Sachdev, Quantum Phase Transitions, 2ed, Springer, 2011[1].pdf

Sachdev, Quantum Phase Transitions, 2ed, Springer, 2011[1].pdf

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ThispageintentionallyleftblankQuantumPhaseTransitionsSecondEditionThisisthefirstbooktodescribethephysicalpropertiesofquantummaterialsnearcriticalpointswithlong-rangemany-bodyquantumentanglement.Readersareintroducedtothebasictheoryofquantumphases,theirphasetransitions,andtheirobservableproperties.Thissecondeditionbeginswithninechapters,sixofthemnew,suitableforanintroduc-torycourseonquantumphasetransitions,assumingnopriorknowledgeofquantumfieldtheory.Thereareseveralnewchapterscoveringimportantrecentadvances,suchastheFermigasnearunitarity,Diracfermions,Fermiliquidsandtheirphasetransitions,quan-tummagnetism,andsolvablemodelsobtainedfromstringtheory.Afterintroducingthebasictheory,itmovesontoadetaileddescriptionofthecanonicalquantum-criticalphasediagramatnonzerotemperatures.Finally,avarietyofmorecomplexmodelsisexplored.Thisbookisidealforgraduatestudentsandresearchersincondensedmatterphysicsandparticleandstringtheory.SubirSachdevisProfessorofPhysicsatHarvardUniversityandholdsaDistinguishedResearchChairatthePerimeterInstituteforTheoreticalPhysics.Hisresearchhasfocusedonavarietyofquantummaterials,andespeciallyontheirquantumphasetransitions.QuantumPhaseTransitionsSecondEditiontSUBIRSACHDEVHarvardUniversityCAMBRIDGEUNIVERSITYPRESSCambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SoPaulo,Delhi,TokyoCambridgeUniversityPressTheEdinburghBuilding,CambridgeCB28RU,UKPublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYorkwww.cambridge.orgInformationonthistitle:

www.cambridge.org/9780521514682c?

S.Sachdev2011Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionsofrelevantcollectivelicensingagreements,noreproductionofanypartmaytakeplacewithoutthewrittenpermissionofCambridgeUniversityPress.Firstpublished2011PrintedintheUnitedKingdomattheUniversityPress,CambridgeAcatalogrecordforthispublicationisavailablefromtheBritishLibraryLibraryofCongressCataloging-in-PublicationDataSachdev,Subir,1961-Quantumphasetransitions/SubirSachdev.Secondedition.p.cmIncludesbibliographicalreferencesandindex.ISBN978-0-521-51468-2（Hardback）1.Phasetransformations（Statisticalphysics）2.Quantumtheory.I.Title.QC175.16.P5S232011530.4?

74dc222010050328ISBN978-0-521-51468-2HardbackCambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.TomyparentsandMenaka,Monisha,andUshaContentsFromthePrefacetothefirsteditionpagexiiiPrefacetothesecondeditionxviiPartIIntroduction11Basicconcepts31.1Whatisaquantumphasetransition?

31.2Nonzerotemperaturetransitionsandcrossovers51.3Experimentalexamples81.4Theoreticalmodels91.4.1QuantumIsingmodel101.4.2Quantumrotormodel121.4.3Physicalrealizationsofquantumrotors142Overview182.1Quantumfieldtheories212.2Whatsdifferentaboutquantumtransitions?

24PartIIAfirstcourse273Classicalphasetransitions293.1Mean-fieldtheory303.2Landautheory333.3Fluctuationsandperturbationtheory343.3.1Gaussianintegrals363.3.2Expansionforsusceptibility39Exercises424Therenormalizationgroup454.1Gaussiantheory464.2MomentumshellRG484.3Fieldrenormalization534.4Correlationfunctions54Exercises56viiviiiContents5ThequantumIsingmodel585.1EffectiveHamiltonianmethod585.2Large-gexpansion595.2.1One-particlestates605.2.2Two-particlestates615.3Small-gexpansion645.3.1d=2645.3.2d=1665.4Review675.5TheclassicalIsingchain675.5.1Thescalinglimit705.5.2Universality715.5.3Mappingtoaquantummodel:

Isingspininatransversefield725.6MappingofthequantumIsingchaintoaclassicalIsingmodel74Exercises776Thequantumrotormodel796.1Large-gexpansion796.2Small-gexpansion806.3TheclassicalXYchainandanO

（2）quantumrotor826.4TheclassicalHeisenbergchainandanO（3）quantumrotor886.5Mappingtoclassicalfieldtheories896.6Spectrumofquantumfieldtheory906.6.1Paramagnet916.6.2Quantumcriticalpoint926.6.3Magneticorder92Exercises957Correlations,susceptibilities,andthequantumcriticalpoint967.1Spectralrepresentation977.1.1Structurefactor987.1.2Linearresponse997.2Correlationsacrossthequantumcriticalpoint1017.2.1Paramagnet1017.2.2Quantumcriticalpoint1037.2.3Magneticorder104Exercises1078Brokensymmetries1088.1Discretesymmetryandsurfacetension1088.2Continuoussymmetryandthehelicitymodulus1108.2.1Orderparametercorrelations112ixContents8.3TheLondonequationandthesuperfluiddensity1128.3.1Therotormodel115Exercises1159BosonHubbardmodel1179.1Mean-fieldtheory1199.2Coherentstatepathintegral1239.2.1Bosoncoherentstates1259.3Continuumquantumfieldtheories126Exercises130PartIIINonzerotemperatures13310TheIsingchaininatransversefield13510.1Exactspectrum13710.2Continuumtheoryandscalingtransformations14010.3Equal-timecorrelationsoftheorderparameter14610.4Finitetemperaturecrossovers14910.4.1LowTonthemagneticallyorderedside,?

0,T?

15110.4.2LowTonthequantumparamagneticside,?

gc17511.2.2Criticalpoint,g=gc17711.2.3Magneticallyorderedgroundstate,ggc,T?

+18611.3.2HighT,T?

+,?

18611.3.3LowTonthemagneticallyorderedside,ggc,T?

18711.4Numericalstudies18812Thed=1,O（N3）rotormodels19012.1Scalinganalysisatzerotemperature19212.2Low-temperaturelimitofthecontinuumtheory,T?

+19312.3High-temperaturelimitofthecontinuumtheory,?

+?

T?

J199xContents12.3.1Field-theoreticrenormalizationgroup20112.3.2Computationofu20512.3.3Dynamics20612.4Summary21113Thed=2,O（N3）rotormodels21313.1LowTonthemagneticallyorderedside,T?

s21513.1.1Computationofc21613.1.2Computationof22013.1.3Structureofcorrelations22213.2Dynamicsofthequantumparamagneticandhigh-Tregions22513.2.1Zerotemperature22713.2.2Nonzerotemperatures23113.3Summary23414Physicsclosetoandabovetheupper-criticaldimension23714.1Zerotemperature23914.1.1Tricriticalcrossovers23914.1.2Field-theoreticrenormalizationgroup24014.2Staticsatnonzerotemperatures24214.2.1d324814.3Orderparameterdynamicsind=225014.4Applicationsandextensions25715Transportind=226015.1Perturbationtheory26415.1.1I26815.1.2II26915.2Collisionlesstransportequations26915.3Collision-dominatedtransport27315.3.1?

expansion27315.3.2Large-Nlimit27915.4Physicalinterpretation28115.5TheAdS/CFTcorrespondence28315.5.1Exactresultsforquantumcriticaltransport28515.5.2Implications28815.6Applicationsandextensions289PartIVOthermodels29116DiluteFermiandBosegases29316.1ThequantumXXmodel296xiContents16.2ThedilutespinlessFermigas29816.2.1Diluteclassicalgas,kBT?

|,030116.2.3High-Tlimit,kBT?

|30416.3ThediluteBosegas30516.3.1dgcandforggc（g0.Itturnsoutthatworkingoutwardfromthequantumcriticalpointatg=gcandT=0isapowerfulwayofunderstandinganddescribingthethermodynamicanddynamicpropertiesofnumeroussystemsoverabroadrangeofvaluesof|ggc|andT.Indeed,itisnotevennecessarythatthesystemofinteresteverhaveitsmicroscopiccouplingsreachavaluesuchthatg=gc:

itcanstillbeveryusefultoarguethatthereisaquantumcriticalpointataphysicallyinaccessiblecouplingg=gcandtodevelopadescriptioninthedeviation|ggc|.Itisoneofthepurposesofthisbooktodescribethephysicalperspectivethatsuchanapproachoffers,andtocontrastitwithmoreconventionalexpansionsaboutveryweak（sayg0）orverystrongcouplings（sayg）.1.2NonzerotemperaturetransitionsandcrossoversLetusnowdiscusssomebasicaspectsoftheT0phasediagram.First,letusaskonlyaboutthepresenceofphasetransitionsatnonzeroT.Withthislimitedcriterion,therearetwoimportantpossibilitiesfortheT0phasediagramofasystemnearaquantumcriticalpoint.TheseareshowninFig.1.2,andwewillmeetexamplesofbothkindsinthisbook.Inthefirst,showninFig.1.2a,thethermodynamicsingularityispresentonlyatT=0,andallT0propertiesareanalyticasafunctionofgnearg=gc.Inthesecond,showningT0T0gcgcg（a）（b）tFig.1.2Twopossiblephasediagramsofasystemnearaquantumphasetransition.Inbothcasesthereisaquantumcriticalpointatg=gcandT=0.In（b）,thereisalineofT0second-orderphasetransitionsterminatingatthequantumcriticalpoint.Thetheoryofphasetransitionsinclassicalsystemsdrivenbythermalfluctuationscanbeappliedwithintheshadedregionof（b）.6BasicconceptsFig.1.2b,thereisalineofT0second-orderphasetransitions（thisisalineatwhichthethermodynamicfreeenergyisnotanalytic）thatterminatesattheT=0quantumcriticalpointatg=gc.Movingbeyondphasetransitions,letusasksomebasicquestionsaboutthedynamicsofthesystem.AverygeneralwaytocharacterizethedynamicsatT0isintermsofthethermalequilibrationtimeeq.Thisisthecharacteristictimeinwhichlocalthermalequi-libriumisestablishedafterimpositionofaweakexternalperturbation（say,aheatpulse）.Hereweareexcludingequilibrationwithrespecttogloballyconservedquantities（suchasenergyorcharge）whichwilltakealongtimetoequilibrate,dependentuponthelengthscaleoftheperturbation:

hencetheemphasisonlocalequilibration.Globalequilibrationisdescribedbytheequationsofhydrodynamics,andweexpectsuchequationstoapplyinallcasesattimesmuchlargerthaneq.WefocushereonthevalueofeqasafunctionofggcandT.FromtheenergyscalesdiscussedinSection1.1,wecanimmediatelydrawanimportantdistinctionbetweentworegimesofthephasediagram.Wecharacter-izedthegroundstatebytheenergy?

in（1.1）.Atnonzerotemperature,wehaveasecondenergyscale,kBT.Comparingthevaluesof?

andkBT,weareimmediatelyledtotheimportantphasediagramin