物理评论快报论文

PRL101,061601(2008)

SelectedforaViewpointinPhysicsPHYSICALREVIEWLETTERS

weekending8AUGUST2008

GravityDualsforNonrelativisticConformalFieldTheories

KoushikBalasubramanianandJohnMcGreevy

CTP,MIT,Cambridge,Massachusetts02139,USA(Received6May2008;published4August2008)

Weattempttogeneralizetheanti–deSitter/conformal eldtheorycorrespondencetononrelativisticconformal eldtheorieswhichareinvariantunderGalileantransformations.Suchsystemsgovernultracoldatomsatunitarity,nucleonscatteringinsomechannels,and,moregenerally,afamilyofuniversalityclassesofquantumcriticalbehavior.Weconstructafamilyofmetricswhichrealizethesesymmetriesasisometries.Theyaresolutionsofgravitywithanegativecosmologicalconstantcoupledtopressurelessdust.Wediscussrealizationsofthedust,whichincludeabulksuperconductor.Wedeveloptheholographicdictionaryand ndtwo-pointcorrelatorsofthecorrectform.Astrangeaspectofthecorrespondenceisthatthebulkgeometryhastwoextranoncompactdimensions.

DOI:10.1103/PhysRevLett.101.061601

PACSnumbers:11.25.Tq,03.75.Ss

Introduction.—Manyattemptshavebeenmadetousetheanti–deSitter/conformal eldtheory(AdS/CFT)cor-respondence[1]tostudysystemsrealizableinalaboratory.Onedoesnotyethaveaholographicdualmatchingtheprecisemicroscopicdetailsofanysuchsystemandisthereforeledtotrytomatchtheuniversalityclassofthesystem.Ingeneral,physicsinthefarinfraredisdescribedbya(sometimestrivial) xedpointoftherenormalizationgroup.Ithasbeenarguedthattheassociatedzero-temperatureCFTcontrolsaswathofthe nite-temperaturephasediagram,namely,theregioninwhichthetempera-tureistheonlyimportantscale,andnodimensionfulcouplingshaveturnedon(see,e.g.,[2]).Intryingtousegauge-stringdualitytostudysuchquestions,itseemsurgent,then,tomatchthesymmetriestothoseofthesystemofinterest.

TheAdS/CFTcorrespondencesofargivesaneffectivedescriptionofrelativisticconformal eldtheoriesatstrongcoupling.Notmanyoftheseareaccessibleexperimentally.However,therearemanynonrelativisticconformal eldtheorieswhichgovernphysicalsystems.Suchexamplesariseincondensedmatterphysics[2],atomicphysics[3],andnuclearphysics[4].Inthe rstsituation,thesearecalled‘‘quantumcriticalpoints’’;thistermreferstoasecond-orderphasetransitionthatisreachednotbyvary-ingthetemperaturebutbyvaryingsomecouplingcon-stantsatzerotemperature.

Aparticularlyinterestingexamplewhereexquisiteex-perimentalcontrolispossibleisthecaseofcoldfermionicatomsatunitarity(e.g.,[3]andreferencesthereinandthereto).Thesearescale-invariantinthefollowingsense.Theinteractionsaretuned(bymanipulatingtheenergyofatwo-bodyboundstatetothreshold)tomakethescatteringlengthin nite(hencethesystemisstronglycoupled).Thematerialisdiluteenoughthattheeffectiverangeofthepotentialcanbetreatedaszero.Whilethemassoftheatomsisadimensionfulquantity,thedependenceoftheenergyandotherphysicalquantitiesonthemassisdeter-minedbythesymmetryalgebra.ArecentpaperstudyingtheconstraintsfromGalileanconformalsymmetryonsuchsystemsisRef.[5].

InthisLetter,wesetoutto ndabulkdualofnon-relativisticCFTs,analogoustotheAdSgravitydescriptionofrelativisticCFTs,atstrongcoupling.Weapproachthisquestionbyconsideringthealgebraofgeneratorsofthenonrelativisticconformalgroup,whichappearsinRef.[5](relatedworkincludesRef.[6]).Weseekasolutionofstringtheory(initslow-energygravityapproximation)whoseasymptoticboundarysymmetrygroupisnotthe

´symmetrygroup(i.e.,Lorentzandtranslations)Poincare

butratherGalileaninvarianceandtranslations.Wealsodemandanonrelativisticversionofscaleinvarianceand,whenpossible,specialconformaltransformations.

Clearly,wewillhavetointroducesomebackgroundenergydensityto ndsuchasolution(i.e.,itwillnotbejustasolutionofgravitywithacosmologicalconstant).Holographically,thisisbecausethetheoriesthatwearetryingtodescribeariseingeneral(perhapsnotalways)fromrelativisticmicroscopictheories(notnecessarilycon-formal)uponaddingsomebackgroundofstuff(which xesapreferredreferenceframe)andthentakingsomelimitfocusingonexcitationswithsmallvelocitiesintheframeofthestuff.

Ananalogywhichmaybeusefulisthefollowing.Theplane-wavesymmetrygroupisacontractionoftheAdSisometrygroup;theplane-wavesolutionarisesfromAdSinconsideringalimitfocusingontheworldlineofafast-movingparticle.SoitmightbepossibletotakesomelimitofAdSanalogoustotheplane-wavelimitto ndoursolution.ThelimitinvolvedwouldbelikethePenroselimitbutwouldreplacethesingleparticleworldlinewithauniformbackgroundnumberdensity.Weleavesuchaderivationfromamoremicroscopicsystemforthefutureandproceedtoguesstheendresult.

Scaleinvariancecanberealizedinnonrelativistictheo-riesinanumberofways.Onefreedomistherelativescaledimensionoftimeandspace,calledthe‘‘dynamicalex-¨dingercaseponent’’z(see,e.g.,[7]).ThefamiliarSchro