeature:thedefectcurrentonlyarisesifsuperuidityispresentwithnon-vanishingrelativevelocity. ReplacingEqs. 3{72 and 3{82 ,intoEq. 3{74 andthetimederivativeofEq. 3{73 ,wegetthefollowingequationsofmotion @wij@i@js0ij@i@2tujs0ij@ @wlk@i@jwlk=0;(3{84) @wjin0ij@ @wij!@j@ji @wlk0@ @wijlk!@jwlk=0:(3{85) Inthecasewhereu=0,wegetthedispersionrelationofthefourthsoundmodes,fromEq. 3{84 @wijqiqj;(3{86) whichisjusttheresultobtainedbyAndreevandLifshitz[ 4 ],since(@=@)wij=(@=@)wij.Ontheotherhand,whendefectuctuationsareabsent(=0),Eqs. 3{84 and 3{85 65

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@wlk@iwlk0@ @wij@jwkk:(3{87) Therefore,therearenoadditionalsoundmodeswithoutdefects. Anotherinterestingcaseisvanishingthesuperuiddensity,i.e.,normalsolids.Insuchacase,wegetthesamesoundspeedasEq. 3{87 inthecaseinwhichthedefectdensityuctuationsvanishbecausethesecondterminEq. 3{85 doesnotcontributetothesoundspeed,butrathertothediusion.Moreover,theconservationequationfordefects,Eq. 3{82 ,impliesthat@t=0whensij=0,whichagreeswiththedissipationlessdescriptionofsupersolid:adefectcurrentarisesonlywhendissipationistakenintoaccount[ 73 ].InChapter4westudythedissipativehydrodynamicsandcalculatethedefectdiusioncoecient. ToconcludethissectionwerewritetheLagrangiandensity,Eq. 3{68 ,intermsofthedefectuctuationusingEq. 3{78 : 2s0ij@i@j@ @wijwij+0@ @wijwii1 2@ @wij21 2@ji @wijwijwkk1 220@ @wijw2ii+1 2n0ij@tui@tuj; wherewehaveintroduced=+0t.Inthefollowingsection,westudythisLagrangiandensityconsideringatwo-dimensionalisotropicsupersolidtocalculatethehydrodynamicmodesandthedensity-densitycorrelationfunction. 4 ].Inthissectionwecalculatethesecondsoundspeedandthedensity-densitycorrelationfunctionwhichcontainsadditionalinformationonthesecondsoundmodes. 66

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76 ].Westartbyenumeratingtheconservationlawsandbrokensymmetriesofathree-dimensionalsupersolid.Sincethermaluctuationsarenotconsidered,theenergyconservationlawcanbeomitted;therefore,thereareonlyconservationlawsofmassandthreecomponentsofmomentum.Inadditiontothese,therearethreebrokentranslationalsymmetries,andonebrokengaugesymmetryduetotheBose-Einsteincondensation.Thus,wehaveeightconservationlawsandthebrokensymmetries,andthereforeeighthydrodynamicmodesforthethree-dimensionalsupersolidwithoutthermaluctuations.Thecorrespondinghydrodynamicmodesaretwopairsoftransversepropagatingmodes,onepairoflongitudinalpropagatingmodes,andonepairoflongitudinalsecondsoundmodes.Theappearanceoftheselongitudinalsecondsoundmodesisanotherdenitesignatureofasupersolid,inadditiontothe-anomalyinthespecicheat[ 43 ],theNCRI[ 6 ],andsoon. Inordertoinvestigatetheappearanceofthesecondsoundmodesinasupersolidweconsideratwo-dimensionalisotropicsupersolid.Thereductionbyonespatialdimensionresultsinonlyonepairoftransversepropagatingmodes;thelongitudinalpartremainsintact.Notethattheindicesonthedensitiesarelostbecauseoftheisotropy.Wehavethefollowingthermodynamicrelations: @wij=ij;(3{90) @wij=@ @wij=1 whereistheisothermalcompressibilityatconstantstrain,isaphenomenologicalcouplingconstantbetweenthestrainandthedensity,and~and~arethebareLamecoecientsatconstantdensity.Then,theEuclideanLagrangian(withtheimaginarytime,=it)ofEq. 3{88 byusingtheaboverepresentationsofthethermodynamicrelations 67

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2s0(@i)2+uii1 21 2~+1 2n0(@ui)2; whereuij=(wij+wji)=2,andwehaveneglectedthelineartermswhichdonotaectthedynamicsbecausetheyarethetotalderivativesofanalyticalvariables.WhenFouriertransformedEq. 3{92 becomes 2(Qn)(Qn)uL(Qn)A0BBBB@(Qn)(Qn)uL(Qn)1CCCCA+1 2n0!2n+~q2uT(Qn)uT(Qn); whereQn(q;!n)withMatsubarafrequencies!n,uL=(qu)=qwithq=jqj,uT=u(uL=q)q,and where~+2~.Aswediscussedearlier,thereisonepairoftransversesoundmodeswhosesoundspeedis n0:(3{95) Inaddition,wegetonepairoflongitudinalrstsoundmodesandanotherpairoflongitudinalsecondsoundmodes.Theirsoundspeedsare 20+1 2s n0+1 202;(3{96) 68

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201 2s n0+1 202:(3{97) Whenthesuperuiddensityvanishes,i.e.,foranormalsolid,thesecondsoundspeedc2vanishes.Therefore,thenormalsolidhasonlyonepairoftransversesoundmodesandonepairoflongitudinalsoundmodeswhosesoundspeedsbecome 0;(3{98) and whichagreewiththeresultsobtainedbyZippeliusetal.[ 73 ]afteridentifying~=Zippelius+2Zippelius+1=Zippelius,~=Zippelius,=(Zippelius+1=Zippelius)=0,and=Zippelius.Moreover,werecoverthesoundspeedofthenormaluidbyassumingthattheLamecoecientsandthecouplingconstantvanish. Ontheotherhand,thecorrelationfunctionscanbeeasilyderivedfromEq. 3{93 .Weobtain and where A=n0!4n++n01 20q4:(3{103) 69

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3{78 ,thedensity-densitycorrelationfunctionbecomes where Wecannowperformtheanalyticcontinuation(i!n=!+i)fromh(Qn)(Qn)itoh(Q)(Q)i,resultingin whereQ(q;!). Then,theresponsefunctioncanbeobtainedbytakingtheimaginarypartofthedensity-densitycorrelationfunctionEq. 3{107 : wherewehaveusedtheidentity 1 Itiseasytoshowthattheresponsefunctionsatisesthesumrules:thethermodynamicsumrule 00(q;!) where(q)isthestaticdensity-densitycorrelationfunction(AppendixC),andthef-sumrule !00(q;!)=0q2:(3{111) 70

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4 ],usingthevariationalprinciple.Startingfromthederivednon-dissipativehydrodynamicswecalculatedboththefourthsoundspeedandthesecondsoundspeed.Inaddition,weobtainedthedensity-densitycorrelationfunctionofamodelsupersolidusingtheLagrangiandensity,andfoundthateachpairoflongitudinalpropagatingmodesproducessingularitiesinthedensity-densitycorrelationfunctionandapairof-functionpeaksintheresponsefunction. 71

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InChapter3weobtainedthenon-dissipativehydrodynamicsforasupersolidusingthevariationalprinciple,andshowedthattheequationsofmotionarethosewhichAndreevandLifshitzhadderivedwithoutdissipation[ 4 ].Usingthelinearizedequationsofmotionwefoundthatforatwo-dimensionalsupersolidthereareintotalsixpropagatingmodes:onepairoftransversesoundmodes,onepairoflongitudinalrstsoundmodes,andonepairoflongitudinalsecondsoundmodes.Thelongitudinalpropagatingmodesappearinthedensity-densitycorrelationfunctionasdoubledelta-functionpeakslocatedat!=cqwithasoundspeedc. Whendissipationistakenintoaccount,themodestructureofthesystemchanges:duetodissipationsomediusivemodesappear,e.g.,athermaldiusionmodearisesduetoviscosity.Ontheotherhand,dissipationdampsthesoundmode.Thedispersionrelationofapropagatingmodeinthelimitofsmallqisgivenby whereDistheattenuationcoecient. Forathreedimensionalnormalsolidthereareveconservationlaws:mass,energyandthreecomponentsofmomentum.Inaddition,thetranslationalsymmetriesarebrokeninthreedierentdirections,andthereareeighthydrodynamicmodesforathreedimensionalnormalsolid.Thecorrespondinghydrodynamicmodesarefourtransversesoundmodes,twolongitudinalrstsoundmodes,onethermaldiusionmodeandonedefectdiusionmode[ 76 ]. Incontrast,aswediscussedinChapter3thereareninehydrodynamicmodesforathree-dimensionalsupersolidbecausetheadditionalbrokengaugesymmetryduetotheBosecondensation.Becauseofthisbrokengaugesymmetryoneofthediusionmodesofthenormalsolidbecomesapairofpropagatinglongitudinalsecondsound 72

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4 ].Theirderivationstartsbyidentifyingninehydrodynamicvariables,andwritingdowntheconservationequationsfortheconserveddensitiesandadditionalequationsforthebrokensymmetryvariables.Themass,themomentumandtheenergyareconserved: whereikisthestresstensorandQtheenergycurrent.Inadditiontotheseconservationlaws,thereareequationsofmotionforthebrokensymmetryvariables.First,fromthebrokentranslationalsymmetryinthethreespatialdirectionswehave whereJisanarbitraryfunctiontobedetermined.Foraperfectsolid,i.e.,solidwithoutanydefects,thedensitychangeislinkedtothelatticeuctuation(ru).However,AndreevandLifshitzpointedoutthatthedensityuctuationisindependentofthelatticedisplacementduetodefects;therefore,thedensitybecomesaseparatehydrodynamicvariable.SincethegaugesymmetryoftheBose-Einsteincondensatewavefunctionis 73

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ThecurrentsinvolvedintheconservationlawsandthearbitraryfunctionsJandaretobedetermined.Inadditiontotheseeighthydrodynamicequations,thereisanadditionalequationwhichistheentropyproductionequation T;(4{7) whereqistheheatcurrentandRisthepositivedenitedissipationfunction.Theredundancycondition,Eq. 4{7 ,combinedwiththethermodynamicsandGalileancovarianceallowsustoobtaintheconstitutiverelationsthatrelatecurrents,J,andtohydrodynamicvariables.Theresultingreversiblecurrentsare Rij=Ts(vnkvsk)pkij+vsjvsi+vsipj+vnjpijijkwik;(4{8) R=+1 2vs2:(4{10) Ontheotherhand,usingthepositivenessofRweobtainthedissipativecurrents: D=@i[pi(vnivsi)]ki@ivnk;(4{12) Dki=kilm@lvnmki@l[pl(vnlvsl)];(4{14) whereikisthethermalconductivitytensor,iklm,,andikaretheviscositytensors,ikisthethermodiusioncoecienttensorforthedefects,andikisthedefectdiusioncoecienttensor.Notethatthedissipativecurrentsarelinkedtothequantitiesofoppositetime-reversalproperties.Thereversiblecurrenthasthesametime-reversal 74

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Wearenowinapositiontoinvestigatethemodestructureofasupersolidwithdissipation.Sinceweareinterestedinthecollectivemotionofdefectsratherthanthermaldiusion,wewillneglectthermaluctuations(ik=ik=0).Alsoweconsider,forsimplicity,anisotropictwo-dimensionalsupersolid.AswediscussedinChapter3,thereductionofdimensionforasolidwithisotropyresultsinremovingapairoftransversepropagatingmodes.Duetoisotropy,wehavefortheviscositycoecients 3iklm;(4{15) ik=ik;(4{16) ik=ik:(4{17) Thenthehydrodynamicequationsforatwo-dimensionalisotropicsupersolid,includingthenonlineartermwhichwasneglectedbyAndreevandLifshitz,are Weconsideructuationsfromtheequilibriumvalues,Eqs. 3{63 through 3{66 and @wij+@ @wijwij=1 75

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wherewehaveusedtherepresentationsEqs. 3{89 through 3{91 andtheMaxwellrelation(@=@wlm)=(@lm=@)wij,andlinearizedEqs. 4{18 through 4{21 .Wenextdividethelinearizedhydrodynamicequationsintothetransverseandlongitudinalparts.First,theFouriertransformedequationsofmotionforthetransversepartare Consequently,wegetthetransversesoundspeedobtainedintheprevioussectionwiththeattenuationconstantDT=+n0~.Second,thelongitudinalequationsofmotionare wheres0,and~+.SoundspeedsandtheattenuationconstantscanbeobtainedbycalculatingthedispersionrelationsafterFouriertransformingEqs. 4{26 through 4{29 .WeLaplace-Fouriertransformtheminsteadforlaterpurposes.AfterLaplace-Fouriertransformingwend 76

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n0iz+q21 FromEq. 4{31 weobtaintwosoundspeedscL,Eq. 3{96 andc2,Eq. 3{97 ,withtwoattenuationconstants,respectively, where Nowwecanseethatwhens=0,i.e.anormalsolid,thesecondsoundmodesdisappearbutthereisthedefectdiusionmodewiththediusionconstant whichagreeswiththeresultobtainedbyZippeliusetal.[ 73 ]. 77 ] 77

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74 78 ].WestartthissectionbyreviewingthelightscatteringexperimentsdoneonnormaluidsandontheHeIIsuperuid.Wethencalculatethedensity-densitycorrelationfunctionforamodelsupersolid,anddescribealightscatteringexperimentfordetectingthesupersolidtransition. 79 ]: @PT1cv @PTcv @PTcv wherecvisthevolumespecicheatatconstantvolume,cPthevolumespecicheatatconstant,thepressure.Thethermaldiusionconstantis cP;(4{39) withthethermalconductivity.Thesoundspeedofnormaluidisc2NF=(@P=@)swithanattenuationconstant 3++ cPcP whereistheshearviscosityandthebulkviscosity.TherstterminEq. 4{38 isaLorentzianofwidth2Dthq2locatedat!=0duetothethermaldiusion:foreachdiusionmodeaLorentzianpeak(theRayleighpeak)atthecenterappears.Theremainingtwotermsarecontributionsfromthesoundmodes.ThesecondterminEq. 4{38 isapairofLorentziansofwidthDNFq2locatedatcNFq(theBrillouindoublet).Thelasttermdoesnotcontributemuchinthecorrelationfunction;however,itsexistence 78

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74 78 ].Figure 4-1 showsthespectrumofliquidargonatT=84:97KmeasuredbyFleuryandBoon[ 80 ]. Thedensity-densitycorrelationfunctionforasuperuidwasobtainedbyHohenbergandMartinusingthetwo-uidhydrodynamics[ 81 ].Therstsoundcontributiontothedensity-densitycorrelationfunctionis[ 81 82 ] whereS1(q)=kBT2KT=withtheisothermalcompressibilityKT=1=(@=@T)P,and 3+:(4{42) Thecontributionofthesecondsoundis[ 81 82 ] (!2c22q2)2+(D+Dth)2q4!2S2(q);(4{43) whereS2(q)=(cP=cv1)S1(q)and where1,2,3and4(=1)arethecoecientsofsecondviscosity,andisthecoecientofrstviscosity.FortemperaturesaboveT,wehaves=0,andEq. 4{43 reducestotherstterminEq. 4{38 ;therefore,theRayleighpeakofthethermaldiusionmodeinnormaluidsplitsintoaBrillouindoubletofthesecondsoundmodes.ThissplittingwasobservedinthelightscatteringexperimentsbyWinterlingetal.[ 83 ]andbyTarvinetal.[ 84 ](Fig. 4-2 ). 4{30 .Inordertoobtain 79

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4{31 .Theexpressionforthedensity-densityKubofunctioncanbefoundinAppendixE,Eq. E{1 .Thenthecorrelationfunction00(q;!)canbeobtainedbytakingtherealpartofEq. E{1 : (!2c2Lq2)2+(!q2DL)2+(!2c2Lq2)I3(q) (!2c2Lq2)2+(!q2DL)2iq4c22D2I2(q) (!2c22q2)2+(!q2D2)2+(!2c22q2)I4(q) (!2c22q2)2+(!q2D2)2; whereI1(q),I2(q),I3(q),andI4(q)aregiveninAppendixE.TherstandthirdtermsinEq. 4{45 givetwoBrillouindoubletscenteredat!=cLq,and!=c2qwithwidthDLq2andD2q2,respectively.ThesecondandfourthtermsinEq. 4{45 arenegligibleneartheBrillouindoublets.Theobtaineddensity-densitycorrelationfunctionsatisesboththethermodynamicsumrule,Eq. 3{110 ,andf-sumrule,Eq. 3{111 Ontheotherhand,wecanshowthatwhendissipationisneglected,thesecondandfourthtermsinEq. 4{45 vanish,anddeltafunctionsareobtainedfromtheremainingtwotermsbytakingthelimitDL;D2!0.Therefore,thesusceptibility,Eq. 4{45 ,reducestothenon-dissipativedensity-densitycorrelationfunction,Eq. 3{108 Moreover,forthenormalsolid(s=0),thesecondterminEq. 4{45 vanishessothatthereisonlyoneBrillouindoubletduetothelongitudinalrstsoundmodes.Atthesametime,thefourthterminEq. 4{45 becomesaLorentziancenteredat!=0whichistheRayleighpeakduetothedefectdiusion.Therefore,weconcludethatthesamesplittingasinthetransitionfromanormaluidtoasuperuidoccursinthetransitionfromanormalsolidtoasupersolid. Wenowstudyqualitativelythedensity-densitycorrelationfunctionEq. 4{45 .Letussetrst===0forsimplicity.Thenforsmalls,thedensity-densitycorrelation 80

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(!2c2Lq2=c2NS)2+(!q2DL=D)2+x1+2(x1)2 (!2c22q2=c2NS)2+(!q2D2=D)2;(4{46) wherex=0c2NS,y==0D,!(D=c2NS)!andq(D=cNS)qwiththedefectdiusionconstantD==andthelongitudinalsoundspeedofnormalsolidcNS,Eq. 3{99 .WeshowinFig. 4-3 thenormalizeddensity-densitycorrelationfunctionsofanormalsolidandasupersolid.Todoso,wehaveusedtherstsoundspeedcNS=550m/s,thedensity=0:19048g/cm3,theisothermalcompressibility=0:29615108cms2/gforsolid4He[ 85 86 ]atthemolarvolume21cm3/mole,theviscosityof4Heuid2105g/cms,thetypicalwavenumberinvolvedinlightscatteringq1=100nm,and=81011cm3s/g.InFig. 4-4 weshowthesplittingoftheRayleighpeakduetothedefectdiusionofnormalsolidintoanadditionalBrillouindoubletofthesecondsoundfordierentvaluesofsupersolidfraction. Therefore,weconcludethatthelightscatteringonasolid4Hecouldprovideuswithveryrichinformationaboutthesupersolidphaseanalogoustothesuperuidphase.ThedetectionoftheadditionalBrillouindoubletinthespectrumofscatteredlightwillgiveanothersignaturefortheexistenceofthesupersolid.However,adetectionofthediusionmodeofdefectsmustbeprecededbythatofthepropagatingmodes.Thedefect-mediatedsupersolidpredictedbyAndreevandLifshitzassumestheexistenceofasucientlylargenumberofdefects.Butthesmallnumberofthermaldefectspresentinasolid4He[ 87 ]andthelargeactivationenergiesofvacanciesandinterstitials[ 32 ]areproblematicinrealizingtheproposedBrillouindoubletofthesecondsoundmodes. 81

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Brillouinspectrumofliquidargon.Reprintedgure3withpermissionfromP.A.FleuryandJ.P.Boon,Phys.Rev.186,244(1969).Copyright(1969)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRev.186.244). Figure4-2. Brillouinspectraof4Hesuperuid.Reprintedgure6withpermissionfromJ.A.Tarvin,F.Vidal,andT.J.Greytak,Phys.Rev.B15,4193(1977).Copyright(1977)bytheAmericanPhysicalSociety(http://link.aps.org/doi/10.1103/PhysRevB.15.4193). 82

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Density-densitycorrelationfunctionsofisothermalandisotropicnormalsolids(dashedline)andsupersolids(solidline). 83

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SplittingoftheRayleighpeak(dashedline)duetothedefectdiusionmodeofanormalsolidintotheBrillouindoubletofthesecondsoundmodes. 84

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InChapter1wediscussedthegrowingtheoreticalandexperimentalinterestindislocationsandgrainboundariesinsolidsandvorticesinsuperuids,duetotheanomalousbehaviorsofsolid4Heatlowtemperatures.InthisChapterwestudythedynamicsofvorticesanddislocationspresentinsupersolids.Ourprimaryobjectiveinthischapteristoderiveaneectiveactionforvorticesanddislocationsinsupersolids,andstudytheirpropertiessuchastheinertialmassofatopologicaldefect. WestartwiththeLagrangiandensityderivedinChapter3,Eq. 3{68 ,forsupersolidswithoutthermaluctuations: 2(@i)2@ @wijwij1 2@ @wij()2+1 2nij(@tui@i)(@tuj@j)1 2@ij wherewehaveshifted=tandomitted,forsimplicity,thesubscript`0'usedtoindicatetheconstantequilibriumvalues.ThedynamicalvariablesinEq. 5{1 are,,andui.AswediscussedinChapter3thersttwotermsgenerallycanbeignoredbecausetheyaretotalderivatives.However,whenvorticesanddislocationsareconsidered,itisnotpossibletosimplydropthesetermsbecauseofsingularitiesduetodefects.InfactaswediscussinSections5.1and5.2,thersttermisresponsiblefortheMagnusforceforvortices[ 70 ]andthesecondtermthePeach-Koehlerforcefordislocations[ 71 72 ].Beforeweproceed,we\integrateout"thedensityuctuationstoobtain 2@ @wij(@t)21 2sij@i@j+1 2nij@tui@tuj1 2@ij 2nij+@ @wij@tui@j+wij@t; wherewehavedroppedconstantswhichdonotaectonthedynamics,andusedEqs. 3{75 and 3{79 .Inaddition,fortheinteractiontermsbetweensuperuidityandelasticitywedistinguishedexplicitly@tui@ifromwij@t,eventhoughtheycan 85

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Weconsideratwo-dimensionalisotropicsupersolidforsimplicity.WiththeassumptionofisotropyandtherepresentationsofEqs. 3{89 through 3{91 ,theEuclideanaction(=it)oftheLagrangianEq. 5{2 canbewrittenas 2ZdZd2x2i@+2ijuij+2(@)2+s(@i)2+n(@ui)2+(~22)u2ii+2~u2ij+i(n2)(@ui@i+uii@); whichisourstartingpointtoderivetheeectiveactionforvorticesanddislocations,andstudytheirdynamicsinsupersolids. me;(5{4) wheremisthemassofparticleofsupersolid.Fortwo-dimensionalsupersolidsvorticesarepoint-likeparticles,andthelineintegralistakeninthecounterclockwisedirectionaroundtheaxis^zperpendiculartothetwo-dimensionalsupersolidsystem.Wenowcanseparateintotwopartssuchthat 86

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5{4 alongapathenclosingvortices mXe;(5{6) whichimplies mXe^z(2)(xx);(5{7) wheree=1isthechargeofthe-thvortexlocatedatx()=[x();y()].Inaddition,thevelocityofthesuper-componentcanbewrittenasthesumofalongitudinalpart@iSandatransversepart@iV.Since@2V=0,forVwetaketheansatz and,inthefollowing,wederiveaneectiveactionforvorticesinsupersolidsintermsofvortexcoordinatesx().Let'srstinsertEq. 5{5 intoEq. 5{3 .Thentheactionbecomes where 2ZdZd2x2(@S)2+s(@iS)2+2(@V)2+s(@iV)2+22(@V)(@S)+n(@ui)2+(~22)u2ii+2~u2ij+2i(n2)@ui@iS+i(n2)(@ui@iV+uii@V): ThelinearterminuijisdroppedinSEbecausetherearenodislocations.AsshowninRef.[ 70 ],fromS1onecanderiveatransverseforce,knownastheMagnusforce,thatactsonavortex.Since@=Psi()@=@xi()withsi()=dxi()=dbeingthevortexvelocity,S1canberewrittenintermsofthevortexvelocityandaneectivevector 87

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wherewehavedened Therefore,the-thvortexmoveswithavelocitys()inaneectivemagneticeldgivenbyrxaA(x)=he^z=m.TheMagnusforce[ 70 88 ]actingonthe-thvortexisthengivenby whichshowsamotionofvortexperpendiculartotheappliedforce. NowweeliminatethedisplacementvectoruandtheanalyticphaseSinS2byusingtheequationsofmotiontoexpressS2intermsofV.Asusual,theequationsofmotioncanbecalculatedbytakingvariationsofSEwithrespecttoSandu.ThedecompositionofuintothelongitudinalpartuLandthetransversepartuTleadsustotheequationsofmotion: 2(n2)@(@V);(5{15) 2(n2)@(@V);(5{17) where=~+2~.TheseequationsofmotionareinhomogeneousdierentialequationsforuandSwithsourcetermsfromV:@Vforlongitudinalwavemodesand@iVfortransversewavemodes.Whenn=2,thereisnosourcetermforthedisplacementvectorasexpectedbecausethereisnocouplingbetweensuperuidityandelasticity.Inthiscase,Eqs. 5{15 through 5{17 becomeordinarywaveequations.TheequationofmotionforthetransverseparthasthesameformasEq.10ofRef.[ 89 ]inwhichthevortexdynamicsinasuperconductorwasstudied.Intheirwork,theyestimatedthesizeofthesheardeformationcausedbyavortexwhichismovingwithavelocitymuchless 88

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89 ]wecanalsoestimatethemaximalsheardeformationduetoavortexforsolid4He: nc2T1:57108[m2/s](n2)s nc2T:(5{18) Takingthephenomenologicalcouplingconstanttobezero,andthevortexspeedtobescT300m/s,themaximalsheardeformationumaxT51011mwhichisnegligiblesimilartotheconclusionobtainedinRef.[ 89 ]. WenowsolveforuL,S,anduT,inFourierspace,resultingin n!2+~q2(@iV)(q;!);(5{21) whereAisgivenbyEq. 3{103 ,and(@iV)(q;!)and(@V)(q;!)arethetemporalFouriertransformsof (@iV)(q;)=ih mikqk (@V)(q;)=ih mikqk whichareeasilycalculatedfromEq. 5{8 .ByreplacingEqs. 5{19 and 5{21 intoEq. 5{11 ,S2reducestothefollowingform: 89

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withcand~cbeingsomesoundspeeds,and InEq. 5{27 ,wehaveD=A,a=ns+(n2)2=4andb=s(22)+(n2)2s=42forS2writteninEq. 5{24 .Therefore,wederivetheactionbyevaluatingtheintegralsdenedinEqs. 5{25 through 5{26 .First,thetermwithF0inEq. 5{24 islocalintimeandhasalogarithmicdivergence[F0(x)ln(x)].However,weshowbelowthatthislogarithmicdivergenceiscanceledoutwithcontributionsfromtheothertwotermsinEq. 5{24 .Second,wenowevaluateG1byseparatingoutthelocalterm,resultingin where cp Third,tocalculateG2wedivideitintotwotermsbyexpandinginpartialfractions:onewiththesoundspeedcandtheotherwith~c G2(x;;c;~c)=A nF2(x;;c)+B nF2(x;;~c);(5{31) 90

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c2~c2;(5{32) ThesecondterminEq. 5{24 containsspatialderivativesofG2.TherstspatialderivativesequalminustheLaplacianofG2because@k=@k.Next,inRef.[ 90 ]itisshownthatthesecondspatialderivativeswithvortexvelocitiescanbeconvertedintotemporalderivatives:fromsi()sj(0)@i@jto@@0.Hence,wecanevaluateG2aswellasthethirdterminEq. 5{24 bycalculating@2F2and@2F2.Infact,theycanbewrittenintermsofF0andF1: and UsingEqs. 5{30 5{35 and 5{36 ,weobtain where c2Lc22;(5{38) 91

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c22c2L:(5{39) Notethatthelocaltermcancelsoutbecausea=sn+(n2)2=4,andwenallygettheactionforvorticesinsupersolids Fromthederivedactionwecaninferthateachwavemodecouplingtovorticesproducesanon-local\Coulomb"potentialterm.Inparticular,awavemodeperpendiculartothevortexvelocitygeneratesanadditionaltermwhichisproportionaltotheproductoftwovortexvelocitiesatdierenttimes.Withoutcoupling,theelasticvariableisnotrelatedtothephasevariablesothatonecanintegrateouttheelasticpartinEq. 5{11 ,andtheremainingpartofactionisofasuperuidwithvortices.Inthiscase(n=2),wegetA=0andB=ns,andthederivedLagrangianforvorticesreducesto whichistheresultobtainedbyEckernandSchmid[ 90 ]. Wecalculatethevortexmassbytakingthelimitasc2(0)2jxi()xi(0)j2fromthederivedaction,Eq. 5{40 .Thenweobtainafrequency-dependentmassofvortexinasupersolid 4(n2)21 ~+1 where~=0:5772:::isEuler'sconstant,andisacut-ointroducedtoregulatethedivergenceintemporalintegrals.Forthecaseinwhich=0,theinertialmassofavortex 92

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~+1 Onceagain,withoutthecouplingofsuperuidtothedisplacementvectorwerecoverthemassofvortexinsuperuid,obtainedbyEckernandSchimd. wherebistheBurgersvector.InacompletecirculationaroundadislocationaBurgersvectorcausesamismatchtothecirculationinanideallatticebyanamountofb.Therearetwotypesofdislocationlines:edgedislocationsandscrewdislocations.Iflisthetangentunitvectortothedislocationline,foredgedislocationsl?bwhileforscrewdislocationslkb.Sinceweworkwithtwo-dimensionalsupersolids,thereareonlyedgedislocationsbecausescrewdislocationsproduceadisplacementperpendiculartotheplaneofthesystem. ThederivationoftheeectiveactionfordislocationsisanalogoustothecalculationdoneforvorticesinSection5.1.Firstwetakeintoaccountdislocationsbyintroducingexplicitlythenon-analyticalpartduetodislocationsinthedisplacementvector: whereuSisanalyticanduDissingular.Consequently,thecontourintegralforasystemofdislocationsresultsin 93

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wherebistheBurgersvectorofthe-thedgedislocationlocatedatx()=[x();y()].ThenuponreplacingthedecompositionofthedisplacementvectorintoEq. 5{3 ,theactioncanbewrittenasthesumoftwoparts: and 2ZdZd2x"n(@uSi)2+2n(@uSi)(@uDi)+n(@uDi)2+uSijSij+uSijDij+uDijSij+uDijDij+~(@)2+s(@i)2+2i~uSii@+i~@uDi@i+i~uDii@#; where and InS2wealsointroducedsomesimplenotations:~n2,~2and~~22.Second,weremovetheanalyticvariablesuSandintheactionbymeansoftheequationsofmotion.TheequationsofmotionareobtainedbytakingthevariationsoftheactionEq. 5{3 withrespecttoanduSi: ~@2+s@2+i~@uSii=i Whenthecouplingbetweenthesuperuidityandtheelasticityisneglected(~=0),uSandbecomecompletelydecoupledfromeachother,anddislocationsproduceonlyelastic 94

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2qjT1i0!@uDj(q;!)+iqkT1ijDjk(q;!)+~ 2!T1i0uDjj;(5{54) and 2qjT100!@uDj(q;!)+iqkT10jDjk(q;!)+~ 2!T100uDjj(5{55) where NowweuseEqs. 5{54 and 5{55 towritetheactionintermsofuD(alongwithDij).ThenthelaststepinderivingtheactionfordislocationsistousetheansatzforuD,Eq. 5{47 ,andexpresstheactionintermsofthecoordinatesofdislocations(AppendixFfordetails).Theresultis 95

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whereG(1)2,G(2)2,G(3)2andG(4)2canbewrittenintermsofF2byusingEqs. 5{31 through 5{33 with 4~;(5{60) 4~;(5{62) ~+~2 96

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InEq. 5{59 thespatialderivativesofthefunctionF2isinvolved.Since@i=[xi()xi(0)]=X@XwithX=jx()x(0)j,wehave cij1 cxixj Asinthecaseofvortices,weobtainednon-local\Coulomb"potentialsforthepropagatingmodesexistinginsupersolids(thesecondandthepenultimatetermsinEq. 5{59 ).Itisworthwhiletonoticethatthederivedactionfordislocationsisanisotropiceventhoughthesolidisisotropic.Thisisbecauseofthevectorialcharacteristicsofthesingularity(Burgersvector)duetodislocationswhichbreaksthediscretesymmetryofthelatticeandleavesthelatticeanisotropic.InthecaseofvorticestheeectiveactionEq. 5{40 remainsisotropicbecausethesingularitycausedbyavortexisascalar.Inaddition,withdislocationsatermlocalintimeappearsintheaction.Moreinterestingly,itdoesnotdependonthesuperuiddensity.Thusthislocaltermseemstobeintrinsictothedynamicsofdislocationsinnormalsolids;however,superuiditymakesacontributionthroughthecouplingwithdislocations(thetermwith~).Ifweneglecttheinteractionbetweenthesuperuidityandtheelasticity(~=0),thecoecientofthelocaltermbecomesnc2Lwhichcouldsuggestthatitsoriginisduetothelongitudinalelasticsoundmodes. Ontheotherhand,wecanobtainthePeach-KoehlerforcefromS1.Sincethestresstensorisconstant,itcanbetakenoutoftheintegral,andtheintegralofuDijbecomesasummationovereachdislocationofa\surface"integraloverthetwosidesofthecutalongwhichthesingulardisplacementvectorofthe-thdislocationundergoesadiscontinuityofb.ThetwosurfacesofthecutaredenotedbyonwhichuD=0and+onwhichuD=b,wherediandd+iaregivenby[^zdx()]iand[^zdx()]i,respectively 97

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5-1 ).Thenwehave whereuisthesingularpartofthe-thdislocation'sdisplacementvector.Thenegativeofcoecientofdx()intheintegrandofEq. 5{69 istheforceactingonthe-thdislocation.Bythecyclicpropertyweobtain whichisthePeach-Koehlerforceperpendiculartothevectorbinthelatticeplane[ 71 72 ]. Similartovorticeswenowcandeneaninertialmassforaslowlymovingdislocation.Inthelimitc2(0)2jxi()xi(0)j2,weobtain (~+2~)2+~2 Theeectivemassofadislocationisatensorbecauseofthereasondiscussedearlier.Ifisneglected,weget 2s(~+2~)2#): AneectiveactionfordislocationsinnormalsolidscanbeobtainedfromEq. 5{59 byneglectingthecouplingbetweensuperuidityandelasticity(~=0).Foranormalsolid, 98

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Insummary,inthischapterwehaveconsideredtwobasictopologicaldefects(vorticesandedgedislocations)whichcouldexistinasupersolidphase.Wehavederivedtheeectiveactionsintermsofthecoordinatesoftopologicaldefects.Theinertialmassassociatedwiththekineticenergyofsuchdefectscouplingwiththeelasticdeformationandsuperuiditywereobtained. 99

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Cutforanedgedislocation.OnsurfaceuD=0whereason+uD=b.Theunitnormalsofaregivenby^zd^x. 100

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Weproposedaviscoelasticsolidmodelasanalternativeexplanationfortherecenttorsionaloscillatorobservationsonsolid4He.Wehaverststudiedthedynamicalresponseofisotropicviscoelasticsolidswithcylindricalsymmetrytoanoscillatoryshearstressusingtheno-slipboundarycondition.Wefoundtheelasticresonanceeectintheeectivemomentofinertiaandtheeectivedampingcoecients.Atlowdrivingfrequencies,theeectivemomentofinertiadecreasesastheshearmodulusincreases.ThisleadsustoapossibleconnectionbetweentheanomalousincreaseinshearmodulusobtainedbyDayandBeamish[ 34 ]andtheNCRIobservedbyKimandChan[ 19 20 ].However,thequantitativeestimateoftheperiodshiftduetothe10%increaseintheshearmoduluswasfoundtobeonlyaboutonehundredthofthetheobservedperiodshiftinTOexperiments.Wealsofoundthatthefrequency-dependentcomplexshearmodulusofaviscoelasticsolidcanexplaintheresultsoftheTOexperiments.Inourmodel,acharacteristictime,whichisrelatedtotheviscosityofsolidsincreasesrapidlyasthetemperatureislowered,andcausesboththedropintheresonantperiodandthepeakintheinverseofQ-factor.OurviscoelasticmodelpredictsQ1max=(P=P0)=1asothertheoreticalworks[ 49 50 ];however,theexperimentalvaluesofthisratioislessthanthepredictedvaluevaryingfrom0.01to0.65.TheconsequenceofthisdiscrepancybetweenthetheoriesandtheexperimentsisthatanytheoreticalmodeldoesnottboththeperiodshiftandthechangeintheinverseofQ-factor.Inthiswork,wefoundthatwecouldidentifythechangeintheinverseofQ-factor,butonlythe10%oftheresonantperiodshift.Thisfactcouldsuggestthattheunexplainedpartoftheperiodshiftmightbeduetosupersolidity. Inthesecondpartofworkweinvestigatedthehydrodynamicpropertiesofamodelsupersolid.WeintroducedthevariationalprincipletoderiveaLagrangiandensityaswellasthenon-dissipativehydrodynamicsforsupersolids.Oneofourmainresultsisthat 101

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Next,wehavecalculatedthedensity-densitycorrelationfunctionforanisotropicsupersolidinlocalthermalequilibriumusinglinearresponsetheory.First,wefoundthatwhendissipationisneglected,eachpairofpropagatingmodeswhosesoundspeediscproducesa-functionpairlocatedat!=cqinthedensity-densitycorrelationfunction.Second,wehaveextendedthestudytoincludedissipation.DuetodissipationtheneglecteddefectdiusionmodeofanormalsolidappearsasacentralRayleighpeakinthedensity-densitycorrelationfunction.Analogoustothesuperuidtransitionofliquid4He,wefoundthattheRayleighpeakduetothedefectdiusionmodeofanormalsolidbecomesaBrillouindoubletofthepairoflongitudinalsecondsoundmodesinthesupersolidtransition.WeproposedaBrillouinlightscatteringexperimenttoobservethissplittingasanalternativewaytodetectthesupersolidphase. Finally,wehavestudiedthedynamicsofvorticesanddislocationsinsupersolidsusingthederivedLagrangianforsupersolids.Aneectiveactionforvorticesandanotherfordislocationswereobtainedintermsofthecoordinatesofvorticesanddislocations.Inthecaseofvortices,wefoundthateachsoundmodeexistinginasupersolidcoupleswithvortices,andgeneratesanon-local\Coulomb"potential.Contrarytoasuperuid,transversepropagatingmodesexistinasupersolidandthesetransverseelasticmodescontributetotheactionforvorticesaswell.FordislocationseachmodeinsupersolidsproducesbothCoulombandnon-Coulombpotentialswhicharenon-localintime.Anothercontrasttovorticesisthatalocaltermwithalogarithmicdivergencearises.Asaresult,wecalculatedthefrequency-dependentinertialmassofvorticesanddislocations.Thevectorialnatureofsingularityofdislocationsdestroystheisotropyofsolidsandthederivedmassbecomesasecondranktensor.Inbothcasestheeectivemasshasa 102

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Thisthesislaysthegroundworkforseveralfuturestudies.First,theviscoelasticmodelforsolid4Heprovideduswitharelaxationtimewhichmightberelatedtodislocationsinsolids.Wehopethatitwouldbepossibletoderiveanalyticallytherelaxationtimeofviscoelasticityfromthedynamicequationsofdislocations.AssuggestedbyDayandBeamish,thedislocationmotioniscontrolledby3Heimpuritiesinsolid4He,andeectivelychangestheshearmodulusand,possibly,therelaxationtimeofsolid4He.Consequently,wewouldunderstandbettertheconnectionbetweentheshearmodulusexperimentandtheTOexperiment.Inthisregard,thederivedactionofamodel(super)solidwithdislocationwillbeusefulbecauseitdescribesthedislocationdynamicsin(super)solids.Second,wecanextendtheviscoelasticmodelforsolid4HeinaTOinseveraldirections.WecaninvestigatetheresponseofasupersolidintheTOexperimentbyusingthederivedhydrodynamicequationofsupersolids.Weexpectthatwhenthesupersolidhydrodynamicsiscombinedwiththeviscoelasticmodel,itwouldbepossibletotalltheTOresultsmoreprecisely.Alternatively,wehopethatthemodelcouldbeimprovedbyconsideringtheinhomogeneityofthesystemand/oradierentboundarycondition.WethinkthatthelocalvariationoftherelaxationtimecouldsmearoutoneoftheTOresponsesofthehomogeneoussystem,andtheslipboundaryconditionwouldprovideuswithalargerperiodshift.Finally,weshowedthatthevariationalmethodissystematicallyeectiveinderivingthenon-dissipativehydrodynamicsandtheLagrangiandensityofacontinuummediumwhichischaracterizedwithconservationlawsandbrokensymmetries.ThederivedLagrangiandensitywasusefulinstudyingthedynamicsoftopologicaldefects.Weplantoapplythevariationalprincipleinothersystems;possible 103

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104

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Inthisappendixweshowthedetailedcalculationtogetthebackactiontermsfordierentgeometriesoftorsioncells.Takingu(t)exp(i!t),Eq. 2{12 reducesto whereqEp A{1 forthreedierentboundaryconditions:innitecylinder,nitecylinder,andinniteannulus.Forthecoordinatesystemusedinthisappendix,refertoFig. 2-1 A{1 becomes ThedierentialequationEq. A{2 canbereducedtotheBesselequationoforderone.Applyingtheboundaryconditionandthenitenessatthecenterwegetthedisplacementeld withqER=!E.Theobtaineddisplacementelduyieldsanon-vanishingstress Thecorrespondingtotaltorqueforaheighthis wherethemomentofinertiaofrigidbodyIRB=R4h=2.UsingEqs. 2{17 and 2{18 ,fromEq. A{5 ,wenallygettheeectivemomentofinertia 105

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Also,wegetthebackactiontermusingEq. 2{6 InthiscaseweneedtosolveEq. A{1 : Wedecomposeuintotwoparts:u=0r+V(r;z)withV=0attheboundary.ThenEq. A{10 reducesto ThesolutionofEq. A{11 is wheretheGreen'sfunctionG(x;x0)satisestheinhomogeneousdierentialequation r(rr0)(0)(zz0);(A{13) 106

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21Xm=eim(0);(A{15) wecanexpandG(x;x0)as 221Xm=Z10dkeim(0)cos[k(zz0)]gm(k;r;r0):(A{16) ReplacingthisintoEq. A{13 ,wegetthemodiedBesselequationoforderm0forgm(k;r;r0): r(rr0);(A{17) wherek02k2q2Eandm02m2+1.Thegeneralsolution,usingthenitenessconditionatthecenter,is wherer<=minfr;r0gandr>=maxfr;r0g.Thentheuseoftheboundaryconditionatr=r0, r0;(A{19) yieldstheGreen'sfunction Finally,usingEq. A{12 wegetthedisplacementeld hzr 2mRI1(mr) 107

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Firstweseethatwerecovertheresultsoftheinnitecylindercasetakingthelimitthath!1.Inthislimit,mbecomesiqE,andtherstterminthesecondsquaredbracketcancelstherstterminEq. A{21 ,andthesecondterminthesecondsquaredbracketbecomesthedisplacementeldforinnitecylinderEq. A{3 Therearetwonon-vanishingcomponentsofthestresstensorwhichare ei!t1Xm=1(1)m+1 hzI2(mr) hzr 2mRI1(mr) Thetotaltorqueis (2m1)2R hm2H(im)8!2R20 where Then,theeectivemomentofinertiaforaviscoelasticcylinderis (2m1)22R2 108

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(2m1)22R2 Thebackactiontermis (2m1)2R hm2H(im)+8!2R2 ThedierentialequationforuisthesameasthatofaninnitecylinderEq. A{2 whosegeneralsolutionsareJ1(qEr)andN1(qEr).Sincethenitenessattheoriginisnolongernecessary,thedisplacementeldisgivenby where Thenon-vanishingstressandthetotaltorqueforthiscaseare 109

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Theeectivemomentofinertiais Theeectivedamptingcoecientis qEAJ2(qER)+BN2(qER))=(2hR2i! qEAJ2(qERi)+BN2(qERi)): Thebackactiontermis 110

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InChapter3wederivedthehydrodynamicsofasupersolidusingthevariationalprinciple,andshowedthatthevelocityofthesuper-componentisirrotational.InthisAppendixweshowthatitispossibletoderivesystematicallythehydrodynamicsofasupersolidwiththetransversepartofvswhichgeneratedbyvorticespresentinthesupersolid.ThetransversepartofvscanbeobtainedbyimposinganotherLin'sconstraint[ 61 ]: ThentheLagrangianoftheisotropicsupersolidwithalltheconstraints,Eqs. 3{22 3{24 3{25 and B{1 becomes 2svs2+1 2(s)vn2USS(;s;s;Rij)+(@t+@isvsi+(s)vni)+@ts+@i(svni)+(@t()+@isvsi+(s)vni)+i@t(sRi)+@j(sRivnj); where,,,andareLagrangianmultipliers,andwehaveusedthesecondLin'sconditioncombinedwiththemassconservationequation. Theequationsofmotionarecalculatedbytakingthevariationswithrespecttothedynamicalvariables: 2vn2@USS DtDn Dt=0(B{3) 2vs21 2vn2@USS Dn Dt+RiDni 111

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@t+svsi@i+(s)vni@i=0(B{9) FromEq. B{6 ,weobtainthefollowingClebschpotentialrepresentationsforthevelocityofthesuper-component Therefore,weget Ontheotherhand,theuseoftheClebschrepresentationofvsleadsusEq. 3{34 TakingthegradientofEq. B{3 ,wegettheEulerequationforvs 112

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(s)Dnvni 2(s)@i(vnjvsj)2@ts+@j(svnj)vsi@t(s)+@j(s)vnjvni+s 2(s)@i(vnjvsj)2(@t(s)+@j(s)vnj)(vnivsi);+s However,themomentumconservationequationforthiscaseinwhichrvs=0,isthesameasthatofthecaseofthelongitudinalvs,Eq. 3{41 113

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GivenEq. 3{19 ,thethermodynamicrelationofthestaticfreeenergydensityFSS=ESSTsiscalculated WithoutthermaluctuationsweexpandthestaticfreeenergydensityFSSuptothesecondorderinuctuationsinthedensityandthedisplacementvectoru: 2@ @wij()2+@ @wijwij+1 2@ij UsingEqs. 3{89 through 3{91 thefreeenergycanbewrittenas,inFourierspace, 2~q2u2T+1 2(q)uL(q)B0B@(q)uL(q)1CA;(C{3) where ThenthestaticcorrelationfunctionscanbeeasilyreadoutfromEq. C{3 .Weobtain 202;(C{5) q(202):(C{6) 114

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Ingeneral,theLaplace-Fouriertransformationsofhydrodynamicequationsofaphysicalsystemwillhaveaform: wherearesomehydrodynamicvariablesandAisanarbitrarymatrixwhichcanbeobtainedfromthehydrodynamicequations.Theindexindicatesthehydrodynamicvariable.Ontheotherhand,thelinearresponsetheorytellsushow(q;!)isrelatedto(q;t=0)throughtheresponsefunction[ 74 ], Notethat(q)arethestaticsusceptibilitieswhichcanbecalculatedfromstatisticalmechanics.Then,Eq. D{1 and D{2 implythat 1 wherewehaveused1=. Now,wedeneaKubofunctionK[ 78 ]as where~00(x;x0;t)isdenedas ~(x;x0;t)=2i(t)~00(x;x0;);(D{5) 115

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i00(x;x0;!) Therefore,usingEq. D{3 ,theKubofunctioncanbeobtained andthesusceptibility00isrelatedtothisKubofunctionthroughEq. D{4 116

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UsingEq. D{7 thedensity-densityKubofunctionisobtainedfromEq. 4{31 wherethestaticsusceptibilitiesaregiveninAppendixC,and n0q2;(E{3) n0+i;(E{4) n0q2s0q2+s0 n0q2;(E{5) n0q3+is0 Wecanrewritethedensity-densityKubofunctionbydoingpartialfractionexpansion.TheneachterminEq. E{1 canbeseparatedintotherstsoundpartandthesecondsoundpartasfollows whereAij,Bij,DijandEijcanbewrittenasafunctionofaij,bij,dijandeij,Eqs. E{3 through E{7 : (c2Lc22)2+q2(DLD2)(DLc22D2c2L)+ibij(c22DLD2c2L)+dij(c2Lc22)+ieij(DLD2) (c2Lc22)2+q2(DLD2)(DLc22D2c2L); 117

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(c2Lc22)2+q2(DLD2)(DLc22D2c2L)+idijc2Lq2(DLD2)+eij[c2Lc22+q2DL(D2DL)] (c2Lc22)2+q2(DLD2)(DLc22D2c2L); (c2Lc22)2+q2(DLD2)(DLc22D2c2L)++ibij(c2LD21c22)+dij(c22c2L)+ieij(D21) (c2Lc22)2+q2(DLD2)(DLc22D2c2L); (c2Lc22)2+q2(DLD2)(DLc22D2c2L)+idijc22q2(D21)+eij[c22c2L+q2D2(1D2)] (c2Lc22)2+q2(DLD2)(DLc22D2c2L): Thefunctionsinthedensity-densitycorrelationfunction,Eq. 2{7 ,aregivenby 118

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InthisappendixweshowthederivationofS2,Eq. 5{59 ,indetailstartingwithEq. 5{49 andtheansatzusedforuD,Eq. 5{47 .TheuseoftheequationsofmotionEqs. 5{54 and 5{55 withEqs. 5{56 through 5{58 leadsustotheactionintermsofthesingulardisplacementvector where 2Zd! 2Zd! 2Zd! 119

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5{47 ,onecaneasilycalculatethespatialFouriertransforms: Usingtheseequationsweget: whereG(1)2isdenedbyEq. 5{31 with 4~;(F{10) 120

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whereG(4)2,G(5)2,andG(6)2aredenedbyEq. 5{31 with ~;(F{15) 121

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5{31 with (F{22) whereG(8)2isdenedbyEq. 5{31 with ~ns+~2 InEqs. F{9 through F{22 ~G2isdenedas ~G2(x;;c;~c)=~A nF2(x;;c)+~B nF2(x;;~c);(F{25) where ~A~ac2~b c2~c2;(F{26) ~B~a~c2~b with~a=ba(c2+~c2)and~b=ac2~c2.InderivingEqs. F{9 through F{22 onecanndusefulthefactthatsi()@icanbechangedto@.NowwecancombineS(1)2,S(2)2,S(3)2,andS(4)2togetherbyusingEqs. 5{29 5{35 and 5{36 .WendthatthelocaltermwithF0

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5{59 123

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Chi-DeukYoowasborninSeoul,KoreaandimmigratedtoArgentinaduringhishighschoolyears.Henished,overcomingculturalshock,highschoolinBuenosAires,Argentina,andenrolledinthePhysicsDepartmentoftheUniversityofBuenosAires,wherehereceivedaLicenciadoenCienciasFsicasin2002.InthesameyearhewasadmittedtothePhysicsDepartmentoftheUniversityofFlorida.DuringtheearlyyearsattheUniversityofFloridahewasinterestedinhigh-energytheory,butin2004hejoinedProf.AlanT.Dorsey'sgrouptostudytheoreticalcondensedmatterphysics. 128