A mathematical proof that the transition to a supe

arXiv:0808.3438v1[math-ph]26Aug2008Amathematicalproofthatthetransitiontoasuperconductingstateisasecond-orderphasetransitionShujiWatanabeDivisionofMathematicalSciencesGraduateSchoolofEngineering,GunmaUniversity4-2Aramaki-machi,Maebashi371-8510,JapanEmail:[e-mail]apfunctionandthethermodynamicalpotentialintheBCS-Bogoliubovtheoryofsuperconductivity,wherethegapfunctionisafunctionofthetemperatureTonly.WeshowthatthesquaredgapfunctionisofclassC2ontheclosedinterval[0,Tc]andpointoutsomemorepropertiesofthegapfunction.Here,Tcstandsforthetransitiontemperature.Onthebasisofthisstudywethengive,examiningthethermodynamicalpotential,amathematicalproofthatthetransitiontoasuperconductingstateisasecond-orderphasetransition.Furthermore,weobtainanewandmorepreciseformofthegapinthespecificheatatconstantvolumefromamathematicalpointofview.MathematicsSubjectClassification(2000):45G10,82D55Keywords:Second-orderphasetransition,superconductivity,gapfunction,thermo-dynamicalpotential1IntroductionLetε0besmallenoughandletusfixitunlessotherwisestated.LetkB0andωD0standfortheBoltzmannconstantandfortheDebyefrequency,respectively.WedenotePlanck’sconstantbyh(0)andsetℏ=h/(2π).Letμ0standforthechemicalpotential.LetN(ξ)≥0standforthedensityofstatesperunitenergyattheenergyξ(−μ≤ξ∞)andletN0=N(0)0.Here,N0standsforthedensityofstatesperunitenergyattheFermisurface(ξ=0).LetU00beaconstant.ItiswellknownthatsuperconductivityoccursattemperaturesbelowthetemperatureTc0calledthetransitiontemperature.Wenowdefineit.Definition1.1.ThetransitiontemperatureisthetemperatureTc0satisfying1U0N0=ZℏωD/(2kBTc)εtanhηηdη.1Generallyspeaking,thegapfunctionisafunctionbothofthetemperatureTandofwavevector.InthispaperwehoweverregardthegapfunctionasafunctionofthetemperatureTonly,anddenoteditbyΔ(T)(≥0).SuchasituationisconsideredintheBCS-Bogoliubovtheory[1,3],andisacceptedwidelyincondensedmatterphysics(seee.g.[6,(7.118),p.250],[12,(11.45),p.392]).Seealso[9]and[10]forrelatedmaterial.Thegapfunctionsatisfiesthefollowingnonlinearintegralequationcalledthegapequation(c.f.[1]):For0≤T≤Tc,(1.1)1=U0N0ZℏωD2kBTcε1pξ2+f(T)tanhpξ2+f(T)2kBTdξ.Here,forlaterconvenience,thesquaredgapfunctionisdenotedbyf,i.e.,f(T)=Δ(T)2.Remark1.2.WeintroducethecutoffεinDefinition1.1andinthegapequation(1.1).Whenε=0,Definition1.1andthegapequation(1.1)reducetothoseintheBCS-Bogoliubovtheory[1,3].Furthermore,whenε=0,thethermodynamicalpotentialΩinDefinition1.4belowreducestothatintheBCS-Bogoliubovtheory(seealso(1.2)and(1.3)below).Seee.g.Niwa[6,sec.7.7.3,p.255].Thegapequation(1.1)isasimplifiedone,andthegapequationwithamoregeneralpotentialisstudiedextensively.Odeh[7]andBillardandFano[2]establishedtheexistenceanduniquenessofthepositivesolutiontothegapequationwithamoregeneralpotentialinthecaseT=0.InthecaseT≥0,Vansevenant[8]andYang[11]determinedthetransitiontemperatureandshowedthatthereisauniquepositivesolutiontothegapequationwithamoregeneralpotential.RecentlyHainzl,Hamza,SeiringerandSolovej[4],andHainzlandSeiringer[5]provedthattheexistenceofapositivesolutiontothegapequationwithamoregeneralpotentialisequivalenttotheexistenceofanegativeeigenvalueofacertainlinearoperatortoshowtheexistenceofatransitiontemperature.Letf(T)beasin(1.1)andsetΩS(T)=ΩN(T)+δ(T),ΩN(T)=−2N0ZℏωD2kBTcεξdξ−4N0kBTZℏωD2kBTcεln1+e−ξ/(kBT)dξ(1.2)+V(T),T0,δ(T)=f(T)U0−2N0ZℏωD2kBTcεnpξ2+f(T)−ξodξ(1.3)−4N0kBTZℏωD2kBTcεln1+e−pξ2+f(T)/(kBT)1+e−ξ/(kBT)dξ,0T≤Tc,V(T)=2Z−ℏωD−μξN(ξ)dξ−2kBTZ−ℏωD−μN(ξ)ln1+eξ/(kBT)dξ(1.4)−2kBTZ∞ℏωDN(ξ)ln1+e−ξ/(kBT)dξ,T0.Remark1.3.SinceN(ξ)=O(√ξ)asξ→∞,theintegralontherightsideof(1.4)Z∞ℏωDN(ξ)ln1+e−ξ/(kBT)dξ2iswelldefinedforT0.Definition1.4.LetΩS(T)andΩN(T)beasabove.ThethermodynamicalpotentialΩisdefinedbyΩ(T)=(ΩS(T)(0T≤Tc),ΩN(T)(TTc).Remark1.5.Generallyspeaking,thethermodynamicalpotentialΩisafunctionofthetemperatureT,thechemicalpotentialμandthevolumeofourphysicalsystem.Fixingthevaluesofμandofthevolumeofourphysicalsystem,wedealwiththedependenceofΩonthetemperatureTonly.Remark1.6.Hainzl,Hamza,SeiringerandSolovej[4],andHainzlandSeiringer[5]studiedthegapequationwithamoregeneralpotentialexaminingthethermodynamicpressure.Definition1.7.WesaythatthetransitiontoasuperconductingstateatthetransitiontemperatureTcisasecond-orderphasetransitionifthefollowingconditionsarefulfilled:(a)ThethermodynamicalpotentialΩ,regardedasafunctionofT,isofclassC1on(0,∞).(b)Thesecond-orderderivative∂2Ω/∂T2
iscontinuouson(0,∞)\{Tc}andisdiscontinuousatT=Tc.Remark1.8.Condition(a)impliesthattheentropyS=−(∂Ω/∂T)iscontinuouson(0,∞)andthat,asaresult,nolatentheatisobservedatT=Tc.Ontheotherhand,(b)impliesthatthespecificheatatconstantvolume,CV=−T∂2Ω/∂T2
,isdiscontinuousatT=Tc.SeeProposition2.4below,whichgivesanewandmorepreciseformofthegapΔCVinthespecificheatatconstantvolumeatT=Tcfromamathematicalpointofview.Fromaphysicalpointofview,itispointedoutthatthetransitionfromanormalstatetoasuperconductingstateisasecond-orderphasetransition.Butamathematicalproofofthisstatementhasnotbeengivenyetasfarasweknow.Inthispaperwefirstshowthatthereisauniquesolution: