GossamerSuperconductivity

R.B.Laughlin

DepartmentofPhysics,StanfordUniversity,Stanford,CA94305

(Dated:September1,2002)

AnnewsuperconductinghamiltonianisintroducedforwhichtheexactgroundstateistheAn-

dersonresonatingvalencebond.Itdiffersfromthet-Jandhubbardhamiltoniansinpossessingapowerfulattractiveforce.Itssuperconductingstateischaracterizedbyafullandintactd-wavetunnelinggap,quasiparticlephotoemissionintensitiesthatarestronglysuppressed,asuppressedsuperfluiddensity,andanincipientMott-Hubbardgap.[PublishedasPhil.Mag.86,1165(2006).]PACSnumbers:74.20.-z,74.20.Mn,74.72.-h,71.10.Fd,71.10.Pm

IthasbeenknownsincetheearlyworkofUemura

1

that

thesuperfluiddensityandtransitiontemperatureofun-derdopedcupratesuperconductorsbothvanishmore-or-lesslinearlywithdopingandareproportional.Thecon-stantofproportionalityisconsistentwiththetransitionbeinganorder-parameterphaseinstabilityanalogoustotheKosterlitz-Thoulesstransitionin2dimensions

3

.This

ideaissupportedbynumerousotherexperiments,includ-ingtheopticalsumrulestudiesofUchida

2

,thegiant

proximityeffectreportedbyDeccaetal

4

,andtherecent

heat-transportmeasurementsofWangetal

5

showing

superconductingvortex-likeeffectsabovethetransitiontemperature.Thetransporttrendscontinueintothein-sulator,whereAndo

6

reportsthathigh-temperaturehall

effectconsistentwithacarrierdensityroughlypropor-tionaltodoping.Atthesametime,however,thed-wavegapinthequasiparticlespectrumgrowsmonotonicallyasthedopingdecreasesandsaturatesatavalueofabout0.3eV

7–10

.Thishasledtospeculationsthatthetunnel-

ingpseudogapistheenergytomakeapre-formedCooperpair,whichthencondensesintoasuperfluidatalowertemperature.Thereisnodirectevidencethatthed-wavenodalstructuresurvivesintotheinsulator,butthereiscircumstantialevidenceforthis,notablytheobservationinLa2xSrxCuO4byYoshidaetal

11

ofdispersingquasi-

particlebandsnearthed-wavenodethatbecomefainterasdopingisreducedbutdonotshiftorchangetheirvelocityscale.ThesebandsarealsodetachedfromthelowerHubbardband,andsimplymaterializeinmid-gapasthedopingisincreasedfromzero.

Allofthisbehaviorisconsistentwiththeideathat

thesuperconductivitypersistsdeepintothe“insulating”state,coexistswithantiferromagnetismthere,andfailstoconductonlybecauseitslong-rangeorderisdisrupted,presumablyonaccountofitslowsuperfluiddensity

12

.

Therearemanywaysthelattercouldoccur,includingimpuritylocalizationorcrystallizationoftheorderpa-rameter,forsuchagossamersuperconductorisphysically

equivalenttoadilutegasofbosonsandthushighlyun-stable.

Theideathatthe“insulator”mightactuallybeathin,

ghostlysuperconductorisimplicitinthemathematicsoftheAndersonresonatingvalencebond(RVB)

13

worked

outbyvariousauthorsinthelate1980s

14–16

andfur-

therextendedrecentlybyParamehanti,Randeria,and

Trivedi

17

.Unfortunately,thisideahasalwaysrunafoul

ofabasicpremiseofRVBtheorythatsuperconductiv-ityshouldbeauniversalaspectofquantumantiferro-magnetism.Thispremiseisbothconfusingandfun-damentallyincorrect,astheconventionalspindensitywavegroundstate,whichcontainsnosuperconductivity,isaperfectlygoodprototypeforaquantumantiferro-magnet.Therealissueisnotwhetherallantiferromag-netsaresuperconductorsbutwhethersomeofthemare-i.e.whetherthereexistsasecondkindofantiferro-magnetismdistinguishedfromthefirstbyatinyback-groundsuperfluiddensity.Onewouldalsoliketoknowwhichhamiltoniansfavorthissecondkindofstateoverthefirst.Itisnotjustthehamiltoniansthatstabilizean-tiferromagnetism,forthesesimplyemphasizetheaspectsofthevacuathatarethesameandde-emphasizetheas-pectsthataredifferent.IthasbeenknownsincetheearlyworkofHsu

18

,forexample,thatthesetwokinds

ofvacuumhavealmostidenticalvariationalenergiesinthecontextofthet-Jhamiltonian.Thisisconsistentwiththerecentnumerical-variationalstudiesofSorellaetal

19

,whoreportthatthet-Jmodelsuperconductsinare-

gionofitsparameterspace,eventhoughpreviousnumer-icalworkonthesamemodelreportedantiferromagneticstripeordering

20

.Beccaetal

21

havearguedthatthe

latterisanartifactlatticeanisotropy.Howevertheim-portantpointitthatsensitivitytoalgorithmicdetailandtheinherentdifficultyofdeterminingtheorder,reectedinlackofagreementamongdifferentgroups,demonstratethatmodelsofthiskindarehighlyconflictedandclosetoaquantumphasetransition

22

.Inotherwords,byex-

aggeratingthemagnetismthesemodelsconfusetheissueratherthanclarifyingit.Thereisnopersuasiveevidenceforsuperconductivityinthehubbardmodel

23–26

.

Thepurposeofthisletteristoproposeanewstrategyforresolvingthecupratedilemma.Ratherthanstruggletodiagonalizeaconflictedhamiltonian,weshallmodifytheequationsofmotiontostabilizethegossamersuper-conductor.Theeasiestwaytodothisisbypostulatingapowerfulattractiveforcebetweenelectrons,justasonewouldinanyothersuperconductor.Thissolutionisnotunique,butitisexperimentallyfalsifiablethroughtheex-citationspectrumofthestate,whichismodel-dependent.Insofarasthesepropertiesmatchexperiment,whichhasyettobeseen,itwouldsuggestthatcoulombinteractions

2

arenotsucienttoexplaincupratesuperconductivity.

Weconsideraplanarsquarelatticeofsitesjonwhich

electronsmaysit.Thesuperconductingvacuumwewishtostabilizeisj>=j>,where

j>=

NY

k

(uk+vkcy

k"cy

k#)j0>;(1)

withck=N1=2PN

jexp(ikrj)cjasusual,and

=

NY

j

z

(nj"+nj#)=2

0(10nj"nj#):(2)

TheBardeen-Cooper-Schrieffer(BCS)pairingampli-tudessatisfy



kk

kk



uk

vk



=Ek



uk

vk



(3)

andarenormalizedbyu

2k+v

2

k=1.Theyarerelatedto

theholedopingby

1

N

X

k

v

2

k=1

1

N

X

k

u

2k=

1

2

:(4)

Theimportantpositiveeigenvalue

Ek=

q

(k)2+2

k(5)

istheenergytomakeaquasiparticle(eitheranelectronorahole)whentheparameter0iszero.Theparameter

z0isafugacityrequiredtokeeptheelectrondensitythe

sameasonevaries0.Assumingthesuperconducting

orderparametertobed-wave,sothereisnoon-sitepair-ingamplitude,thechargestatesofasitearestatisticallyindependentandcharacterizedbyafugacityz.Thecon-ditionthatthetotalchargeonthesitebe1,whereisthedoping,is[2z+2(1)z

2

]=[1+2z+(1)z

2

]=1,

where1=(10)

2

,or

z=

p

1(12)

(1)(1+)

=(

1

1+

)z0:(6)Theparameterz0isthefactorbywhichzexceeds(1)=(1+),itsvaluefor0=0.

ThehamiltonianisconstructedusingtheBCSannihi-

lationoperators

bk"=ukck"+vkcy

k#bk#=ukck#vkcy

k":(7)

Solongas06=1,thepartialprojectorhasaninverse

1

=

NY

j

z

(nj"+nj#)=2

0(1+0nj"nj#);(8)

where0=0=(10).Thisenablesustoconstruct

themodifiedannihilationoperators

~b

k"=bk"1

=

1

p

N

NX

j

e

ikrj





z

1=2

0uk(1+0nj#)cj"+z

1=2

0vk(10nj")cy

j#



;(9)

andlikewisefor~b

k#,forwhich~b

kj>=0.Thusj>is

aneigenstateofthehamiltonian

H=

X

k

Ek~by

k

~b

k(10)

witheigenvalue0.However,thishamiltonianhasonlynon-negativeeigenvalues,sinceforanywavefunctionj>

<|Hj>=

X

k

Ek<~b

kj~b

k>0:(11)

Thusj>isagroundstateofH.However,itisalsothe

groundstatebyvirtueofadiabaticcontinuity.ThestateinquestionmaybecontinuouslydeformedintoaBCSstatebytaking0slowlytozero.Sinceitdoesnotcross

aphaseboundaryintheprocess,thegroundstateandlow-lyingexcitationsmusttrackinaone-to-oneway.

Letusnowconsiderthequasiparticleexcitationsof

thissuperconductor.Theoperators~b

knolongeran-

ticommuteproperlywiththeirhermitianadjointsandthuscannotbeusedtocreatequasiparticles.Thephys-icalmeaningofthisisthatthequasiparticlesinter-act.Insteadwewillusethevariationalwavefunctionsjk>=by

kj>borrowedfromtheRVBliterature.

Theexpectedenergyis

<k|Hjk>

<kjk>

=Ek

<j>

<kjk>'Ek:(12)

Thelaststeprequiresevaluatingtherelevantnorms,whichcanbedonebyhandonlyapproximately

17,18

.We

repeattheargumentshereforcompleteness.From

by

k"j>=

1

u2

k

cy

k"j>=

1

v2

k

ck#j>(13)

wefindthat

1

N

NX

k

u

2k

<jbk"

2by

k"j>

<j2

j>

=

<jcj"

2cy

j"j>

<j2

j>

3

=z0



1+(1)z

1+2z+(1)z2



=

1+

2

(14)

and

1

N

NX

k

v

2

k

<jbk"

2by

k"j>

<j2

j>

=

<jcy

j"

2cj"j>

<j2

j>

=

1

z0



z+z

2

1+2z+(1)z2



=

1

2

:(15)

Forj6=j0weassumethattheamplitudeforagivenconfigurationisweightedbythesquarerootofitscorre-spondingprobability,andthattheseweightsaddequally.Wethenhave

<jcj"

2cy

j0"j>

<j2

j>

<j>

<jcj"cy

j0"j>

'

4

12(zz0)

1+(1)z

1+2z+(1)z2

2

=1(16)

and

<jcy

j"

2cj0"j>

<j2

j>

<j>

<jcy

j"cj0"j>

'

4

12(

z

z0

)



1+z

1+2z+(1)z2

2

=1:(17)

Letusnowconsiderthelow-energyspectroscopicprop-

ertiesofthismodel.Reasoningasabove,wendthematrixelementforphotoemissiontobe

<k#jck"j>p

<k#jk#><j>

=gvk;(18)

where

g

2

'

20





1

0





1p

1(12)

12

]}

:(19)

Inversephotoemissionisthesameexceptwithuksub-

stitutedforvk.Thesuppressionofthephotoemissionintensityismatchedbyasimilarsuppressionofthesu-perfluidorderparameter:

<jcj"cj0#j>

<j>'g

2<jcj"cj0#j>

<j>

:(20)

Weneednotconsiderthecaseofj=j0forad-wavesuperconductor.Thusunderstrongprojectionnearhalf-fillingthismodelexhibitsthepseudogapphenomenon:

ThequasiparticleenergiesremainattheirunperturbedvaluesEkas0increasesfrom0to1,butthesuperfluiddensitydecreasesfrom1to2jj=(1+jj).

LetusnowconsidertheformationoftheMott-

Hubbardgap.Photoemissionofaquasiparticleaccountsforonlyasmallfractionofthesumrule

<jcy

kckj>

<j>'g

2

v

2

k+(1g

2

)

1

2

:(21)

Therestmustoccuratahigherenergyscale,thevalueofwhichmaybeestimatedbycomputingtheexpectedenergyofahole.Fromtheanticommutators

f~b

k";cj"g=

1

p

N

e

ikrj



z

1=2

00vkcj"cy

j#



(22)

f~b

k#;cj"g=

1

p

N

e

ikrj



z1=2

00ukcj"cj#

+z

1=2

0vk(10nj#)



(23)

weobtain

<jcy

kHckj>=z0



10

1

2

2

v

2

kEk

+

1

N

NX

q

Ek+q



z0

2

0Aqv

2

k+q+



2

0

0

Bqu

2k+q



;(24)

where

Aq=

NX

j



<jS0Sjj>

<j>

+

14

<jn0njj>

<j>

(1)

2

4



e

iqrj(25)

Bq=

NX

j

<jcy

0"cy

0#cj#cj"j>

<j>

e

iqrj:(26)

Proceedingsimilarlywiththestatecy

k"j>,wefindthat

theenergytoinjectanelectronistheexactparticle-holeconjugateofthisexpression,producedfromitbyinter-changingukandvk,negating,andsubstitutingof1nj

fornj.Thusathalf-fillingtheelectronspectralfunction

issymmetric.Thedensityandsuperfluidcorrelationsbecomesuppressedathalf-fillinginthe0!1limit,

whilethemagneticcorrelationsbecomeenhanced.Ap-proximatingthelatterbythecorrelationfunctionoftheNeelvacuum,weobtainat=0

4

lim

0!1

<jcy

kHckj>

<jcy

kckj>

=lim

0!1

<jckHcy

kj>

<jckcy

k>

'

1

10



1

N

NX

q

Eq+

1

2

Ek



:(27)

ThusthespectralfunctionconsistsofMott-Hubbard“lobes”athighenergieswithafaintbandofstatesatmid-gapassociatedwiththegossamerquasiparticles.Thechemicalpotentialdoesnotjumpinthismodelwhenistunedfromnegativetopositive,asitdoesinthehub-bardmodel

26

.

Letusnowconsiderantiferromagnetism.Athalf-lling

thelargestterminEq.(10)isanon-sitecoulombre-pulsionofmagnitude2

PN

kEk=N(10).Ifthisre-

pulsionismadeslightlylarger,thesystembecomesun-stabletospindensitywaveformationontopofthesuperconductivityatthenestingwavevectorofthed-wavenodes.Thecaseofhalf-llinghasbeenworkedoutindetailbyHsu

18

andweonlyquotetheresult

here.ThequasiparticledispersionrelationEq.(5)be-comesmodifiedtoEk=

p

(k)2+2

k+2

0,where

theenergygap0isrelatedtothesitemagnetization

<jS

z

jj>=<j>=mby

m'

m0

10(14m2

0)

m0=0=

2

N

NX

k

Ek:(28)

ItwasobservedbyHsu

18

thatstrongprojectionenhances

themagnetizationbyroughlyafactorof2overitsun-projectedvalue,enablingtheHeisenbergmodelvalueofm=0:307tobeachievedwitharathermodestvalueof0,about1/3thezoneaverageofEk.Incontrastto

thecaseworkedoutbyhim,however,thegapherehasphysicalmeaningandcanbedetectedinatunnelingorphotoemissionexperiment.Thissmallquasiparticlegapisakeycharacteristicofthegossamersuperconductor.

PhasefluctuationshavebeenleftoutofEq.(10)on

thegroundsthattheyareirrelevanttothefermispec-trum,whichischaracterizedbyanenergyscalemuchhigherthanthesuperconductingTc.However,theyare

essentialforaccountingforboththeUemuraplotandstrange-metaltransportaboveTc.Thetransportofadilutegasofbosonsformedwhentheorderparameterde-phaseswouldnotexhibitanytraditionalmetallicbehav-iorandindeedwouldtendto“shortout"conductionbythefermions.Lattice-mediatedcrystallizationofagos-samersuperconductorwouldalsoprovideareadyexpla-nationforwhythestaticstripes

27

areobservedat=1=8

issomecupratesandnotothers,whythestripecommen-surationwavevectorissopeculiar,whystripescanbede-stroyedbymoderatepressure

28

,andwhythematerials

insulatewhensubjectedtostrongmagneticelds

29

.

IwishtoexpressparticularthankstoZ.-X.Shenand

myotherNEDOcollaboratorsH.Eisaki,A.Fujimori,S.Maekawa,N.Nagaosa,N.P.Ong,Y.Tokura,andS.Uchidafortheircollegialityandinnumerablehelpfuldiscussions.ThisworkwassupportedbytheDepartmentofEnergyundercontractNo.DE-AC03-76SF00515.

R.B.Laughlin:http://large.stanford.edu

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