Overlap Dirac Operator, Eigenvalues and Random Matrix Theory

The properties of the spectrum of the overlap Dirac operator and their relation to random matrix theory are studied. In particular, the predictions from chiral random matrix theory in topologically non-trivial gauge field sectors are tested.

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OverlapDiracOperator,EigenvaluesandRandomMatrixTheory

RobertG.Edwards

a

,UrsM.Hellera,JoeKiskisb,RajamaniNarayananc

aSCRI,FloridaStateUniversity,Tallahassee,FL32306-4130,USAb

Dept.ofPhysics,UniversityofCalifornia,Davis,CA95616,USAc

AmericanPhysicalSociety,OneResearchRoad,Ridge,NY11961,USA

ThepropertiesofthespectrumoftheoverlapDiracoperatorandtheirrelationtorandommatrixtheoryarestudied.Inparticular,thepredictionsfromchiralrandommatrixtheoryintopologicallynon-trivialgauge eldsectorsaretested.

AnimportantpropertyofmasslessQCDisthespontaneousbreakingofchiralsymmetry.TheassociatedGoldstonepionsdominatethelow-energy, nite-volumescalingbehavioroftheDiracoperatorspectruminthemicroscopicregime,1/ΛQCD<<L<<1/mπ,withLthelengthofthesystem[1].Thisbehaviorcanbecharacterizedbychiralrandommatrixthe-ory(RMT).TheRMTdescriptionofthelow-energy, nite-volumescalingbehaviorisspeci- edbysymmetrypropertiesoftheDiracoper-atorandthetopologicalchargesectorbeingcon-sidered[2,3].TheRMTpredictionsareuniversalinthesensethatthesymmetryproperties,butnottheformofthepotentialmatters[4].Fur-thermore,thepropertiescanbederiveddirectlyfromthee ective, nite-volumepartitionfunc-tionsofQCDofLeutwylerandSmilga,withoutthedetourthroughRMT[3],thoughRMTnicelyandsuccinctlydescribesandclassi esalltheseproperties.ThetopologicalchargeenterstheRMTpredictionviathenumberoffermioniczeromodes,relatedtothetopologicalchargethroughtheindextheorem.ThesymmetrypropertiesoftheDiracoperatorfallintothreeclasses,corre-spondingtothechiralorthogonal,unitary,andsymplecticensembles[3].Examplesare,respec-tively,fermionsinthefundamentalrepresentationofgaugegroupSU(2),fermionsinthefundamen-talrepresentationofgaugegroupSU(Nc)withNc≥3,andfermionsintheadjointrepresenta-

The properties of the spectrum of the overlap Dirac operator and their relation to random matrix theory are studied. In particular, the predictions from chiral random matrix theory in topologically non-trivial gauge field sectors are tested.

2

Figure1.PlotsofPmin(z)versuszforthevariousensemblesinthelowesttwotopologicalsectors.Thecurveineachplotisa ttothepredictionfromrandommatrixtheorywiththebestvalueforthechiralcondensate.

shalldescribefurthervalidatesthechiralRMTpredictionsandstrengthensthecasefortheuse-fulnessoftheOverlapregularizationofmasslessfermions.

ThemasslessoverlapDiracoperator[6]isgivenbyD=

1

The properties of the spectrum of the overlap Dirac operator and their relation to random matrix theory are studied. In particular, the predictions from chiral random matrix theory in topologically non-trivial gauge field sectors are tested.

3

Acollectionofthenecessaryformulaeforthedis-tributionofthelowesteigenvalue,Pmin(z),canbefoundin[9].

WecomparetheRMTpredictionswithourdatainFig.1.IfΣisknown,theRMTpre-dictionsforPmin(z)areparameterfree.Ontherathersmallsystemsthatweconsideredhere,wedidnotobtaindirectestimatesofΣ.Instead,wemadeone-parameter tsofthemeasureddis-tributions,obtainedfromhistogramswithjack-knifeerrors,totheRMTpredictions,withΣthefreeparameter.OurresultsandsomeadditionalinformationaregiveninTable1.WenotetheconsistencyofthevaluesforΣobtainedintheν=0andν=1sectorsofeachensemble.Alter-natively,wecouldhaveusedthevalueofΣob-tainedintheν=0sector,toobtainaparameterfreepredictionforthedistributionoftherescaledlowesteigenvalueintheν=1sector.Obviously,thepredictionswouldhavecomeoutverywell.Withthefermionsinthefundamentalrepre-sentation,wefound81(forSU(2)),and147(forSU(3))con gurationswithtwozeromodesand1and3withthreezeromodes.Fortheorthog-onalensemble,wearenotawareofapredictionforPmin(z)intheν=2sector,whilefortheuni-taryensembleourdata,albeitwithverylimitedstatistics,agreesreasonablywellwiththeparam-eterfreepredictionwithΣfromTable1.

Forfermionsintheadjointrepresentation,wekeeponlyoneofeachpairofdegenerateeigen-valuessoν=1isthesectorwithtwoexactzeromodes.Suchcon gurationscannotbeassignedanintegertopologicalchargesinceintegerchargesgiverisetozeromodesinmultiplesoffour[10],andwenotethereareasigni cantnumberofcon- gurationswithtwozeromodesasseeninTa-ble1.ThegoodagreementwiththeRMTpre-dictionfoundinthiscaselendsfurthersupporttotheexistenceofcon gurationswithfractionaltopologicalcharge[10].

WehavetestedthepredictionsofchiralrandommatrixtheoryusingtheoverlapDiracoperatoronpuregauge eldensembles.We ndthedis-tributionofthelowesteigenvalueinthedi erenttopologicalsectors tswellwiththepredictionsofchiralRMT,withcompatiblevaluesforthechiralcondensatefromthedi erenttopologicalsectors.

Table1

Thechiralcondensate,Σ,from tsofthedistribu-tionofthelowesteigenvaluetotheRMTpredic-tions.ThethirdcolumngivestheWilson-Diracmassparameterused,thefourththenumberofcon gurations,Nν,ineachtopologicalsector.Repr.mNνSU(2)fund.2.31293

1.810.2155(37)5.100.1655(16)

SU(3)fund.2.02136SU(2)adj.2.31251

2.010.2931(45)

ThisresearchwassupportedbyDOEcon-tractsDE-FG05-85ER250000andDE-FG05-96ER40979.REFERENCES

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