Some New Results on Complex-Temperature Singularit

arXiv:cond-mat/9605176v24Jun1996ITP-SB-96-18May,1996SomeNewResultsonComplex-TemperatureSingularitiesinPottsModelsontheSquareLatticeVictorMatveev∗andRobertShrock∗∗InstituteforTheoreticalPhysicsStateUniversityofNewYorkStonyBrook,N.Y.11794-3840AbstractWereportsomenewresultsonthecomplex-temperature(CT)singularitiesofq-statePottsmodelsonthesquarelattice.WeconcentrateontheproblematicregionRe(a)0(wherea=eK)inwhichCTzerosofthepartitionfunctionaresensitivetofinitelatticeartifacts.Fromanalysesoflow-temperatureseriesexpansionsfor3≤q≤8,weestablishtheexistence,inthisregion,ofcomplex-conjugateCTsingularitiesatwhichthemagnetizationandsusceptibilitydiverge.Fromcalculationsofzerosofthepartitionfunction,weobtainevidenceconsistentwiththeinferencethatthesesingularitiesoccuratendpointsae,a∗eofarcsprotrudingintothe(complex-temperatureextensionofthe)FMphase.Exponentsforthesesingularitiesaredetermined;e.g.,forq=3,wefindβe=−0.125(1),consistentwithβe=−1/8.Byduality,theseresultsalsoimplyassociatedarcsextendingtothe(CTextensionofthe)symmetricPMphase.Analyticexpressionsaresuggestedforthepositionsofsomeofthesesingularities;e.g.,forq=5,ourfindingisconsistentwiththeexactvalueae,a∗e=2(−1∓i).Furtherdiscussionsofcomplex-temperaturephasediagramsaregiven.∗email:[e-mail]∗∗email:[e-mail]1IntroductionandModelInthispaper,wereportsomenewresultsoncomplex-temperaturesingularitiesoftheq-statePottsmodel[1,2]onthesquarelattice.ThePottsmodelhasbeenofinterestbothasanexampleofaparticularuniversalityclassforcriticalphenomenaandasamodelforphysicalphenomenasuchastheadsorptionofcertaingasesonsubstrates[3].However,incontrasttothe2DIsingmodel(equivalenttotheq=2case)thefreeenergyofthePottsmodelforgeneralqhasneverbeencalculatedinclosedform,evenforzeroexternalfield(s).Someexactresultshavebeenestablishedforthemodel:fromadualityrelation,thecriticalpointseparatingthedisordered,Zq-symmetrichigh-temperaturephasefromthelow-temperaturephasewithspontaneouslybrokenZqsymmetryandassociatednonzeroferromagnetic(FM)long-rangeorderisknown[1].Thefreeenergyandlatentheat[4],andmagnetization[5]havebeencalculatedexactlybyBaxteratthiscriticalpoint,establishingthatthemodelhasacontinuous,second-ordertransitionforq≤4andafirst-ordertransitionforq≥5.Baxterhasalsoshownthatalthoughtheq=3modelhasnophasewithantiferromagnetic(AFM)long-rangeorderatanyfinitetemperature,thereisanAFMcriticalpointatT=0[5].Thevaluesofthecriticalexponents(fortherangeofqwherethetransitioniscontinuous)havebeendetermined[6].Subsequently,furtherinsightintothecriticalbehaviorwasgainedusingthemethodsofconformalfieldtheory[7].Areviewofworkupthrough1982wasgiveninRef.[8].Ingeneral,ifoneknewtheexact(zero-field)freeenergy,onewouldbeabletodeterminethefullphasediagramasafunctionofcomplextemperature.TheideaofgeneralizingavariableonwhichthefreeenergydependsfromrealphysicalvaluestocomplexvalueswaspioneeredbyYangandLee[9].Theseauthorsconsideredthegeneralizationoftheexternalmagneticfieldtocomplexvalues[9]andprovedacelebratedtheoremthatthecomplex-fieldzerosoftheIsingmodelpartitionfunctionlieontheunitcircleintheμplane,whereμ=e−2βH,pinchingtherealaxisasthetemperatureTdecreasesthroughthecriticalpointTc.Complex-temperature(CT)singularitiesofIsingmodels,firstconsideredinRef.[10],wereinvestigatedbothbymeansofCTzerosofthepartitionfunction[11]-[13]andviatheireffectsonlow-temperatureseriesexpansions[14].Aswellasbeingofhistoricalinterest,thesearerelevantherebecauseoftheequivalenceofthe(spin1/2)Isingmodelandq=2Pottsmodel.Thereiscontinuinginterestinsuchcomplexificationsbecauseofthedeeperinsightwhichtheygiveoneintothepropertiesofstatisticalmechanicalmodels(fortheIsingmodel,see,e.g.,Refs.[15]-[26]).Fromgeneralargumentsandcomparisonswithexactsolutionsfor2DIsingmodelswithisotropiccouplings,oneknowsthatinthethermodynamiclimit,CTzerosmergetogethertoformcurves(includingpossiblelinesegments)acrosswhichthefree1energyisnon-analytic.Thesecurvesformthecomplex-temperaturephaseboundary(CTPB)Bofthemodel.Onecandefinenotionsofcomplex-temperatureextensions(CTE’s)ofthephysicalparamagnetic(PM),ferromagnetic(FM)and(ifitoccurs)antiferromagnetic(AFM)phases.Incertaincasesthereareother(labelled“O”)complex-temperaturephaseswhichdonothaveanyoverlapwithanyphysicalphase.ThesevariousCTphasesareseparatedbyboundariescomprisingB.ThelocusofpointsmakingupBmayalsocontainpart(s)consistingofcurves(arcs)orlinesegmentswhichprotrudeintoandterminatein,certainphases.Therehavebeenseveralcalculationsofcomplex-temperaturezerosofthepartitionfunc-tionforthePottsmodelonthesquarelattice[27]-[32].Sincetheearlycalculationsforq=3,4,ithasbeenrecognizedthatthezerosshowoneclearfeature:ifoneusesduality-preservingboundaryconditions,thenintheRe(a)0region(wherea=eK;seebelowfornotation),thesezeroslieonaportionoftheunitcircle|x|=1,wherex=(a−1)/√q[28]-[31].Inpassing,wenotethatintheq→∞limitithasbeenshown(assumingthattheq→∞limitandthethermodynamiclimitcommute)thattheCTzeroslieontheunitcircle|x|=1[32,33].However,foragiven(finite)q,thesituationintheRe(a)0regionhasprovedtobemuchmo