固体量子理论基础

ThispageintentionallyleftblankBASICASPECTSOFTHEQUANTUMTHEORYOFSOLIDSOrderandElementaryExcitationsAimedatgraduatestudentsandresearchers,thisbookcoversthekeyaspectsofthemodernquantumtheoryofsolids,includinguptodateideassuchasquantumfluctuationsandstrongelectroncorrelations.Itpresentsthemainconceptsofthemodernquantumtheoryofsolids,aswellasageneraldescriptionoftheessentialtheoreticalmethodsrequiredwhenworkingwiththesesystems.Diversetopicssuchasthegeneraltheoryofphasetransitions,harmonicandanharmoniclattices,Bosecondensationandsuperfluidity,modernaspectsofmagnetismincludingresonatingvalencebonds,electronsinmetals,andstrongelectroncorrelations,aretreatedusingtheunifyingconceptsoforderandelementaryexcitations.Themaintheoreticaltoolsusedtotreattheseproblemsareintroducedandexplainedinasimpleway,andtheirapplicationsaredemonstratedthroughconcreteexamples.danieli.khomskiiisaProfessorintheIIPhysikalischesInstitut,CologneUniversity.Hismainresearchinterestsarethetheoryofsystemswithstronglycorrelatedelectrons,metal–insulatortransitions,magnetism,orbitalorderingandsuperconductivity.BASICASPECTSOFTHEQUANTUMTHEORYOFSOLIDSOrderandElementaryExcitationsDANIELI.KHOMSKIICAMBRIDGEUNIVERSITYPRESSCambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo,Delhi,Dubai,TokyoCambridgeUniversityPressTheEdinburghBuilding,CambridgeCB28RU,UKFirstpublishedinprintformatISBN139780521835213ISBN139780521542920ISBN139780511788321©D.Khomskii20102010Informationonthistitle:www.cambridge.org9780521835213Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionofrelevantcollectivelicensingagreements,noreproductionofanypartmaytakeplacewithoutthewrittenpermissionofCambridgeUniversityPress.CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurlsforexternalorthirdpartyinternetwebsitesreferredtointhispublication,anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYorkwww.cambridge.orgPaperbackeBook(EBL)HardbackContentsForewordandgeneralintroductionpageix1Somebasicnotionsofclassicalandquantumstatisticalphysics11.1Gibbsdistributionfunctionandpartitionfunction11.2Thermodynamicfunctions21.3Systemswithvariablenumberofparticles;grandpartitionfunction42Generaltheoryofphasetransitions62.1Secondorderphasetransitions(Landautheory)62.2(Weak)Firstorderphasetransitions112.2.1Anotherpossibilityofgettingafirstorderphasetransition132.3Interactionwithotherdegreesoffreedom142.4Inhomogeneoussituations(Ginzburg–Landautheory)162.5Fluctuationsatthesecondorderphasetransitions192.5.1Criticalindicesandscalingrelations212.6Quantumphasetransitions232.7Generalconsiderations252.7.1Differenttypesoforderparameters252.7.2Generalprinciple252.7.3Brokensymmetryanddrivingforceofphasetransitions262.7.4TheGoldstonetheorem272.7.5Criticalpoints283BoseandFermistatistics314Phononsincrystals344.1Harmonicoscillator344.2Secondquantization354.3Physicalpropertiesofcrystalsintheharmonicapproximation38vviContents4.4Anharmoniceffects414.4.1Thermalexpansion434.4.2Melting454.4.3Anotherapproachtomelting.Quantummelting484.4.4Lowdimensionalsolids;whyisourworldthreedimensional?515GeneralBosesystems;Bosecondensation545.1Bosecondensation545.2WeaklyinteractingBosegas585.3Bosecondensationandsuperfluidity625.3.1Landaucriterionofsuperfluidity655.3.2Vorticesinasuperfluid676Magnetism706.1Basicnotions;differenttypesofmagneticresponse706.1.1Susceptibilityofnoninteractingspins746.2Interactinglocalizedmoments;magneticordering766.2.1Meanfieldapproximation776.2.2Landautheoryforferromagnets806.2.3Antiferromagneticinteractions846.2.4Generalcase876.3Quantumeffects:magnons,orspinwaves916.3.1Magnonsinferromagnets926.3.2Antiferromagneticmagnons.Zeropointoscillationsandtheirrole986.4Somemagneticmodels1046.4.1Onedimensionalmodels1056.4.2Resonatingvalencebonds,spinonsandholons1096.4.3Twodimensionalmodels1176.5Defectsandlocalizedstatesinmagneticandothersystems1237Electronsinmetals1277.1GeneralpropertiesofFermisystems1277.1.1Specificheatandsusceptibilityoffreeelectronsinmetals1298Interactingelectrons.GreenfunctionsandFeynmandiagrams(methodsoffieldtheoryinmanyparticlephysics)1338.1Introductiontofieldtheoreticalmethodsincondensedmatterphysics)1338.2Representationsinquantummechanics1368.3Greenfunctions1398.4Greenfunctionsoffree(noninteracting)electrons141Contentsvii8.5SpectralrepresentationofGreenfunctions1438.5.1PhysicalmeaningofthepolesofG(p,)1448.5.2PhysicalmeaningofthespectralfunctionA(p,)1468.6PhononGreenfunctions1478.7Diagramtechniques1498.7.1Dysonequations,selfenergyandpolarizationoperators1538.7.2Effectivemassoftheelectronexcitation1569ElectronswithCoulombinteraction1599.1Dielectricfunction,screening:randomphaseapproximation1599.2NestingandgiantKohnanomalies1669.3Frequencydependentdielectricfunction;dynamiceffects16910Fermiliquidtheoryanditspossiblegeneralizations17510.1ThefoundationsoftheFermiliquidtheory17510.2NonFermiliquidstates18310.2.1MarginalFermiliquid18310.2.2NonFermiliquidclosetoaquantumcriticalpoint18410.2.3MicroscopicmechanismsofnonFermiliquidbehaviour;Luttingerliquid18611Instabilitiesandphasetransitionsinelectronicsystems18811.1Peierlsstructuraltransition18811.1.1Qualitativeconsiderations18811.1.2Peierlsinstabilityinthegeneralcase19011.1.3DifferenttheoreticalwaystotreatPeierlsdistortion19211.1.4Peierlsdistortionandsomeofitsphysicalconsequencesinrealsystems19811.2SpinPeierlstransition20211.3Chargedensitywavesandstructuraltransitions,higherdimensionalsystems20611.4Excitonicinsulators20711.5Intermezzo:BCStheoryofsuperconductivity21211.6Spindensitywaves21611.7DifferenttypesofCDWandSDW22011.8Weaklyandstronglyinteractingfermions.Wignercrystallization22212Stronglycorrelatedelectrons22912.1Hubbardmodel23012.2Mottinsulators23012.3MagneticorderinginMottinsulators234viiiContents12.4Oneparticlespectrumofstronglycorrelatedsystems23512.4.1Aproximatetreatment(HubbardIdecoupling)23612.4.2DealingwithHubbardbands.Spectralweighttransfer23812.4.3Motionofelectronsandholesinanantiferromagneticbackground23912.5FerromagnetismintheHubbardmodel?24412.6PhasediagramoftheHubbardmodel24412.7Phaseseparation24712.8t–Jmodel25112.9OrbitalorderinginthedegenerateHubbardmodel25212.10Chargetransferinsulators25812.11Insulator–metaltransition26513Magneticimpuritiesinmetals,Kondoeffect,heavyfermionsandmixedvalence27213.1Localizedmagneticmomentsinmetals27213.2Kondoeffect27613.3Heavyfermionandmixedvalencesystems28213.4Kondoinsulators28813.5FerromagneticKondolatticeanddoubleexchangemechanismofferromagnetism290Bibliography296Index298ForewordandgeneralintroductionForewordTherearemanygoodbooksdescribingthefoundationsandbasicsofsolidstatephysics,suchasIntroductiontoSolidStatePhysicsbyC.Kittel(2004)or,onasomewhathigherlevel,SolidStatePhysicsbyN.W.AshcroftandN.D.Mermin(1976).Howeverthereisadefinitelackofbooksofamoreadvancedlevelwhichwoulddescribethemodernproblemsofsolidstatephysics(includingsometheoreticalmethods)onalevelaccessibleforanaveragegraduatestudentorayoungresearchworker,includingexperimentalists.Usuallythereexistsaratherwidegapbetweensuchbookswrittenfortheoreticiansandthoseforawideraudience.Asaresultmanynotionswhicharewidelyusednowadaysandwhichdetermine‘theface’ofmodernsolidstatephysicsremain‘hidden’andarenotevenmentionedintheavailableliteraturefornonspecialists.Theaimofthepresentbookistotrytofillthisgapbydescribingthebasicnotionsofpresentdaycondensedmatterphysicsinawayunderstandableforanaveragephysicistwhoisgoingtospecializeinbothexperimentalandtheoreticalsolidstatephysics,andmoregenerallyforeveryonewhoisgoingtobeintroducedtotheexcitingworldofmoderncondensedmatterphysics–asubjectverymuchaliveandconstantlyproducingnewsurprises.InwritingthisbookItriedtofollowaunifyingconceptthroughout.Thisconcept,whichisexplainedinmoredetailbelow,maybeformulatedastheconnectionbetweenanorderinasystemandelementaryexcitationsinit.Thesearethenotionswhichplayacrucialroleincondensedmatterphysicsingeneralandinsolidstatephysicsinparticular.Ihopethatthisgenerallinewillhelpthereadertoseedifferentpartsofcondensedmatterphysicsasdifferentsidesofaunifiedpictureandnotasacollectionofseparateunrelatedtopics.Theplanofthebookisthefollowing.Afterdiscussingthegeneraltheoryofphasetransitions(Chapter2)whichformsthebasisfordescribingorderinsolids,ixxForewordandgeneralintroductionIgostepbystepthroughdifferentspecificsituations:systemsofbosons(phononsincrystals–Chapter4,andgeneralBosesystems,includingBosecondensationandsuperfluidity–Chapter5).Thenfollowstheimportantchapteronmagnetism,Chapter6(strictlyspeakingdealingneitherwithbosons,norwithfermions),andafterthatweswitchtothediscussionsoffermions–electronsinsolids,Chapters7–13.IneachtopicIhavetriedtofollowthegenerallinewhichIhavealreadydescribedabove:todiscussfirstthetypeoforderwehaveinoneortheothersituation,thenintroducedifferenttypesofelementaryexcitationsinthem,firstindependentexcitations,butthenpayingmostattentiontotheinteractionbetweenthemandtotheirquantumnature.Thusaltogetherthematerialpresentedinthebookissupposedtocoverthemainsituationsmetinsolids.Thetheoreticalmethodsusedtodescribethesephenomenaareintroducednotsomuchseparately,assuch,butintheappropriateplaceswheretheyareneeded,andinawaywhichimmediatelyshowshowtheyworkinspecificproblems.Thus,instudyingBosesystemsIintroducethewidelyusedBogolyubovcanonicaltransformation,whichlateronisalsousedfortreatingmagnonsinantiferromagnetsandforcertainproblemsforelectrons.Discussingspinwaves,Iintroducethemethodofequationsofmotionwithcorrespondingdecoupling,lateronalsoused,e.g.forstudyingcorrelatedelectrons(theHubbardmodel).Whengoingtoelectronsystems,IdescribetheGreenfunctionmethodandtheFeynmandiagramtechnique–withoutcompleteandrigorousderivations,butwiththeaimofdemonstratinghowthesemethodsreallyworkindifferentsituations.Ihopethematerialcoveredinthisbookwillgivethereaderarelativelycompletepictureofthemainphenomenainmodernsolidstatephysicsandofthemaintheoreticalmethodsusedtostudythem.Butofcourseitisimpossibletocoverinonebookofmodestsizethiswholefield.Themostimportantandevidentomissionsare:Idonotpracticallytouchonthebroadandimportantfieldoftransportphenomena(resistivity,thermalconductivity,thermopower,theHalleffect,etc.)Thisisaverybigtopicinitself,butitliessomewhatoutsidethemainscopeofthisbook.Ialsodonotdiscussspecificfeaturesofsuchimportant,butwellknownmaterialsassemiconductors,ferroelectrics,etc.Alsothewidefieldofsuperconductivityistouchedupononlytotheextentitisrequiredtoillustratethegeneraltreatment.Yetanotherrelativelyrecentandverybeautifultopicismissing–thephenomenonofthequantumHalleffect.HopefullyIcan‘repair’thisomissionlater.Onthetheoreticalsideprobablytwoimportantmethodsarenotsufficientlydiscussedinthebook.Oneistherenormalizationgroupmethodusedtotreatcomplicatedsituationswithstronginteraction.Ionlybrieflymentionthismethod,butdonotdescribeitindetail.Interestedreadersmayfinditsdescription,e.g.inthebooksbyChaikinandLubensky(2000)orStanley(1987).ForewordandgeneralintroductionxiAnothertheoreticaltechniquewidelyusednowadaysistheuseofdifferenttypesofnumericalcalculations.Thisisaverybroadandrapidlydevelopingfieldwhichproveditsefficiencyforstudyingrealmaterialsandfortheoreticalinvestigationsofmanysituationsnotaccessibletoanalyticalcalculations.Thisisquiteaspecialfield,anditrequiresspecialtreatment–althoughwhenappropriateIpresentsomeoftheresultsobtainedinthisway.Withalltheseomissions,Istillhopethatthematerialwhichisincludedwillbeusefulforabroadaudienceandwillgivethereaderarelativelycompletepictureofthemainphenomenaandmainproblemsinmodernsolidstatephysics.Afewwordsaboutthestyleofthepresentation.ThisbookhasgrownoutofalecturecoursewhichIgaveforseveralyearsatGroningenUniversityandatCologneUniversity.Thereforeitstillhassomefeaturesofthelecturenotes.Ipresentinitallthemainideas,butoftennotthefullderivationsofcorrespondingresults.Thisisalsocausedbythefactthatthematerialtoucheduponinthisbookinfactcoversahugefield,anditisimpossibletopresentallthedetailsinonebook.Therearemanymonographsandtextbooksdiscussingindetailtheseparatesubfieldspresentedbelow.HoweverIhavetriedtochoosethetopicsandpresenttheminsuchawaythatthegeneralideasunderlyingmodernsolidstatephysicsandtheinternallogicofthisfieldbecomeclear.Formoredetaileddiscussionsofparticularproblemsandorcorrespondingmethodsoftheirtheoreticaltreatmentthereadersshouldgotothespecializedliterature.Inaccordancewiththisgeneralconceptofthebook,Ididnotincludeinitaspecial‘problems’section.Insomeplaces,however,especiallyinthefirstpartofthebook,Iformulatepartsofthematerial,asProblems.Thosewhowanttogetadeeperunderstandingofthesubjectarerecommendedtostopreadingthetextattheseplacesandtrytofindtheanswersthemselves;thematerialpresentedbeforeusuallymakesthistasknottoodifficult.Theanswers,however,areusuallygivenrightaftertheproblems,sothatreaderscanalsogoonalongthetextiftheydonothaveadesire,ortime,todotheseexercisesthemselves.Actuallymostoftheproblems,withtheiranswers,formanintegralpartofthetext.InseveralplacesinthetextIhavealsoputsomemorespecialpartsofthetextinsmallertype.Thesepartsusuallyrelatetomorespecialized(althoughuseful)material.InadditiontothemainmaterialIhavealsoincludedthreeveryshortchapters(Chapters1,3and7)withashortsummaryofsomeofthebasicfactsfromstatisticalmechanics.Ithinkitwouldbeusefulforreaderstohavethisinformationathand.Someimportantnotionsarementionedseveraltimesindifferentpartsofthetext.Ididthisintentionally,sothatdifferentchapterswouldbecomesomewhatmoreindependent–althoughofcoursetherearealotofcrossreferencesinthetext.xiiForewordandgeneralintroductionIhopethatthebookgivesacoherentpresentationofthemainideasandmethodsofthequantumtheoryofsolids.Therearemanygoodbookswhichcoverpartsofthematerialcontainedinthepresentbook(andactuallymuchmore!).Icanrecommendthefollowing,alreadyclassicalbooks:1.J.M.Ziman(1979),PrinciplesoftheTheoryofSolids.Averygoodandclearbookcoveringthemaintopicsinsolidstatephysics.Highlyrecommended.However,itdoesnotcontainmoremodernmethods.2.N.W.AshcroftandN.D.Mermin(1976),SolidStatePhysics.Alsoaverygoodandwidelyusedbook,coveringthetopicsinmoredetail,onasomewhatmoreelementarylevel.Verytransparentanduseful.3.M.P.Marder(2000),CondensedMatterPhysics.Arathercompletebookdescribingthemainphenomenainsolidstatephysics,butnotgoingintomuchtheoreticaldetail.4.C.Kittel(1987),QuantumTheoryofSolids.Containsdetaileddiscussionofmanyproblemsinquantumtheory,usingmoremodernmethodssuchasdiagramtechniques.Somewhatmoretheoretical.5.G.D.Mahan(2000),ManyParticlePhysics.Givesaverycompletetreatmentofthetopicsdiscussed;itisakindof‘encyclopedia’.ItusestheGreenfunctionmethodallthewaythrough.Veryusefulfortheoreticians,andcontainsallthenecessarydetailsandderivations,etc.Howevernotalltopicsarediscussedthere.6.L.D.LandauandI.M.Lifshits,CourseofTheoreticalPhysics,especiallyStatisticalPhysics(oldonevolumeedition1969,orneweditionv.I1980),andQuantumMechanics(1977).Theseclassicalbookscontainvirtuallyallthebasicmaterialnecessary,andmanyparticulartopicsimportantforourproblems.IfonecancallthebookbyMahananencyclopedia,thenthecourseofLandauandLifshitsisa‘bible’foralltheoreticians,especiallythoseworkingincondensedmatterphysics.Butthesebooksareveryusefulnotjustfortheoreticians,butforeveryonelookingforclearandprecisedescriptionofallthebasicideasoftheoreticalphysics.7.J.R.Schrieffer(1999),TheoryofSuperconductivity.Averyclearbook;containsinparticularaverygoodandcondensedtreatmentoftheGreenfunctionmethodanddiagramtechnique,inaformusednowbymosttheoreticians.8.A.A.Abrikosov,L.P.Gor’kovandE.Dzyaloshinsky(1975),MethodsoftheQuantumFieldTheoryinStatisticalPhysics.Oneofthefirst(andstillthebest)booksonthemodernmethodsappliedtocondensedmatterphysics.Itgivesaverydetailedtreatmentofthetheoreticalmethodsandagooddiscussionofspecificproblems(FermiandBoseliquids;plasma;electron–phononinteractionandthebasicsofthetheoreticaltreatmentofsuperconductivity).Forewordandgeneralintroductionxiii9.P.M.ChaikinandT.C.Lubensky(2000),PrinciplesofCondensedMatterPhysics.Averygoodbookcontaininginparticulardetaileddiscussionofdifferentquestionsconnectedwithphasetransitions.Theaccentisongeneralstatisticalmechanics;specificallyitcontainsalotofmaterialonsoftcondensedmatterphysics,butdoesnotdiscusssuchtopicsaselectronsinmetals,magnetism,etc.Someotherreferenceswillbegivenlater,inthemainbodyofthebook.But,keepinginmindthecharacterofthebook(whichispracticallyexpandedlecturenotesandwhichstillretainthatcharacter),Ideliberatelyrefrainedfromincludingtoomanyreferences–itwouldbesimplyimpossibletocitealltherelevantworks.ThereforeImostlyrefernottooriginalpublicationsbutrathertomonographsandreviewpapers.Interestedreadersmayfindmoredetailedinformationonparticularsubjectsinthesereferences.GeneralintroductionTheunifyingconceptinthisbookistheconceptoforderandelementaryexcitations;thesearethekeywords.Onecanargueasfollows.Ingeneralinmacroscopicsystemswithmanydegreesoffreedomtheinternalstate,orinternalmotiononthemicroscopicscale,israndom.HoweverasT0theentropyofthesystemshouldgotozero,S0;thisisthewellknownNernsttheorem,orthethirdlawofthermodynamics.Accordingly,atT=0thereshouldexistperfectorderofsomekind.SuchorderingsetsinatacertaincharacteristictemperatureT∗,oftenwithaphasetransition,butnotnecessarily.AtT�T∗wecandescribethestateofthesystemaspredominantlyordered,ormaybeinaweaklyexcitedstate.Suchrelativelyweaklyexcitedstateswillbethermallyexcited,butcanappearalsoduetosmallexternalperturbations.Usuallyinsuchastatewecanspeakofasmallnumberofelementaryexcitations,orquasiparticles.Examplesare,e.g.phononsincrystals,magnonsorspinwavesinferromagnets,excitonsinsemiconductors,etc.Sometimessuchelementaryexcitationsareratherstrange:insteadofelectronstheymaybeexcitationswithspin,butnocharge(spinons),orviceversa(holons).Thereexistalsotopologicalexcitations(solitons,vortices,etc.).Inafirstapproximationwecanconsidertheseexcitationsasnoninteracting.Suchare,e.g.phononsinaharmoniccrystal,etc.Howeverinthenextstepwehavetoincludeingeneralaninteractionbetweenquasiparticles.Theremayexistinteractionsbetweenthesamequasiparticles.Theylead,e.g.toanharmoniceffectsincrystals(phonon–phononinteractions);theyareincludedintheLandauFermiliquidtheory,andgiverisetoscreeningforelectronsinmetals;themagnon–magnoninteractioncanlead,e.g.totheformationofboundstatesxivForewordandgeneralintroductionofmagnons,etc.Therealsoexistinteractionsbetweendifferentquasiparticles:electron–phononinteractions,pairinginconventionalsuperconductors,andinteractionsbetweenmanyotherelementaryexcitations.Oftenduetotheseinteractionsthepropertiesofquasiparticlesarestronglychanged,orrenormalized:anexampleistheformationofpolarons(electron+stronglatticedistortion).Alsonewquasiparticlesmaybeformed(plasmonsduetotheCoulombinteractionofelectrons;excitons–boundstatesofelectronsandholesinsemiconductors).Eventhegroundstateitself,theverytypeorordering,maychangebecauseofsuchinteractions.Anexampleisthesuperconductingstateinsteadofthenormalstateofametal.Itisimportantthatthesequasiparticles,orelementaryexcitations,arequantumobjects.Consequently,oneshouldnotvisualizetheorderascompletelyclassical:therearequantumfluctuations(zeropointmotion,orzeropointoscillations)evenatT=0.Sometimestheyleadonlytominornumericalchanges,buttherearecases,especiallyinlowdimensionalorfrustratedsystems,whentheycancompletelymodifythepropertiesofasystem,e.g.destroyingthelongrangeordertotally.Theycanalsomodifythepropertiesofthephasetransitionsthemselves,e.g.leadingtoquantumphasetransitions.Thustheclassicalpictureisalwaysveryuseful,butoneshouldbecautiousandawareofitspossiblefailuresinsomecases–butveryoftenthesecasesarethemostinteresting!Intreatingtheseproblemsalotofdifferentapproacheswereused,anddifferentmethodsdeveloped.Thesemethodsareoftenusednotonlyinsolidstatephysicsorcondensedmatterphysicsingeneral;manyofthemarealsowidelyused(andoftenhavebeendeveloped!)inotherpartsofphysics:inelementaryparticlephysics,infieldtheory,andinnuclearphys