ParallelsBetweenQuantumAntiferromagnetismandtheStrongInteractions

R.B.Laughlin

DepartmentofPhysics,StanfordUniversity,Stanford,CA94305

(Dated:October1,1997)

Isuggestthatthegreatbodyofknowledgegainedoverthepast10yearsaboutsimplespin-1/2

quantumquantumantiferromagnetspointstoaconnectionbetweencupratesuperconductivityandthestronginteractions.Theunderlyingphysicalideaisthatthephasediagramofsuchamagnetconsistsofcompetingorderedphasesregulatedbyanearbyquantumcriticalpoint.Exactlyatthiscriticalpointthelow-lyingelementaryexcitationsofthemagnetaregaugefieldsandparticleswithfractionalquantumnumbersanalogoustothespinonandholonexcitationsfoundinspinchains.Anarbitrarilysmalldistanceaway,however,thesebindatlowenergyscalestomakethefamiliarcollectivemodesoftheorderedstatesintowhichonerenormalizes.Vestigesofthese“parts”ofthecollectivemodesmaybeseeninconventionalmaterialsandmodelsinhigh-energyspectroscopyandinconsistenciesinsumrulesexactlythewayquarksareseeninparticlephysics.[PublishedinProc.oftheInauguralConf.oftheAPCTP,ed.byY.M.Cho,J.B.Hong,andC.N.Yang(WorldSci.,Singapore,1998).]PACSnumbers:71.27.+a,74.20.+a,74.20.-z,11.15.Ha

cccc

cccc

cccc

cccc

66

66

66

66

??

??

??

??t

t









Quark

Antiquark

Gluon

Antiferromagnet

)

6





-



FIG.1:Thephysicalbehaviorofthestronginteractionsgen-eratedspontaneouslyinsimpleantiferromagnets.

ThepremiseofthisarticleisillustratedinFig.1.I

wishtoarguethattherearephysicallyidentifiableob-jectsinsimpleHeisenbergantiferromagnetswhichbehavelikeU(1)quarksandcouldconceivablybeaptanaloguesofthem.Thesearenottheelementaryexcitationsofthesysteminmostcasesbutratherobjectsoutofwhichtheelementaryexcitationsarebuilt.Itismycurrentbeliefthatthequark-likeobjectsandthegaugefieldsthroughwhichtheyinteractarethetrueelementaryexcitationsatsomenearbyquantumcriticalpoint,butIshallmostlysidestepthisissueandconcentrateonthephysicalmean-ingfulnessoftheparticlesinthecommonly-studiedcases.

Theantiferromagnetsinquestionaredescribedbythe

t-JHamiltonian

H=PGHoPG;(1)

where

PG=

Y

j



1cy

j"cy

j#cj#cj"



(2)

istheGutzwillerprojectorand

0

0.5

1

1.5

/b0/b

E/J

0123

/b0/b

E/t

FIG.2:Spinon(top)andholon(bottom)dispersionrelationsobtainedbyBares,Blatter,andOgata

1

fromtheBetheansatz

solutionofthesupersymmetricspinchain.

Ho=

X

<j,k>



t

X



cy

jck+

J

2

SjSk



;(3)

thesum<j,k>beingovernear-neighborpairsofalat-tice,witheachpaircountedtwicetomaintainhermitic-ity.Whenthedimensionofthelatticeis2orgreater,thephasediagramofthismodeliscomplexandbeyondourmeanstocomputereliably.Whenthedimensionofthelatticeis1,however,thereisanexactsolutionatthesupersymmetricpointJ=2t,thegroundstateofwhichisanondegeneratesinglet,i.e.hasnoorder,andtheelementaryexcitationsofwhicharespin-1/2,charge-0

2

0

0.5

1

1.5

2

2.5



E/J

qx

qy

ttttt

M



X



-3-2-1

0123



E/t

FIG.3:2-dspinon(top)andholon(bottom)dispersionrela-tionsgivenbyEqs.(4)and(5).Theinsetlabelsthespecialpointsinthebrillouinzone.particlesknownas“spinons”andspin-0,charge-1parti-clesknownas“holons”.Thesearethequark-likeobjectsoftheproblem.TheirdispersionrelationsareplottedinFig.2

1

.Theexistenceofspinonsandholonsisin-

timatelyconnectedwiththelackoforderinthegroundstate,andisthuscommonin1dimension,wherecontin-uoussymmetrybreakingisimpossible,butuncommoninhigherdimension,whereorderseemstooccuralmostalways.Butifthehigher-dimensionalgroundstateisforcedtobedisorderedbymeansofavariationalansatz,whichisequivalenttoaddingasmalllong-rangeinter-actiontotheHamiltoniantodestabilizetheorder,thenspinonsandholonsmakessenseandweobtainin2-dthedispersionrelations

2

E

spinon

q=1:6J

q

cos2(q

x)+cos2(q

y)(4)

E

holon

q=2t

q

cos2(q

x)+cos2(q

y)(5)

plottedinFig.2.ThesearealsoimplicitintheU(1)gaugetheorydescriptionsofthet-Jmodelbasedonthecommensuratefluxsaddlepoint

3,4

.Iwishnowtoestab-

lishthattheseparticleshavephysicalmeaningatinter-mediateenergyscalesevenwhenthesystemisallowed

0

0.05

0.1

01234

E/J

E/t

6

.0.2.4.6.8

1.0

.0.1.2.3.4.5

E/J

E/t

3

3

3

3

3

33

+2

2

2

2

2

22

3

3

3

3

+

+

+

+

2

2

2

2

FIG.4:Top:Opticalconductivitycomputedbyexactdiago-naflizationbyMoreoandDagotto

5

ona44clusterforthe

caseof=1=16andJ/t=0:4.Thearrowat!=0indi-catestheDrudeoscillator.Bottom:Totaloscillatorstrength<T>=NtdefinedbyEq.(9)anditsDrudecontributioncomputedbyexactdiagonalizationona44cluster.The

symbols2,+,and3correspondtoJ/t=0.1,0.4,and1.0,

respectively.ThesolidcurveisaplotofEq.(11).Thedottedcurveisaguidetotheeye.toorder,givingrisetoforcesthatbindthematlowen-ergyscalesintothewell-knownexcitationsoftheorderedphases.

InFig.4Ishowtheopticalconductivity

xx(!)=

1

N



!

X



j<jjxj0>j

2

(¯hωE+E0)(6)

computedbyDagottoandMoreo

5

forasingleholeina

44cluster,whichisrepresentativeofsuchcalculations.Herej>indicatesanexacteigenstateofenergyEandjxistheelectriccurrentoperator

jx=PGi

t

h

X

j

X





cy

jckcy

kcj



PG;(7)

wherekdenotesthenearneighborofjinthex-direction.

Thiscalculationprovidesevidencefortheexistenceoftheholonandmeasuresthesizeofitsmass.Thef-sumrule

Z1

0

xx(!)dω=



4

<0jTj0>

N

(8)

isplottedversusdopinginthelowerpartofthefigure,asisits“Drude”contribution.Thewidthandshapeofthe

3

0.00.1

-6-4-20246

E/J

E/t

M

0.00.1

-6-4-20246

E/J

E/t

0.00.1

-6-4-20246

E/J

E/t

FIG.5:Aq(E)=ImGq(E)asdefinedbyEq.(12),com-putedbyexactdiagonalizationona44clusterbyDagotto

6

forthecaseofJ/t=0.latercannotbecomputedaccuratelybutitsintegratedareacan.Bothsumrulesarestraightlinesatlowdop-ing,theslopeofwhichdoesnotdependonJ.Thisis

thebehaviorofadopedsemiconductor.Fromtheusualexpression

Z1

0

xx(!)dω=



2

h

2

m

n;(9)

wefindamassof

m=0:77

h

2

tb2;(10)

wherebisthebondlength.Thiscomparesfavorablywiththe1=

p

2inthesesameunitsobtainedfromthecurvature

ofEq.(5)nearitsminimumandthe0.54obtainedin1-dfromtheBethesolution.Thefactthatthe“Drude”weightisalwaysabouthalfthetotalindicatesthatthisparticleisthecarrier.Thefullsumrule

0.00.1

-6-4-20246

E/J

E/t

M

0.00.1

-6-4-20246

E/J

E/t

0.00.1

-6-4-20246

E/J

E/t

FIG.6:SameasFig.5exceptwithJ/t=0:2

<T>=2:6Ntδ(11)

alsoagreeswithEq.(5)inequalingthep

8tperhole

associatedwiththeholonbandminimum.Thisnumberhasnoconnectiontothemassingeneral,andisthusanadditionalconstraintonthebandstructure.

Furtherevidencefortheexistenceoftheholonmaybe

foundintheelectronpropagatorinthelimitofsmallJ/t.FollowingthenotationofEq.(7),wedefinetheelectronpropagatorbyGq(E)=

X



j

<jcy

qj0>j

2

EE+E0+iη

+

j<jcqj0>j

2

E+EE0iη



(12)

where

cq=

1

p

N

NX

j

exp(iqrj)cj:(13)

InFigs.5and6Ishowtheimaginarypartofthisfunc-tionathalf-fillingcalculatedbytheexactdiagonalization

4

0.00.20.40.60.8

-3-2-10123

D(E)

E/t

FIG.7:HolondensityofstatesdefinedbyEq.(14).

methodforJ/t=0:0and0.2byDagotto

6

.Ineithercase

thespectrumisabroadcontinuumabout6twidewithapronounceddipinthecenterandaweightthatmovesfromlowtohighenergyasthemomentumisadvancedfromtoM.IntheJ!0limitthebroadcontin-

uummaybeascribedtothedecayoftheinjectedholeintospinon-holonpairinthelimitthatthespinonisveryheavy,forthenthespectrumshouldbetheholondensityofstates

D(E)=

b

2

22

X



Z/b

/b

Z/b

/b

(EE

holon

q)dqxdqy

(14)

weightedbyadecaymatrixelement.ThedensityofstatescomputedfromEq.(5)isplottedinFig.7.Anyreasonablemodelwillgivethemotionoftheweightsince

<0jcy

qcqj0>=

1

2

(15)

<0jcy

qHcqj0>=<0|Hj0>

2t



<~S

1~S

2>

3



14

]{

cos(qx)+cos(qy)



(16)

athalf-filling,where1and2denotenear-neighborsites.

TheJ/t=0:2curvesalsohaveapeakatlowbinding

energywhichisthequasiparticleofthemagneticinsu-lator.InFig.8Ishowthedispersionrelationofthisquasiparticlefoundnumericallybyanumberofauthors

7

.

IthasadeepminimumatandanoverallbandwidthW,thedifferencebetweenthemaximumandminimumofthedispersionrelation,thatdoesnotdependont.Thiswidth,measuredinmultiplesoft,isplottedagainstJ/tinFig.8

8

.Fromtheslopeofthelineoneobtains

W=2:2J;(17)

or1:6

p

2J,whichisthespinonbandwidthgivenbyEq.

(4).Theprefactor1.6inEq.(4)hasthephysicalsig-nificanceofamagneticstiffness.Itcausesthespinon

-2.5

-2

MX

E/t

W/t

22

2

2

2

2

2

222

2

2

2

2

2

2222

2

2

2

2

2

2

2

22222222222222

2

2

2

2

2

222

0

0.5

1

1.5

2

00.20.40.60.81

J/t

W/t

3

3

3

3

3

++

+

+

+

+

+

2

2

2

2

2

2

FIG.8:Top:QuasiparticledispersionrelationcalculatedbyLiuandManousakis

7

usingspin-waveperturbationtheoryfor

thecaseofJ/t=0:2.Bottom:QuasiparticlebandwidthW/tcalculatedbyPoilblanc

8

usingexactdiagonalizationon

clustersofvarioussizes.ThedottedlineisaplotofEq.(17).velocityattobethespin-wavevelocityoftheorderedantiferromagnet

9

vs=1:6

JB

h

:(18)

Thequasiparticlepeakisaccompaniedbyscattering

resonances.ThesecannotbeseeninFig.4becausethesampleistoosmall,buttheymaybeseenclearlyinFig.9,whichisthespectralfunctionatforJ/t=0:2,calculatedusingspinwaveperturbationtheory

7

.The

quasiparticlepeakandthefirsttworesonancesarela-beledbyromannumerals.TheirenergiesareplottedasafunctionofJ/tinFig.9.Thelinesthroughthedatapointsrepresenttheformula

En/t=3:28+(J/t)

2=3



"2

:03;n=15:46;n=27:81;n=3

#

:(19)

Theseenergiesareexactlythespectrumexpectedalightparticleinorbitaboutaheavyone,providedtheattrac-tiveforcebetweenthetwoisastring,i.e.V(r)jrj.Moreprecisely,theHamiltonian

H=

h

2

2m

r

2

+2:2Jj

r

b

j3:28t;(20)

wheremisthemassderivedbytheconductivitysumruleandgivenexplicitlybyEq.(10),hasenergyeigenvalues

5

0.00.10.20.3

-4-3-2-101234

J/t

E/t

6

I

II

III

I

II

III

-4-2

01

0.00.10.20.30.40.5

E/t

J/t

33333333333

+

+

++

+

+++

+

+

+

2









FIG.9:Top:SpectraldensityatcalculatedbyspinwaveperturbationtheorybyLiuandManousakis

7

forthecaseof

J/t=0:1inthelimitoflargesamplesize.Bottom:Energiesofquasiparticle(I)andfirsttwostringresonances(IIandIII)asafunctionofJ/t.ThedashedlinesareplotsofEq.(19).givenbyEq.(19)exceptforsubstitution(2.63,5.54,7.81)!(2.03,5.46,7.81).

Thesefactshavethefollowingphysicalinterpretation.

Thequasiparticleisaboundstateofaspinonandtheholonanalogoustothehydrogenatom.Itsbandstruc-turetracksthatofthespinonbecausethespinonis“heav-ier”thantheholoninthesenseofhavinganarrowerband.Theopticalsumrule,bycontrast,issensitivetothelightparticle,andthusmeasurestheholonproperties.Thesamethingistrueinhydrogen,wheretheacceler-ationmassisdominatedbytheprotonbuttheopticalpropertiesaredominatedbytheelectron.Thepotentialbindingtheseparticlestogetherisastringatlowdoping,whichmeansthattheycanneverseparateanddonotexistasseparateentitiesinthislimit,butalreadyatadopingofoneholeina44lattice,or=1=16,some-thingoccurstoallowthestringtobreakandtheholontoionizeofftobecomeafreecarrier.

Thet-JHamiltonianisformallyequivalenttothe

Lagrangian

3,4

L=

NX

j

X



fy

j



ih∂t+j



fj+by

j



ih∂t+j



bj

j



+

X

<j,k>





J

4

jjkj

2

+

J

2



jk

X



fy

jfk

2t

J

by

jbk





t

2

J

by

jby

kbkbj



;(21)

wherefjandbjarefictitiousfermionandbosonopera-torsonsitejintermsofwhichtheelectroniswritten

cj=fjby

j;(22)

jisaLagrangemultiplierwhichwhenintegratedoutforcestheconstraint

X



fy

jfj+by

jbj=1;(23)

andjkisaHubbard-Stratonovichvariable.Thisisa

U(1)gaugetheorytotheextentthatjkmaybeap-proximatedashavingafixedlength,forthenthephasefunctionsasavectorpotentialalongthebond<j,k>,thescalarpotentialonsitejbeingj.ThisturnsouttobeabadapproximationforthisparticularLagrangian,butwecanimagineadiabaticallytransformingitintooneforwhichjjkjisfixedandforwhichasmallMaxwellterm

LMax=

1

g



J

X

<j,k,`m>

jkk``mmj

+

1

J

X

<j,k>

jh@tjk+(jk)jkj

2



(24)

isaddedasaregulator.Thentheclassicalsaddlepointhasmagneticfluxperplaquette,thef-andb-particlesbecomefreeparticleswiththerelationsofEqs.(4)and(5),althoughwithdifferentcoefficients,andweob-tainconventionallatticeQEDwithdoubledfermions.Thelimitrelevanttothelow-dopingnumericalworkisg!1,whichisstronglyconfining.Thusthestringforcesmaybeassociatedwithconfinementinstrongly-coupledQED,theantiferromagneticorderinlimitmaybeassociatedwiththechiralsymmetrybreakingknowntoaccompanyconfinementinthisproblem,andthespinwave,whichisboththeGoldstoneofthebrokensymme-tryandaboundpairofspinons,maybeassociatedwiththepion.

Thecorrectappearanceofastringforceinthean-

tiferromagneticallyorderedphasesuggeststhattheun-bindingofthequasiparticleseeninFig.4mightindi-cateafirst-ordertransitiontoasuperconductingphasecorrespondingtothecoulombicphaseofthegaugethe-ory.Themagneticorderisknowntodisappearatabout=0:05,whichisconsistentwithdeconfinementby=1=16.Howeveritisonlyasuggestion,fortheaboveLagrangianislessaccurateandmoredifficulttosolvethanthespinHamiltonianfromwhichitwasderived,

6

andallthemajororderingquestionsfortheformerarestillunresolved.Itshouldbeviewednotasacompu-tationaltoolbutasmeansforunderstandinghowthephysicsofthestronginteractionsmightmaterializeinaquantumantiferromagnetwithoutbeingpostulated.

Acknowledgments

IwishtoexpressspecialthankstoE.Dagottoforpro-

vidingmehisunpublishedJ!0spectralfunctionsand

toA.M.Tikofskyfornumeroushelpfuldiscussions.ThisworkwassupportedprimarilybytheNSFundergrantNo.DMR-9421888.AdditionalsupportwasprovidedbytheCenterforMaterialsResearchatStanfordUniversityandbyNASACollaborativeAgreementNCC2-794.

R.B.Laughlin:http://large.stanford.edu1

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2

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3

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4

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5

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6

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7

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8

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9

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