Phase transitions and topology changes in configur

arXiv:cond-mat/0303200v1[cond-mat.stat-mech]11Mar2003PhasetransitionsandtopologychangesinconfigurationspaceLapoCasetti∗IstitutoNazionaleperlaFisicadellaMateria(INFM),Unit`adiRicercadiFirenze,viaG.Sansone1,I-50019SestoFiorentino(FI),ItalyMarcoPettini†IstitutoNazionalediAstrofisica,OsservatorioAstrofisicodiArcetri,LargoE.Fermi5,I-50125Firenze,ItalyE.G.D.Cohen‡TheRockefellerUniversity,Box28,1230YorkAvenue,NewYork,NewYork10021-6399(Dated:February2,2008)1AbstractTherelationbetweenthermodynamicphasetransitionsinclassicalsystemsandtopologicalchangesintheirconfigurationspaceisdiscussedfortwophysicalmodelsandcontainsthefirstexactanalyticcomputationofatopologicinvariant(theEulercharacteristic)ofcertainsubmani-foldsintheconfigurationspaceoftwophysicalmodels.Themodelsarethemean-fieldXYmodelandtheone-dimensionalXYmodelwithnearest-neighborinteractions.Theformermodelunder-goesasecond-orderphasetransitionatafinitecriticaltemperaturewhilethelatterhasnophasetransitions.ThecomputationofthistopologicinvariantisperformedwithintheframeworkofMorsetheory.Inbothmodelstopologychangesinconfigurationspacearepresentasthepotentialenergyisvaried;however,inthemean-fieldmodelthereisaparticularly“strong”topologychange,correspondingtoabigjumpintheEulercharacteristic,connectedwiththephasetransition,whichisabsentintheone-dimensionalmodelwithnophasetransition.Thecomparisonbetweenthetwomodelshastwomajorconsequences:i)itlendsnewandstrongsupporttoarecentlyproposedtopologicalapproachtothestudyofphasetransitions;ii)itallowsustoconjecturewhichparticulartopologychangescouldentailaphasetransitioningeneral.Wealsodiscussasimplifiedillustrativemodelofthetopologychangesconnectedtophasetransitionsusingoftwo-dimensionalsurfaces,andapossibledirectconnectionbetweentopologicalinvariantsandthermodynamicquantities.PACSnumbers:75.10.Hk;02.40.-k;05.70.Fh;64.60.CnKeywords:Phasetransitions;topology;configurationspace;mean-fieldmodels∗Electronicaddress:casetti@fi.infn.it†Electronicaddress:[e-mail];Alsoat:INFM,UdRFirenze,andINFN,SezionediFirenze,viaG.Sansone1,I-50019SestoFiorentino(FI),Italy.‡Electronicaddress:[e-mail]whetherthecurrentmathematicaldescriptionofthermodynamicphasetransitions(basedonthelossofanalyticityofthermodynamicobservables[1,2,3])istheultimatepossibleone,orwhetherareductiontoadeepermathematicallevelispossible.Besidesapurelytheoreticalmotivation,thereareotherreasonsforthinkingofsuchapossibility.Amongtheothers,wementionthegrowingexperimentalevidencethatphasetransitionsoccurinverysmallNsystems,likenuclearclustersaswellasatomicandmolec-ularclusters,innanoandmesoscopicsystems,inpolymersandproteins,inverysmalldropsofquantumfluids(BEC,superfluidsandsuperconductors).Moreover,newmathematicalcharacterizationsofthermodynamicphasetransitionscouldwellbeofinterestforthetreatmentofotherimportanttopicsinstatisticalphysics,asisthecaseofamorphousanddisorderedsystems(likeglassesandspin-glasses),ortoincorporatealsofirst-orderphasetransitions.AdifferentattemptthenisdiscussedherehasbeenmadeinmacroscopicparameterspaceinsteadofinmicroscopicphasespaceinRef.[4].Inanumberofrecentpapers[5,6,7,8,9,10]aproposalhasbeenputforwardforanewmathematicalapproachtothestudyofphasetransitions.Thisappliestophysicalsystemsdescribedbycontinuousvariables–qiandpi,i=1,...,N–enteringastandardHamiltonianH=12NXi=1p2i+V(q1,...,qN),(1)whereV(q1,...,qN)isthepotentialenergy.ThemainissueofthisnewapproachisaTopologicalHypothesis(TH).ThecontentoftheTHisthatattheirdeepestlevelphasetransitionsofasystemareduetoachangeofthetopologyofsuitablesubmanifoldsinitsconfigurationspace.Moreprecisely,oncethemicroscopicinteractionpotentialV(q1,...,qN)isgiven,theconfigurationspaceofthesystemisautomaticallyfoliatedintothefamily{Σv}v∈Rofequipotentialhypersurfacesindependentlyofanystatisticalmeasurewemaywishtouse.Now,fromstandardstatisticalmechanicalargumentsweknowthatthelargerthenumberNofparticles,theclosertosomeΣvarethemicrostateswhichsignificantlycontributetothestatisticalaveragesofthermodynamicobservables.AtlargeN,andatanygivenvalueoftheinversetemperatureβ,theeffectivesupportofthecanonicalmeasureisnarrowedverycloselytoasinglehypersurfaceΣv≡{q∈RN|V(q)=v}⊂RN,withvasuitablefunctionofβ.3Now,theTHconsistsinassumingthatsomesuitablechangeofthetopologyofthe{Σv},orequivalently,atlargeN,ofthesubmanifoldsMv={q∈RN|V(q)≤v}(themanifoldsΣvaretheboundariesoftheMv,i.e.,Σv=∂Mv),occurringatsomevc=vc(βc)(orvc=vc(Ec)),isthedeeporiginofthesingularbehaviorofthermodynamicobservablesataphasetransition;(bychangeoftopologywemeanthat{Σv}vvcarenotdiffeomorphic[23]tothe{Σv}vvc,orequivalentlythat{Mv}vvcarenotdiffeomorphictothe{Mv}vvc).InthefollowingofthepaperweshallconsiderthetopologychangesoftheMv,becausethesearenaturallyinvestigatedusingMorsetheory.Thepresentpapercontributestothisnewapproachtophasetransitionswithacrucialstepforward.Thisconsistsofanexactanalytictreatmentofthetopologychangesintheconfigurationspaceofthemean-fieldXYmodelandoftheirrelationtothethermodynamicphasetransition,reportedinSec.II.InSec.IIIwepresen