Anomalous thermal conductivity of frustrated Heisenberg spin-chains and ladders

arXiv:cond-mat/0201300v3 [cond-mat.str-el] 9 Sep 2002AnomalousthermalconductivityoffrustratedHeisenbergspin-chainsandladders

J.V.AlvarezandClaudiusGros

Fakult¨at7,TheoretischePhysik,UniversityoftheSaarland,66041Saarbr¨ucken,Germany.

Westudythethermaltransportpropertiesofseveralquan-

tumspinchainsandladders.Wefindindicationsforadiverg-ingthermalconductivityatfinitetemperaturesforthemodelsexamined.Thetemperatureatwhichthenon-divergingpref-

actorκ(th)(T)peaksisingeneralsubstantiallylowerthanthetemperatureatwhichthecorrespondingspecificheatcV(T)ismaximal.Weshowthatthisresultofthemicroscopicap-

proachleadstoasubstantialreductionforestimatesofthemagneticmean-freepathλextractedbyanalyzingrecentex-periments,ascomparedtosimilaranalysesbyphenomenolog-icaltheories.PACSnumbers:

Introduction-Thenatureofthermaltransportin

systemswithreduceddimensionsisalongstandingprob-

lemandhasbeenstudiedintensivelyinclassical1sys-

temseitherbydirectnumericalout-of-equilibriumsimu-

lations2orbyinvestigationofsoliton-solitonscattering

processes3.Therehasbeen,ontheotherhand,con-

siderablylessprogressforquantumsystems.Huber,in

oneofthefirstsworksonthesubject4,evaluatedthe

thermalconductivityκ(T)fortheHeisenberg-chainwith

anequation-of-motionapproximationandfoundafinite

κ(T),aresultwhichis,bynow,knowntobewrong.It

hasbeenshownrecently5,thattheenergy-currentop-

eratorcommuteswiththehamiltonianforthespin-1/2

Heisenbergchain.Thethermalconductivityisconse-

quentlyinfiniteforthismodel.Theintriguingques-

tion,“underwhichcircumstancesdoesaninteracting

quantum-systemshowaninfinitethermalconductivity”,

isto-datecompletelyopen,theanalogousquestionfor

classicalsystemsbeingintensivelystudied1.

Asecondmotivationtostudyκ(T)forquantumspin-

systemscomesfromexperiment.Ananomalouslarge

magneticcontributiontoκhasbeenobserved6forthe

hole-dopedspin-laddersystemSr14−xCaxCu24O41.For

Ca9La5Cu24O41,whichhasnoholesintheladders,an

evenlargerthermalconductivityhasbeenmeasured7

raisingthepossibilityofballisticmagnetictransportlim-

itedonlybyresidualspin-phononandimpurityscatter-

ing.Thereis,however,up-to-datenomicroscopiccalcu-

lationforthethermalconductivityofspin-ladders.

InthisLetterwepresentafinitesizeanalysisofthe

thermalconductanceinHeisenbergJ1−J2chainsand

ladderssuggestingballisticthermaltransportinthese

twofamiliesofspinmodels.Followingthisresultwe

proposeapossiblescenariothatwouldaccountanum-

berofanomaliesfoundinthethermalconductivityin

spinsystems.Weapplytheseideastothecaseof

Ca9La5Cu24O41,obtaininggoodagreementwithrecentexperimentalmeasurements.

Models-Weconsiderquasione-dimensionalsystems

forwhichthehamiltoniantakestheformH=󰀋Lx=1Hx,

whereListhenumberofunitsalongthechain.Wewill

considertwomodels,theisotropicHeisenbergchainwith

dimerizednearestandhomogeneousnext-nearestneigh-

borexchangecouplings,

H(ch)x=J󰀆

(1+δ(−1)x)Sx·Sx+1+αSx·Sx+2󰀈

,

andthetwo-legHeisenberg-ladder:

H(lad)x=J󰀆(S1,x·S1,x+1+S2,x·S2,x+1)+J⊥S1,x·S2,x.

H(ch)hasbeenproposedtomodelthemagneticprop-

ertiesofthespin-PeierlscompoundCuGeO3.Thenon-

frustrateddimerizedHeisenbergchain(α=0)models

themagneticbehavior8of(VO)2P2O7.Ca9La5Cu24O41containsdopedspin-chainsandundopedspin-ladders7,

describedbyH(lad).

Method-Ourgoalistomakeconnectionbetweenthe

microscopictransport-propertiesofthelowdimensional

modelspresentedaboveandtheexperimentalmeasure-

mentsofthermalconductivitymentionedintheintro-

duction.

Thethermalconductivityisdefinedastheresponse

oftheenergy-currentdensityjExtoathermalgradient

jEx=−κ∇Tandithasunitsof[κ]=W

cjEx,

wherecisthelatticeconstantalongthechain-direction.

Intheabsenceofappliedmagneticfieldsspin-inversion

symmetryholdsandtheKuboformulaforκ=

limω→0κ(ω)reducesto4

κ=limω→0β2α=0=δandaspin-anisotropy).Thetotalthermal

currentcommuteswiththehamiltonianinthiscase5and

thecurrent-currentcorrelationfunctionisthereforetime-

independent.Eq.(1)reducesthentothestaticexpecta-

tionvalueκ(ω=0)=β2󰀎(JE)2󰀏/(Ls),aquantitywhich

canbeevaluatedbyBethe-Ansatz10.

Ingeneralwehave[JE,H]=0,butthethermaltrans-

portwillbeballisticif

κ(th)=β2

vs(3)

toholdforthethermalconductivityκ.

LetusdiscusstherelationofEq.(3)totheusualphe-

nomenologicalformula13(hereinonedimension)

κ(ph)=cVvsλ,(4)

wherecVisthespecificheat.Whenappliedinorder

toanalyzeexperimentaldata,themicroscopicequation

(3)willingeneralyielddifferentvaluesforthemagnetic

mean-freepathλ.Ontheotherhand,wemightexpect

Eq.(4)toholdatlowtemperaturefortheHeisenberg-

chaininthegaplessphase,whentheLuttinger-liquid

quasiparticlesarewelldefinedandaBoltzmannapproach

isjustified.Consequentlyweexpectκ(th)λ/vs≡cVvsλ

inthiscase,i.e.weexpectκ(th)/cV≡v2sinthelimit

T→0.Allthreequantitiesinthisequation(κ(th),cVandvs)canbecomputedbyBethe-AnsatzfortheXXZ-

chain.Onefindsthatthisequationisexact10inthelimit

T→0.

Numerics-Thecomputationofκ(th)(T)demandsthe

wholespectrumofthehamiltonian,restrictingthemax-

imumlatticesizeandacarefulfinite-sizeanalysisisre-

quired.Ingeneralonefinds,e.g.fortheXXZ-modelfor

whichtheexactBethe-Ansatzresultisknown10,thatthe

κ(th)inchainswithodd(even)numberofsiteschainsis

aupper(lower)boundoftheexactresult.Thevalueof

κ(th)(T)decreases(increases)forchainswithodd(even)

numberofsiteswhenthelatticeisenlarged.Thisobser-

vationhasledustoconsidertheaverage

κ(th)(Leff)=L0κ(th)(L0)+L1κ(th)(L1)