Phase Transition in Anyon Superconductivity at Finite Temper(13)

The magnetic response of the charged anyon fluid at temperatures larger than the fermion energy gap is investigated in the self-consistent field approximation. In this temperature region a new phase, characterized by an inhomogeneous magnetic penetration,

1Dµν(k)=iN

α2kµkν+Πµν(k)(4.4)

Πµνistheone-loopfermionpolarizationoperatorgivenfrom(2.20)-(2.21)by A1k2A1kω iΠ1k iΠ1ωΠµν= A1kω A1ω2

22iΠ1k iΠ1ω A1ω+Π2k(4.5)

In(4.5)weareconsideringtheframek=(k,0),andthenotationω=k0hasbeenused.

Thee ectivetheoryfortheelectromagnetic eldinthechargedanyon uid,isfoundintegratingtheCS eldsinthepartitionfunction(2.3)withLagrangiandensity(4.2).Thenewe ectiveLagrangiandensitysoobtainedis

L′eff= 1

2± 2

Furthermore,inthepresenceofabackgroundmagnetic eld,thedistinctionbetweenmassandrest

energybecomesessential.Forinstance,inthecontextofstringtheoryinabackgroundmagnetic eld,ithasbeenshown[37]thatthemass(de nedinagreementwithWigner’sde nition)ofhigherspin(s≥1)chargedbosonparticlesdoesnotcoincidewiththeirrestenergy.Thisisduetothemodi cationofthealgebraoftheglobalsymmetriesbythebackground eld.

Fromtheabovediscussion,itisalsoclearthatincalculatingthemagneticmass(eq.(4.10))andtherestenergy(eq.(4.11))at nitetemperature,wehavetotakethepolarizationoperatorscoe cientsinthestaticlimit(ω=0,k～0)andintheplasmonlimit(k=0,ω～0)respectively.Now,becauseofthelackofanalyticityoftheGreen’sfunctionaboutkµ=0atT=0,itisknownthatinQFTtheselimitsdonotcommute[35],[38].InanyontheoryatT=0onefacesasimilarsituation,asitwasshowninref.[31].Then,ineachcasewehavetoconsiderthepolarizationoperatorcoe cientsevaluatedinthecorrespondinglimit.

InanyontheorytheCSinteractiongivesrisetoadispersionequationwithastructuremorecomplicatedthanthatcorrespondingtoaKlein-Gordon-liketheory.Asshownbelow,inthiscasethemagneticmassesandtherestenergiesoftheelectromagneticmodesaredi erent.

2+M2(4.13)