Graduate Texts in Physics Takafumi Kita Statistical Mechanics of Superconductivity Graduate Texts in Physics SeriesEditors ProfessorRichardNeeds CavendishLaboratory JJThomsonAvenue CambridgeCB30HE,UK [email protected] ProfessorWilliamT.Rhodes DepartmentofComputerandElectricalEngineeringandComputerScience ImagingScienceandTechnologyCenter FloridaAtlanticUniversity 777GladesRoadSE,Room456 BocaRaton,FL33431,USA [email protected] ProfessorSusanScott DepartmentofQuantumScience AustralianNationalUniversity ScienceRoad Acton0200,Australia [email protected] ProfessorH.EugeneStanley CenterforPolymerStudiesDepartmentofPhysics BostonUniversity 590CommonwealthAvenue,Room204B Boston,MA02215,USA [email protected] ProfessorMartinStutzmann WalterSchottkyInstitut TUMünchen 85748Garching,Germany [email protected] Graduate Texts in Physics publishes core learning/teachingmaterial for graduate- andadvanced-levelundergraduatecoursesontopicsofcurrentandemergingfields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions,derivations,experimentsandapplications(asrelevant)fortheirmastery and teaching, respectively. International in scope and relevance, the textbooks correspondtocoursesyllabisufficientlytoserveasrequiredreading.Theirdidactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledgeof,aresearchfield. Moreinformationaboutthisseriesathttp://www.springer.com/series/8431 Takafumi Kita Statistical Mechanics of Superconductivity 123 TakafumiKita DepartmentofPhysics HokkaidoUniversity Sapporo,Japan ISSN1868-4513 ISSN1868-4521 (electronic) GraduateTextsinPhysics ISBN978-4-431-55404-2 ISBN978-4-431-55405-9 (eBook) DOI10.1007/978-4-431-55405-9 LibraryofCongressControlNumber:2015939274 SpringerTokyoHeidelbergNewYorkDordrechtLondon Translation from the Japanese language edition: TOUKEIRIKIGAKU KARA RIKAI SURU TYODENDOURIRON by Takafumi Kita, (cid:2)c Saiensusha Co., Ltd. 1-3-25, Sendagaya, Shibuyaku, Tokyo1510051,Japan2013.AllRightsreserved ©SpringerJapan2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerJapanKKispartofSpringerScience+BusinessMedia(www.springer.com) Preface The purpose of this book is to present the fundamentals of the theory of superconductivityin a self-containedmanner by developingand illustrating every required technique of advanced equilibrium statistical mechanics. It is addressed to graduate and undergraduate students who have finished elementary courses of thermodynamics and quantum mechanics. No further background knowledge is requiredinreadingthroughallthechapters. Superconductivity is one of the most spectacular phenomena in nature and typical of broken symmetries. The Bardeen–Cooper–Schrieffer(BCS) theory that hasclarifiedithashadatremendousimpactonthewholefieldofphysics,ranging from condensed-matter physics itself to nuclear and particle physics. Hence, one may expect that learning superconductivity enables one to reach and acquire key concepts and techniques of modern theoretical physics. This book treats this fascinating topic from the viewpoint of statistical mechanics to clarify both mathematicalandlogicalstructuresofthetheoryastransparentlyaspossible. Standardtextbooksonthetopicusuallybeginbydescribingbasicexperimental results such as the Meissner effect, show subsequently that electron–phonon interactionsmay establish virtualattractive forces between electrons, and proceed to present the BCS theory for homogeneous systems. Descriptions of the phe- nomenologicalLondonandGinzburg–Landautheoriesareofteninsertedpriortothe microscopicBCStheory.Inthisway,onemayseethatthesetheoriescandescribe experiments exceedingly well and also acquire basic skills to use them for one’s ownpurposes.However,itmaynotbeentirelyclearinthisstandardapproachwhere superfluidity(flowwithoutdissipation)originates,whatcausestheMeissnereffect to expel the magnetic field from the bulk, or how phase coherence responsible for superfluidity is established. There is also a high threshold in learning about superconductivityforthosewhoarenotwellacquaintedwithelectromagnetismor especiallywellversedintopicsinsolid-statephysics. With these observations, this book adopts an alternative approach based on statistical mechanics. Specifically, it starts from statistical mechanics of quantum ideal gases, adding one by one every new element that is required in under- standing superconductivitytogether with relevanttechniquesof modern statistical v vi Preface mechanics.Thetheoryofsuperconductivityisdevelopedonthisbasisbytakingfull advantageofthesecond-quantizationmethodsothatmacroscopiccondensationinto atwo-particleboundstateismanifest.ThestartingpointistheBCSwavefunction inrealspace,whichiscloselyconnectedwiththecoherentstateforlasersandBose– Einsteincondensates.Adefiniteadvantageofthisapproachisthatphasecoherence isquiteapparent.Thebasicformulationistherebyperformedinrealspacetoderive the Bogoliubov–de Gennes equations so that inhomogeneous cases and arbitrary pairingsymmetrycanbestudiedonanequalfooting.TheBCStheoryispresented subsequentlyasanapplicationofittohomogeneouss-wavepairing. Itwouldbringgreatpleasuretome,theauthor,ifthebookishelpfulforstudents fullofcuriosityandpioneeringspirit.Finally,Iwouldliketoexpressmygratitude to Professor Koh Wada for a critical and careful reading of the manuscript and consequentusefulcomments. Sapporo,Japan TakafumiKita February2015 Contents 1 ReviewofThermodynamics............................................... 1 1.1 ThermodynamicsandHiking........................................ 1 1.2 EquationofState..................................................... 3 1.3 LawsofThermodynamics ........................................... 4 1.4 EquilibriumThermodynamics....................................... 5 1.4.1 BasicEquation.............................................. 6 1.4.2 EquilibriumConditions .................................... 6 1.4.3 LegendreTransformationandFreeEnergy ............... 7 1.4.4 ParticleNumberasaVariable.............................. 8 1.5 ThermodynamicConstructionofEntropyandInternal Energy ................................................................ 10 Problems ..................................................................... 11 2 BasicsofEquilibriumStatisticalMechanics............................. 13 2.1 EntropyinStatisticalMechanics .................................... 13 2.2 DerivingEquilibriumDistributions ................................. 16 2.2.1 MicrocanonicalDistribution ............................... 17 2.2.2 CanonicalDistribution ..................................... 18 2.2.3 GrandCanonicalDistribution.............................. 21 Problems ..................................................................... 23 References.................................................................... 23 3 QuantumMechanicsofIdenticalParticles.............................. 25 3.1 Permutation........................................................... 25 3.2 PermutationSymmetryofIdenticalParticles....................... 26 3.3 EigenspaceofPermutation........................................... 29 3.4 Bra-KetsforMany-BodyWaveFunctions.......................... 31 3.5 OrthonormalityandCompletenessofBra-Kets..................... 32 3.6 MatrixElementsofOperators....................................... 33 3.7 SummaryofTwoEquivalentDescriptions.......................... 33 3.8 SecondQuantizationforIdealGases................................ 34 3.9 CoherentState........................................................ 39 vii viii Contents Problems ..................................................................... 41 References.................................................................... 41 4 StatisticalMechanicsofIdealGases...................................... 43 4.1 BoseandFermiDistributions........................................ 43 4.2 Single-ParticleDensityofStates .................................... 45 4.3 MonoatomicGasesinThreeDimensions........................... 46 4.3.1 Single-ParticleDensityofStates........................... 46 4.3.2 ConnectionBetweenInternalEnergyandPressure....... 47 4.3.3 IntroducingDimensionlessVariables...................... 48 4.3.4 TemperatureDependencesofThermodynamic Quantities ................................................... 50 4.4 High-TemperatureExpansions ...................................... 51 4.5 FermionsatLowTemperatures...................................... 52 4.5.1 FermiEnergyandFermiWaveNumber................... 52 4.5.2 SommerfeldExpansion..................................... 53 4.5.3 ChemicalPotentialandHeatCapacity .................... 55 4.6 BosonsatLowTemperatures........................................ 56 4.6.1 CriticalTemperatureofCondensation..................... 56 4.6.2 ThermodynamicQuantitiesofT <T .................... 57 0 4.6.3 ChemicalPotentialandHeatCapacityforT&T ......... 57 0 4.7 Bose-EinsteinCondensationandDensityofStates................. 58 Problems ..................................................................... 59 References.................................................................... 60 5 DensityMatricesandTwo-ParticleCorrelations ....................... 61 5.1 DensityMatrices ..................................................... 61 5.2 Bloch–DeDominicisTheorem...................................... 62 5.3 Two-ParticleCorrelationsofMonoatomicIdealGases ............ 67 Problems ..................................................................... 71 References.................................................................... 71 6 Hartree–FockEquationsandLandau’sFermi-LiquidTheory........ 73 6.1 VariationalPrincipleinStatisticalMechanics ...................... 73 6.2 Hartree–FockEquations ............................................. 74 6.2.1 DerivationBasedontheVariationalPrinciple ............ 74 6.2.2 DerivationBasedonWickDecomposition................ 77 6.2.3 HomogeneousCases........................................ 78 6.3 ApplicationtoLow-TemperatureFermions......................... 80 6.3.1 FermiWaveNumberandFermiEnergy................... 80 6.3.2 EffectiveMass,DensityofStates,andHeatCapacity.... 80 6.3.3 EffectiveMassandLandauParameter..................... 82 6.3.4 SpinSusceptibility.......................................... 84 6.3.5 Compressibility............................................. 86 6.3.6 LandauParameters ......................................... 87 Contents ix Problems ..................................................................... 89 References.................................................................... 89 7 AttractiveInteractionandBoundStates................................. 91 7.1 AttractivePotentialinTwoandThreeDimensions................. 91 7.1.1 BoundStateinThreeDimensions......................... 92 7.1.2 BoundStateinTwoDimensions........................... 93 7.2 ConsiderationinWaveVectorDomain.............................. 94 7.3 Cooper’sProblem .................................................... 97 Problems ..................................................................... 98 References.................................................................... 99 8 Mean-FieldEquationsofSuperconductivity ............................ 101 8.1 BCSWaveFunctionforCooper-PairCondensation................ 101 8.2 QuasiparticleFieldforExcitations.................................. 103 8.3 Bogoliubov–deGennesEquations................................... 105 8.3.1 DerivationBasedonVariationalPrinciple ................ 106 8.3.2 DerivationBasedonWickDecomposition................ 113 8.3.3 MatrixRepresentationofSpinVariables.................. 115 8.3.4 BdGEquationsforHomogeneousCases.................. 117 8.4 ExpansionofPairingInteraction .................................... 119 8.4.1 IsotropicCases.............................................. 119 8.4.2 AnisotropicCases .......................................... 120 Problems ..................................................................... 122 References.................................................................... 122 9 BCSTheory ................................................................. 125 9.1 Self-ConsistencyEquations.......................................... 125 9.2 EffectivePairingInteraction......................................... 128 9.3 GapEquationandItsSolution....................................... 132 9.4 ThermodynamicProperties.......................................... 135 9.4.1 HeatCapacity............................................... 135 9.4.2 ChemicalPotential ......................................... 138 9.4.3 FreeEnergy................................................. 138 9.5 LandauTheoryofSecond-OrderPhaseTransition................. 139 Problems ..................................................................... 141 References.................................................................... 141 10 Superfluidity,MeissnerEffect,andFluxQuantization ................ 143 10.1 SuperfluidDensityandSpinSusceptibility......................... 143 10.1.1 SpinSusceptibility.......................................... 146 10.1.2 SuperfluidDensity.......................................... 147 10.1.3 Leggett’sTheoryofSuperfluidFermiLiquids............ 149 10.2 MeissnerEffectandFluxQuantization ............................. 151 10.2.1 Ampère’sLaw .............................................. 152 10.2.2 LondonEquation ........................................... 153