Complex-Temperature Phase Diagrams for the q-State

arXiv:cond-mat/0107012v2[cond-mat.stat-mech]4Jul2001Complex-TemperaturePhaseDiagramsfortheq-StatePottsModelonSelf-DualFamiliesofGraphsandtheNatureoftheq→∞LimitShu-ChiuanChang∗andRobertShrock†C.N.YangInstituteforTheoreticalPhysicsStateUniversityofNewYorkStonyBrook,NY11794-3840WepresentexactcalculationsofthePottsmodelpartitionfunctionZ(G,q,v)forarbitraryqandtemperature-likevariablevonself-dualstripgraphsGofthesquarelatticewithfixedwidthLyandarbitrarilygreatlengthLxwithtwotypesofboundaryconditions.LettingLx→∞,wecomputetheresultantfreeenergyandcomplex-temperaturephasediagram,includingthelocusBwherethefreeenergyisnonanalytic.ResultsareanalyzedforwidthsLy=1,2,3.Weusetheseresultstostudytheapproachtothelarge-qlimitofB.05.20.-y,64.60.C,75.10.H∗email:[e-mail]†email:[e-mail]-statePottsmodelhasservedasavaluablemodelforthestudyofphasetransitionsandcriticalphenomena[1,2].Onalattice,or,moregenerally,ona(connected)graphG,attemperatureT,thismodelisdefinedbythepartitionfunctionZ(G,q,v)=X{σn}e−βH(1.1)withthe(zero-field)HamiltonianH=−JXhijiδσiσj(1.2)whereσi=1,...,qarethespinvariablesoneachvertexi∈G;β=(kBT)−1;andhijidenotespairsofadjacentvertices.ThegraphG=G(V,E)isdefinedbyitsvertexsetVanditsedge(bond)setE;wedenotethenumberofverticesofGasn=n(G)=|V|andthenumberofedgesofGase(G)=|E|.WeusethenotationK=βJ,a=eK=u−1,v=a−1(1.3)sothatthephysicalrangesare(i)a≥1,i.e.,v≥0correspondingto∞≥T≥0forthePottsferromagnet(FM),withJ0,and(ii)0≤a≤1,i.e.,−1≤v≤0,correspondingto0≤T≤∞forthePottsantiferromagnet(AFM),withJ0.Onedefinesthe(reduced)freeenergypersitef=−βF,whereFistheactualfreeenergy,viaf({G},q,v)=limn→∞ln[Z(G,q,v)1/n](1.4)whereweusethesymbol{G}todenotelimn→∞Gforagivenfamilyofgraphs.Onatwo-dimensionallattice,fortheq=2Isingcase,andforq=3,4,thePottsferromag-netexhibitsasecond-orderphasetransitionfromaparamagnetic(PM)high-temperaturephasetoalow-temperaturephasewithspontaneouslybrokensymmetryandlong-rangefer-romagneticorder(magnetization).Forq4,thistransitionisfirst-order,withalatentheatthatincreasesmonotonicallywithq,approachingalimitingconstantasq→∞[2,3].ThebehaviorofthePottsantiferromagnetdependsonthevalueofqandthetypeoflattice,aswillbediscussedfurtherbelow.LetG′=(V,E′)beaspanningsubgraphofG,i.e.asubgraphhavingthesamevertexsetVandasubsetoftheedgeset,E′⊆E.ThenZ(G,q,v)canbewrittenasthesum[4]Z(G,q,v)=XG′⊆Gqk(G′)ve(G′)(1.5)wherek(G′)denotesthenumberofconnectedcomponentsofG′.SinceweonlyconsiderconnectedgraphsG,wehavek(G)=1.Theformula(1.5)enablesonetogeneralizeqfromZ+toR+(keepingvinitsphysicalrange).Theformula(1.5)showsthatZ(G,q,v)isapolynomialinqandv(equivalently,a).ThePottsmodelpartitionfunctiononagraphGis1essentiallyequivalenttotheTuttepolynomial[5]-[7]andWhitneyrankpolynomial[2],[8]-[10]forthisgraph,andthisconnectionwillbeusefulbelow.Usingtheformula(1.5)forZ(G,q,v),onecangeneralizeqfromZ+notjusttoR+buttoCandvfromitsphysicalferromagneticandantiferromagneticranges0≤v≤∞and−1≤v≤0tov∈C.AsubsetofthezerosofZinthetwo-complexdimensionalspaceC2definedbythepairofvariables(q,v)formanaccumulationsetinthen→∞limit,denotedB,whichisthecontinuouslocusofpointswherethefreeenergyisnonanalytic.TheprogramofstudyingstatisticalmechanicalmodelswithexternalfieldgeneralizedfromRtoCwaspioneeredbyYangandLee[11],andthecorrespondinggeneralizationofthetemperaturefromphysicaltocomplexvalueswasinitiatedbyFisher[12].Hereweallowbothqandthetemperature-likevariablevtobecomplex.Foragivenvalueofv,onecanconsiderthislocusintheqplane,andweshallsometimesdenoteitasBq,andsimilarly,foragivenvalueofq(notnecessarily∈Z+),onecanconsiderthislocusintheplaneofacomplex-temperaturevariablesuchasvorζ=v√q.(1.6)Itwillbeconvenienttointroducepolarcoordinates,lettingζ=|ζ|eiθ.InthispaperweshallpresentexactcalculationsofthePottsmodelpartitionfunctionZ(G,q,v)forarbitraryqandvonself-dualstripgraphsGofthesquarelatticewithfixedwidthLyandarbitrarilygreatlengthLxwithtwotypesofboundaryconditions.LettingLx→∞,wecomputetheresultantfreeenergyandcomplex-temperaturephasediagram.ResultsareanalyzedforwidthsLy=1,2,3.Weshallusetheseresultstostudytheapproachtothelarge-qlimitofB.Thereareseveralmotivationsforthisstudy.Clearly,newexactcalculationsofPottsmodelpartitionfunctionsonlatticestripswitharbitrarilylargenumbersofverticesareofvalueintheirownright.ThisisespeciallythecasesincethefreeenergyofthePottsmodelhasneverbeencalculatedexactlyford≥2exceptintheq=2Isingcasein2D.Justasthestudyoffunctionsofacomplexvariablecanyieldadeeperunderstandingoffunc-tionsofarealvariable,soalsotheinvestigationofcomplex-temperaturephasediagramsofspinmodelscanprovidefurtherunderstandingofthephysicalbehaviorofthesemodels.Besides[12],complex-temperaturesingularitieswerenoticedinearlyseriesanalyses(e.g.,[13]),andmanystudieshavebeencarriedoutoncomplex-temperature(Fisher)zerosofthepartitionfunctionoftheIsingmodelanditsgeneralizationtotheq-statePottsmodel[14]-[61].Inparticular,severalexactdeterminationsofcomplex-temperaturephasedia-gramsofthePottsmodeloninfinite-length,finite-widthlatticestrips[41,52–54,56–58],incompa