Index theoretic characterization of d-wave superconductors in the vortex state

arXiv:cond-mat/0509258v2 [cond-mat.supr-con] 9 Apr 2006Indextheoreticcharacterizationofd-wavesuperconductorsinthevortexstate

OskarVafek1andAshotMelikyan2

1StanfordUniversityInstituteforTheoreticalPhysicsandDepartmentofPhysics,Stanford,CA943052InstituteofFundamentalTheory,DepartmentofPhysics,UniversityofFlorida,Gainesville,FL32611(Dated:February2,2008)

Weemployindextheoreticmethodstostudyanalyticallythelowenergyspectrumofalatticed-wavesuperconductorinthevortexlatticestate.Thisallowsustocomparesinglyquantizedhc/2eanddoublyquantizedhc/evortices,thefirstofwhichmustalwaysbeaccompaniedbyZ2branchcuts.Foraninversionsymmetricvortexlatticeandinthepresenceofparticle-holesymmetryweproveanindextheoremthatimposesalowerboundonthenumberofzeroenergymodes.Genericcasesareconstructedinwhichthisboundexceedsthenumberofzeromodesofanequivalentlatticeofdoublyquantizedvortices,despitetheidenticalpointgroupsymmetries.ThequasiparticlespectrumaroundthezeromodesisdoublydegenerateandexhibitsaDirac-likedispersion,withvelocitiesthatbecomeuniversalfunctionsof∆0/tinthelimitoflowmagneticfield.Forweakparticle-holesymmetrybreaking,thegappedstatecanbecharacterizedbyatopologicalquantumnumber,relatedtospinHallconductivity,whichgenerallydiffersinthecasesofthehc/2eandhc/evortexlattices.

Understandingtheinteractionsamongfermionicquasi-

particlesandtopologicalexcitationsoftwodimensional

superconductorsconstitutesoneofthemainchallengesin

moderncondensedmatterphysics.Itisalmostcertainly

truethatatleastsomeofthepeculiarphenomenaob-

servedintheunderdopedcupratesarerelatedtotheex-

istenceofnodalfermionicexcitationsofadx2−y2wavesu-

perconductorwithlowsuperfluiddensity,whichiseven-

tuallysusceptibletodephasingbyproliferationofmobile

vortices[1].

Whenanexternalmagneticfieldinducesafiniteden-

sityofvorticeswhichformanAbrikosovlattice,themo-

tionofvorticescanbeneglectedandtheproblemofthe

quasiparticlespectrumissimplified.Theinitialtheoret-

icalanalysiswasbasedonnumericalcomputations[2],

semiclassicalapproximation[3],andscalingarguments

[4].AspointedoutbyFranzandTesanovic[5,6,7],the

quasiparticlesinteractwiththehc/2evorticesbyboth

the(semiclassical)couplingofthechargecurrentstothe

superflowaswellasapurelyquantumcouplingofthe

spincurrentstotheZ2branchcutsemanatingfromthe

vortices.Marinelli,Halperin,andSimon[8]thenshowed

thatwithinthelinearizedapproximationthespectrumat

eachnoderemainsgaplesstoallordersofperturbation

theoryinphasegradients.Thisresultwasrederivedby

aninsightfulsymmetryanalysisinRef.[9],whereitwas

concludedthattheeffectofthehc/2evorticesismerelya

possiblerenormalizationofthenodalquasiparticleveloc-

ities,butthatthepresenceandparticularlythenumber

ofthezeromodesisthesameasinthelatticeofdoubly

quantizedhc/evortices.Varioustopologicalaspectsof

theproblemwerediscussedin[7,9,10].

InthisLetter,werevisittheproblemoftheFermionic

(quasiparticle)spectrumofalatticed-wavesuperconduc-

torinthevortexstate.Ourmainresultisthatthere

canariseafundamentaltopologicaldifferencebetween

thesinglyanddoublyquantizedvortexlatticewhichisbeyondperturbativeanalysis[8,9].Inthecaseofanin-

versionsymmetricvortexlattice,thisdifferencerelates

tothenon-trivialtransformationpropertiesoftheZ2branchcutsunderinversion,absentforhc/evortices.

Ourpointofdepartureistheinversionsymmetricvor-

texlatticestateofthed-wavesuperconductoronatight-

bindinglatticewithperfectparticle-hole(p-h)symmetry.

Weusethep-hsymmetrytoconductnon-perturbative

analysisofthelowenergyspectrumandfindconditions

sufficientfortheexistenceofzeroenergynodalstates.

Armedwiththisanalysis,westudyweakbreakingofthe

p-hsymmetry,whichgapsthenodalpoints,andshow

thattheresultingstateistopologicallynon-trivial.Itis

characterizedbyatopologicalquantumnumberpropor-

tionaltothespin-Hallconductivity[7,9,11,12].

Remarkably,incaseswhenthebranchcutstransform

non-triviallyunderinversion,thenumberofthezero

modesisdoubledforhc/2evorticesrelativetothehc/e

counterpart,despiteidenticaldiscretetranslationalsym-

metryofbothvortexlatticesinquestion.Thedifference

alsoholdsforalatticeofhc/2evortices,whenthebranch

cutsareignored.Wedemonstratethisbyprovingaform

ofanindextheorem[13,14],whichputsalowerbound

onthenumberofzeromodes,andbyexplicitanalyti-

calandnumericalsolutions,whichrevealthatthelower

boundis,infact,saturated.

Wenowprovidejustificationfortheaboveclaims.

ConsidertheHamiltonianH=H0−µN,where

H0=󰀁

󰀍rr′󰀌󰀈

−trr′c†rσcr′σ+∆rr′thevortexlatticestatereads∆rr′=ηr−r′∆0exp(iθrr′);

ηδ=+(−)ifδ󰀗ˆx(ˆy).Thebondphasefactorsexp(iθrr′)

resultfromaself-consistentcalculation,butthemain

topologicalfeatureofθrr′isits2πwindingaroundthe

magneticfieldinducedvortices.AstheinitialAnsatzfor

theself-consistentsolution,wechoosethelatticebond

variablesθrr′asfollows:firstwesolvethecontinuum