Superconductivity and Chiral Symmetry Breaking with Fermion Clusters

Cluster variables have recently revolutionized numerical work in certain models involving fermionic variables. This novel representation of fermionic partition functions is continuing to find new applications. After describing results from a study of a two

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SuperconductivityandChiralSymmetryBreakingwithFermionClusters

ShaileshChandrasekharana

a

DepartmentofPhysics,DukeUniversity,Box90305,DurhamNC27708,USA

arX

iv:hep-lat/0110125v1 16 Oct 2001

Clustervariableshaverecentlyrevolutionizednumericalworkincertainmodelsinvolvingfermionicvariables.Thisnovelrepresentationoffermionicpartitionfunctionsiscontinuingto ndnewapplications.AfterdescribingresultsfromastudyofatwodimensionalHubbardtypemodelthatcon rmasuperconductingtransitionintheKosterlitz-Thoulessuniversalityclass,weshowhowaclustertypealgorithmcanbedevisedtostudythechirallimitofstronglycoupledlatticegaugetheorieswithstaggeredfermions.

1.INTRODUCTION

Duringthelastfewyearsanewclassoffermionalgorithmshaveemerged.Theessentialprogressisaresultofourabilitytorewritecer-tainfermionicpartitionfunctionsasasumovercon gurationsofbondvariableswithpositivedef-initeweights[1,2],i.e.,

W[b]Z=

[b]

Sign[b]≥0isanentropyfactorthat

takesintoaccountdegreesoffreedomotherthanthebondvariables.Typically,thePauliprincipleisencodedinthetopologyofclustersformedbylatticesitesconnectedthroughthebonds.Clus-tersalsocarryavarietyofinterestingphysicalin-formation.Forexample,sizesofcertainclustersarerelatedtocondensates,thesquaresofthesizesofclustersyieldsusceptibilities.Further,clustersareusefulinbuildinge cientalgorithmsclosetocriticalpointswherethecorrelationlengthsdi-vergesincetheyallownon-localupdateswithareasonableacceptance.Thispropertyhashelpedinstudyingcriticalphenomenainfermionicmod-elswithunmatchedprecision.2.SUPERCONDUCTIVITY

TherecentsuccessofclustermethodsinfermionicsystemsoriginatesfromtheHamilto-

Sign[b]=0.

Recently,superconductivityinatwodimen-sionalattractiveHubbardtypemodelwasstudiedusingthemeronclusterapproach.Thefermionpairingsusceptibility χ isausefulobservable.Itisexpectedtosatisfythe nitesizescalingfor-mula

2 η(T)

LT<Tc

χ =(2)

Const.T>TcifthesuperconductingtransitionbelongstotheKosterlitz-Thouless(KT)universalityclass,with0≤η(T)≤0.25,η(Tc)=0.25andη(0)=0.Inthespeci cmodelstudied, χ turnsouttobeasumoverthesquareofthesizeofeachclusterinthezeromeronsectorandtheproductofthesizeofthemeronsinthetwomeronsector.Figure1showsaplotofthesusceptibilityasafunctionoflatticesizeforvarioustemperatures.ConsistencywithKTpredictionsisclear.

Anotherobservablerelevanttothestudyofsuperconductivityisthespatialfermionwindingnumbersusceptibility W2 .Althoughthisisdif- culttoevaluatewithconventionalalgorithms,it

Cluster variables have recently revolutionized numerical work in certain models involving fermionic variables. This novel representation of fermionic partition functions is continuing to find new applications. After describing results from a study of a two

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Figure1.Pairingsusceptibilityasafunctionofsystemsizeforvarioustemperatures.

isrelativelystraightforwardinthemeronclusterapproach.Eachclustercanbeassignedaspa-tialfermionwindingnumber.Thesusceptibilitythenturnsouttobethesumoverthesquareofeachcluster’sspatialwindingnumberinthezeromeronsectorandtheproductofthespatialwind-ingnumberofmeronclustersinthetwomeronsector.Inthein nitevolumelimitbelowthecrit-icaltemperature,onecancombineknownresultstoobtain2πη(T) W2 =1.Resultsagainshowconsistencywiththisexpectation.

Preliminaryresultsfromthisstudywaspresentedin[3]andthe nalanalysisin[4].

3.CHIRALSYMMETRYBREAKINGAlthoughtherecentsuccesshasbeenappliedtoHamiltonianmodelsofchiralsymmetrybreaking[5],clustermethodsareapplicabletomorecon-ventionalLagrangianmodelsaswell.Forexam-ple,considerstronglycoupledlatticegaugethe-orywithmasslessstaggeredfermionsinwhichtheU(1)chiralsymmetryisexpectedtobebrokenspontaneouslyinfourdimensions[6].Thisre-sultwasobtainedbymappingthemassivemodelintoastatisticalmechanicsofmonomer-dimer-polymer(MDP)systemwithpositivede niteBoltzmannweightsandextrapolatingtheresultstothechirallimit.Unfortunately,asfarasweknow,ithasbeendi culttodevisealgorithmsinthechirallimitwherethesystemsbecomecon-strained.Localmetropolisupdateswhichcanbeformulatedinthemassivecasebecomeexponen-tiallyine cientinthechirallimit.HerewearguethatclusterrepresentationsoftheMDPsystemsyieldusefulalgorithmsdirectlyinthechirallimit.Tounderstandtheclusterrepresentationcon-siderforsimplicitythestronglycoupledU(1)gaugesystem.Thepartitionfunctioninthiscaseisgivenbythenumberofclosely-packed-dimer(CPD)con gurationsonalattice.AtypicalCPDcon gurationintwodimensionsisshowninFig.2.

Figure2.AtwodimensionalCPDcon gurationSuchcon gurationsarealsoofinterestinsta-tisticalmechanicsandplayanimportantroleinthesolutiontothe2-dIsingmodel[7].Thechi-ralsymmetryofstaggeredfermionsismanifestinthisrepresentationbythefactthatthechi-ralcondensatevanishessinceitisimpossibleto ndaCPDcon gurationwithonedefect(onesitehasnodimerlinesattachedtoit).Thechiralsusceptibilityontheotherhandisnon-zeroandproportionaltotheratioofthetotalnumberofCPDcon gurationswithtwodefects(twositesarenotconnectedbydimers)andthepartitionfunction.

ItispossibletoextendCPDcon gurations

Cluster variables have recently revolutionized numerical work in certain models involving fermionic variables. This novel representation of fermionic partition functions is continuing to find new applications. After describing results from a study of a two

Figure3.RulesforextendingtheCPDcon gu-rationstoincludeadditionalbondvariables.tocon gurationsofloopsmadeupofbondswhichincludetheoriginalor“ lled”dimers(rep-resentedhereby“solid”bonds)and“empty”dimers(representedby“dashed”bonds)suchthatthepartitionfunctioncanbeexpressedasa

sumoverweightsofnewloopcon gurations.Fig-ure3showstherulesofonesuchextensionintwodimensions.EachshadedplaquetteoftheCPDcon gurationofFig.2carriesoneofthesevenplaquettecon gurationsgivenontheleftsideofequationsinFig.3.Itiseasytocheckthatallconstraintsaresatis edifeachloopismadeupofarepeatingsequenceof lledandemptydimers.Theusefulnessoftheloopvariableisthatadimersystemcanbeupdatedby“ ipping”aloopwhere lleddimersareemptiedandviceversa.Theac-ceptanceofsucha ipisreasonableandleadstoausefulalgorithm.

Thechiralsusceptibilitygetscontributionswhenapartoftheloopis ippedandcanbemeasuredeasilyalongwiththeupdate.Theal-gorithmwas rstappliedtothetwodimensionalmodel.AlthoughaU(1)chiralsymmetrycan-notspontaneouslybreakintwodimensions,longrangecorrelationscanariseaspredictedbytheKosterlitz-Thoulessuniversalityclassasdiscussedintheprevioussection.Figure4plotsthechiral

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susceptibilitywithsystemsize.Surprisingly,al-thoughthedataisnotinconsistentwiththepres-enceoflongrangecorrelations,thesusceptibilitydoesnotseemtofollowthepredictionsofeq.(2).Thispuzzleiscurrentlybeinginvestigatedalongwithextensionstohigherdimensions.

Figure4.Chiralsusceptibilityasafunctionoflatticesizeintwodimensions.

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