Locality Properties of a New Class of Lattice Dira

arXiv:hep-lat/0102012v213Apr2001UT-926LocalityPropertiesofaNewClassofLatticeDiracOperatorsKazuoFujikawaandMasatoIshibashiDepartmentofPhysics,UniversityofTokyoBunkyo-ku,Tokyo113,JapanAbstractAnewclassoflatticeDiracoperatorsDwhichsatisfytheindextheoremhavebeenrecentlyproposedonthebasisofthealgebraicrelationγ5(γ5D)+(γ5D)γ5=2a2k+1(γ5D)2k+2.Herekstandsforanon-negativeintegerandk=0correspondstotheordinaryGinsparg-Wilsonrelation.WeanalyzethelocalitypropertiesofDiracoperatorswhichsolvetheabovealgebraicrelation.WefirstshowthatthefreefermionoperatorisanalyticintheentireBrillouinzoneforasuitablechoiceofparametersm0andr,andthereexistsawell-defined“massgap”inmomentumspace,whichinturnleadstotheexponentialdecayoftheoperatorincoordinatespaceforanyfinitek.Thismassgapinthefreefermionoperatorsuggeststhattheoperatorislocalforsufficientlyweakbackgroundgaugefields.WeinfactestablishafinitelocalitydomainofgaugefieldstrengthforΓ5=γ5−(aγ5D)2k+1foranyfinitek,whichissufficientforthecohomologicalanalysesofchiralgaugetheory.WealsopresentacrudeestimateofthelocalizationlengthdefinedbyanexponentialdecayoftheDiracoperator,whichturnsouttobemuchshorterthantheonegivenbythegeneralLegendreexpansion.1IntroductionWehaverecentlywitnessedaremarkableprogressinthetreatmentoflatticefermions[1]-[4].Thefirstbreakthroughmaybetracedtothedomain-wallfermion[5][6],whichwasfollowedbytheoverlapfermion[7].Seealsorelatedworksin[8].Itiswellknownthattheoverlapfermion[2],whichdevelopedindependentlyoftheGinsparg-Wilsonrelation[1],satisfiesthesimplestversionoftheGinsparg-Wilsonrelation.Therecognitionthatthefermionoperator,whichsatisfiesthesimplestversionoftheGinsparg-Wilsonrelation,givesrisetotheindextheoremonthelattice[3]andthusmodifiedbutexactlatticechiralsymmetry[4],wascrucialintherecentdevelopments.ThelocalitypropertiesoftheNeuberger’soverlapoperatorhavealsobeenestablished[9][10];theoperatorisnotultra-local[11]butexponentiallylocal,whichisconsideredtobesufficienttoensurelocalityinthecontinuumlimitwiththelatticespacinga→0.Inthemeantime,anewclassoflatticeDiracoperatorsDhavebeenproposedonthebasisofthealgebraicrelation[12]γ5(γ5D)+(γ5D)γ5=2a2k+1(γ5D)2k+2(1.1)1wherekstandsforanon-negativeinteger,andk=0correspondstotheordinaryGinsparg-Wilsonrelation[4]forwhichanexplicitexampleoftheoperatorfreeofspeciesdoublingisknown[2].Ithasbeenshownin[12]thatwecanconstructthelatticeDiracoperator,whichisfreeofspeciesdoublersandsatisfiesalatticeversionofindextheorem[3][4],foranyfinitek.Anexplicitcalculationofthechiralanomalyfortheseoperatorshasalsobeenperformed[13].Hereγ5isahermitianchiralDiracmatrixandγ5Dishermitian.Inasense,thisnewclassoffermionoperatorsareregardedasthefirstgenerationoflatticefermionoperatorswhicharedirectlymotivatedbytheGinsparg-Wilsonrelation.IncontrasttothemostgeneralformoftheGinsparg-Wilsonrelation[1],ouralgebra(1.1)allowsanexplicitsolutionwhichsatisfiestheindextheorem:OurexplicitsolutionsillustratethefeaturesofpossiblesolutionsofthegeneralGinsparg-Wilsonrelation.AsalientfeatureofDiracoperatorscorrespondingtolargerkisthatthechiralsym-metrybreakingtermbecomesmoreirrelevantinthesenseofWilsonianrenormalizationgroup.ForH=aγ5Dinthenearcontinuumconfigurationswehave,forexample,H≃γ5ai6D+γ5(γ5ai6D)2fork=0,H≃γ5ai6D+γ5(γ5ai6D)4fork=1(1.2)respectively.Thefirsttermsintheseexpressionsstandfortheleadingtermsinchiralsymmetricterms,andthesecondtermsintheseexpressionsstandfortheleadingtermsinchiralsymmetrybreakingterms,respectively.Thisshowsthatonecanimprovethechiralsymmetryforlargerk,thoughtheoperatorspreadsovermorelatticepointsforlargerk.Asanothermanifestationofthisproperty,thespectrumoftheoperatorswithk0isclosertothatofthecontinuumoperatorinthesensethatthesmalleigenvaluesofDaccumulatealongtheimaginaryaxis(whichisaresultoftakinga2k+1-throot),comparedtothestandardoverlapoperatorforwhichtheeigenvaluesofDdrawaperfectcircleinthecomplexeigenvalueplane.Inthispaper,weanalyzethelocalitypropertiesofthisgeneralclassoflatticeDiracopeartors.AsfortheoverlapDiracoperator[2],whichcorrespondstok=0,theverydetailedanalysesoflocalitypropertieshavebeenperformedbyHernandez,JansenandL¨uscher[9],andNeuberger[10].AsimilarlocalityanalysisofthedomainwallfermionhasbeengivenbyKikukawa[14].Weestablishthelocalitypropertiesofthegeneralclassofoperators(1.1)forallfinitek,whicharesufficientforthecohomologicalanalysesofchiralgaugetheory[15][16],forexample.2ABriefSummaryoftheModelandNotationTheexplicitconstructionoftheoperator,whichsatisfiestherelation(1.1),proceedsbyfirstdefiningH(2k+1)≡(γ5aD)2k+1=12γ5[1+D(2k+1)W1q(D(2k+1)W)†D(2k+1)W].(2.1)2TheoperatorD(2k+1)WisinturnexpressedasageneralizationoftheordinaryWilsonDiracoperatorasD(2k+1)W=i(6C)2k+1+(B)2k+1−(m0a)2k+1.(2.2)TheordinaryWilsonDiracoperatorDW,whichcorrespondstoD(1)W,isgivenbyDW(x,y)≡iγμCμ(x,y)+B(x,y)−1am0δx,y,Cμ(x,y)=12a[δx+ˆμa,yUμ(y)−δx,y+ˆμaU†μ(x)],B(x,y)=r2aXμ[2δx,y−δy+ˆμa,xU†μ(x)−δy,x+ˆμaUμ(y)],Uμ(y)=exp[iagAμ(y)],(2.3)whereweaddedaconstantmasstermtoDW.Ourmatrixconventionisthatγμareanti-hermitian,(γμ)†=−γμ,andthus6C≡γμCμ(n,m)ishermitian6C†=6C.(2.4)Toavoidtheappearanceofspeciesdouble