Exact edge singularities and dynamical correlations in spin-12 chains

Exact formulas for the singularities of the dynamical structure factor, S^{zz}(q,omega), of the S=1/2 xxz spin chain at all q and any anisotropy and magnetic field in the critical regime are derived, expressing the exponents in terms of the phase shifts wh

Exactedgesingularitiesanddynamicalcorrelationsinspin-1/2chains

RodrigoG.Pereira,1StevenR.White,2andIanA eck1

1

DepartmentofPhysicsandAstronomy,UniversityofBritishColumbia,Vancouver,BC,CanadaV6T1Z1

2

DepartmentofPhysicsandAstronomy,UniversityofCalifornia,IrvineCA92697,USA

(Dated:February1,2008)

Exactformulasforthesingularitiesofthedynamicalstructurefactor,Szz(q,ω),oftheS=1/2xxzspinchainatallqandanyanisotropyandmagnetic eldinthecriticalregimearederived,expressingtheexponentsintermsofthephaseshiftswhichareknownexactlyfromtheBetheansatzsolution.

zz

Wealsostudythelongtimeasymptoticsoftheself-correlationfunction 0|Sj(t)Sj(0)|0 .Utilizingtheseresultstosupplementveryaccuratetime-dependentDensityMatrixRenormalizationGroup(DMRG)forshorttomoderatetimes,wecalculateSzz(q,ω)toveryhighprecision.

08

02PACSnumbers:75.10.Pq,71.10.Pm

anJ The“xxz”S=1/2spinchain,withHamiltonian91 ]le LH=J

[SxjSx

j+1+Syj

Syj+1+ SzjSzj+1 hSzj],(1)

j=1

-isoneofthemoststudiedmodelsofstronglycorrelated

rtsystems.ItisequivalentbyaJordan-Wignertransforma-s.tiontoamodelofinteractingspinlessfermions,withthe

tacorrespondingFermimomentumkF=π(1/2+ 0|Sz

m[1].Themodelwith =1describesHeisenbergantifer-j|0 )-romagnets.Theregime0< <1isalsoofexperimentaldinterest;forexample,themodelwith =1/2canbere-nalizedinS=1/2spinladdersnearthecritical eld[2].coInopticallattices,itshouldbeevenpossibletotunethe[anisotropy andexploretheentirecriticalregime 2WhilesomeaspectsofthemodelhavebeensolvedforexactlybyBetheansatzithasbeenverydi cultto0vobtaincorrelationfunctionsthatway.Fieldtheory(FT)6methodsgivethelowenergybehavioratwave-vectors9near0and2kFFromtheexperimentalviewpoint0.arelevantquantityisthedynamicalstructurefactor907Szz

(q,ω)=

0: Le

iqj

j=1

+∞

dteiωt 0|Szj(t)Sz

0(0)|0 .(2)

∞

viThisistheFouriertransformofthedensitycorrelation

Xfunctioninthefermionicmodel.For =1andh=0,rtheexacttwo-spinoncontributiontoSzz(q,ω)wasob-atainedfromtheBetheansatz[6],partiallyagreeingwiththeM¨ullerconjecture[7].Morerecentlyanumberofnewmethodshaveemergedwhichnowmakethisprob-lemmuchmoreaccessible.Theseincludetime-dependentDMRG[8,9,calculationofformfactorsfromBetheansatz[11,12]andnew eldtheoryapproacheswhichgobeyondtheLuttingermodel[13,14].TheresultspointtoaverynontriviallineshapeatzerotemperatureforSzz(q,ω)ofthexxzmodelandofone-dimensionalmodelsingeneralIntheweakcouplinglimit 1andforsmallq,thesingularitiesatthethresholdsofthetwo-particlecontinuumhavebeenexplainedbyanalogywiththex-rayedgesingularityinmetalsInthisLetterwecombinethemethodsofRef.[13]withtheBetheansatztoinvestigatethesingularityexpo-nentsofSzz(q,ω)forthexxzmodelfor niteinteractionstrength andgeneralmomentumq.Inaddition,wede-terminetheexponentsofthelong-timeasymptoticsofthespinself-correlationfunction,whichisnotdominatedbylowenergyexcitations.WecheckourpredictionsagainsthighaccuracynumericalresultscalculatedbyDMRG.Inthenon-interacting, =0case,onlyexcitedstateswithasingleparticle-holepaircontributetoSzz(q,ω).Allthespectralweightiscon nedbetweenthelowerandupperthresholdsωL,U(q)ofthetwo-particlecontinuum.ThechoicesofmomentacorrespondingtothethresholdsdependonbothkFandq.Forzero eld,kF=π/2,ωL(q)foranyq>0isde nedbytheexcitationwithaholeatk1=π/2 qandaparticlerightattheFermisurface(oraholeattheFermisurfaceandaparticleatk2=π/2+q),whileωUisde nedbythesymmetricexcitationwithaholeatk1=π/2 q/2andaparticleatk2=π/2+q/2.For nite eldandq<|2kFωL,U(q)arede nedbyexcitationswitheitherahole πat|,kFandaparticleatkF+qoraholeatkFparticleatkF.Forh=0andq>|2kF qandaathird“threshold”betweenωLandω Uπwhere|,thereSzzis(evenq,ω)hasastepdiscontinuity(seeFor =0,Szz(q,ω)exhibitsatailassociatedwithmultipleparticle-holeexcitations[14].However,thethresholdsofthetwo-particlecontinuumareexpectedtoremainaspointsatwhichpower-lawsingular-itiesdevelopInordertodescribetheinteractionofthehighenergyparticleand/orholewiththeFermisurfacemodes,weintegrateoutallFouriermodesofthefermion eldψ(x)exceptthosenear±kFandnearthemomentumofthehole,k1,orparticle,k2,writingψ(x)～eikFxψR+e ikFxψL+eik1xd1+eik2xd2.

(3)

Linearizingthedispersionrelationabout±kFweobtainrelativisticfermion eldswhichwebosonizeintheusualwayWealsoexpandthedispersionofthed1,2parti-clesaroundk=k1,2uptoquadraticterms.Thisyields

Exact formulas for the singularities of the dynamical structure factor, S^{zz}(q,omega), of the S=1/2 xxz spin chain at all q and any anisotropy and magnetic field in the critical regime are derived, expressing the exponents in terms of the phase shifts wh

2

thee ectiveHamiltoniandensity

2 x

H=dαεα iuα x

α=1,2

dαimp

=

B

dλ

ραimp(λ)

2

,(8)

∞

22

( x L)+( x R)+V12d 1d1d2d22

1α

+(κα x R+κ x L)dd.(4)

whereBistheFermiboundaryandραimp(λ)isthesolu-tiontotheintegralequation

+B′

dλΦα(λ)α

ρimp(λ) =

2πKαThisHamiltoniandescribes RLαα=1,2

aLuttingerliquidcoupledtooneortwomobileimpurities[15,16].InthederivationofEq.(4)fromEq.(1),wedroptermsoftheform

(d αdα)2

becauseweonlyconsiderprocessesinvolvingasingled1and/orasingled2particle.Here R,Laretherightandleftcomponentsoftherescaledbosonic eld.

Thelongwavelength uctuationpartofSz

jisgivenSz

j～ by

1 2/arccos andK=[2 2arccos( )/π)] 1(wesetJ=1).To rstorderin ,thecouplingconstantsdescribingthescatteringbetweenthedparticlesandthebosonsareκαR,L=2 [1 cos(kF kα)].Thedirectd1-d2interactionV12isalsooforder .TheexactvaluesofκR,Lplayacrucialroleinthesingularitiesandwillbedeterminedbelow.

Wemayeliminatetheinteractionbetweenthedparti-clesandthebosonicmodesbyaunitaryU=exp

transformation

i α

dx2πK(γαR R γα

L L)d αdα ,(5)withparametersγα

R,L=καR,L/(v uα).Intheresulting

HamiltonianH

=U HU, R,Larefreeuptoirrelevantinteractionterms[15].Asinthex-rayedgeproblem,γα

mayberelatedtothephaseshiftsattheFermipointsdueR,Ltothecreationofthehighenergydαparticle.

Fortunately,wehaveaccesstothehighenergyspec-trumofthexxzmodelbymeansoftheBetheAnsatz.FollowingtheformalismofRef.[16],wecalculatethe -nitesizespectrumfromtheBetheansatzequationswithanimpuritytermcorrespondingtoremoving(adding)aparticlewithdressedmomentumk1=k(λ1)(k2=k(λ2)),whereλ1,2arethecorrespondingrapidities.ThetermofO(1)yieldsεα= (kα),thedressedenergyoftheparticle.Forzero eld,wehavetheexplicitformula (k)= vcosk.TheexcitationspectrumforasingleimpuritytoO(1/L)reads

E=

2πv

4K

+n++ N nαimpn ],

2+K D dα

imp

2

(6)

withaconventionalnotationfor N,Dandn±[4].Thephaseshiftsnαimpanddα

imparegivenby

+B

nα

imp=dλραimp(λ),

(7) B

B

dλ(1 cosq)

π(v u1)

≈

2

Exact formulas for the singularities of the dynamical structure factor, S^{zz}(q,omega), of the S=1/2 xxz spin chain at all q and any anisotropy and magnetic field in the critical regime are derived, expressing the exponents in terms of the phase shifts wh

cancellationoftheqdependenceofκ1case.Momentum-independentexponentsRandv u1inthelatterhavealsobeen

derived

fortheCalogero-Sutherlandmodel[18].Wenowconsiderathresholdde nedbyhigh-energyparticleandholeatk1,2=π/2 q/2.Therelevantcor-relationfunctionisthepropagatorofthetransformedd 2d1.Forsimplicity,herewefocusonthezero eldcase,inwhichε2= ε1=vsin(q/2),u2=u1and m2=m1=[vsin(q/2)] 1.Particle-holesymmetry

thenimpliesthatγ1R,L=γ2

R,Landd 2d1isinvariantundertheunitarytransformationofEq.(5).Inthenoninteractingcase,thereisasquarerootsingularityattheupperthresholdduetothejointdensityofstates:Szz(q,ω)∝

divergenceoftheωU(q) ω.Thisbehav-iorcontradictstheM¨ulleransatz[7],butisconsistentwiththeanalytictwo-spinonresultfor =1[6].Unliketheoriginalexcitonproblem,aboundstateonlyappearsforV12<0( <0)[21],becausetheparticleandholehaveanegativee ectivemass.For =0,theupperedgecuspshouldintersectahigh-frequencytaildominatedbyfour-spinonexcitationsasproposedin[22].Thispicture

mustbemodi edforh=0,sincethenγ1thebosonicexponentials.R,L=Theγ2

R,Landoneneedstoincludeupperedgesingularitythenbecomes -andq-dependent.Thegeneral nite eldcase,includingthemiddlesingularity[7]forq>|2kFWecanapply theπ|,HamiltonianwillbediscussedofEq.elsewhere.

(4)tostudythe

self-correlationfunctionG(t)≡ 0|Szj(t)Sz

j(0)|0 .Eveninthenoninteractingcase,thelongtimeasymptoticsisahighenergyproperty,sinceitisdominatedbyasaddlepointcontributionwithaholeatthebottomandapar-ticleatthetopoftheband[23].Inthiscase,k1=0and

k2=πandd1,2impvanishbysymmetry(γαR=γα

zero eld,butthemethodcanL).Herewerestricttobeeasilygeneralized.Forh=0and ≥0,G(t)takestheform

G(t)～Be iWt

B3

1

tη2

+t2

,

(15)

whereW= (0)=v.Thelasttwotermsarethestandardlow-energycontributions,withσ=2K.TheamplitudesB3andB4areknown[24].The rsttermisthecontributionfromtheholeatthebottomofthebandandtheparticleatkF=π/2,withexponent

η=(1+K)/2+(1 n1imp)2

/2K=K+1/2.

(16)

3

Thetermoscillatingat2Wcomesfromaholeatk=0andaparticleatk=π.For =0,wehaveη2=1.Theexponentη2isconnectedwiththesin-gularity

the attheupperthresholdofSzz(q,ω)byG(t)～dωeiωtdqSzz(q,ω)forq≈πandω≈ωU(π)=2v.DuetodiscontinuityoftheexponentatωU,η2jumpsfromη2=1toη2=2foranynonzero .Thisbehavior

shouldbeobservedfort 1/(m1V12

2

)～1/ 2.Asare-sult,theasymptoticsofG(t)isgovernedbytheexponentη<3/2for0< <1.For <0,wemustaddtoEq.(15)thecontributionfromtheboundstate.

WecanalsostudySzz(q,ω)withtime-dependentDMRG(tDMRG)[8,9].ThetDMRGmethodsdi-rectlyproduceSzz(x,t)anditsspatialFouriertransformSzz(q,t)forshorttomoderatetimes.Thisinformationnicelycomplementstheasymptoticinformationavailableanalytically.TheDMRGcalculationbeginswiththestandard nitesystemcalculationofthegroundstateφ(t=0)ona nitelatticeoftypicallengthL=200-400,whereafewhundredstatesarekeptforatruncationer-rorlessthan10 10.Oneofthesitesatthecenterofthe

latticeisselectedastheorigin,andtheoperatorSz

0isappliedtothegroundstatetoobtainastateψ(t=0).Subsequently,thetimeevolutionoperatorforatimestepτ,exp(i(H E0)τ)whereE0isthegroundstateenergy,isappliedviaafourthorderTrotterdecomposition[10]toevolvebothφ(t)andψ(t).AteachDMRGstepcenteredonsitejweobtainadatapointfortheGreen’sfunction

G(t,j)byevaluating φ(t)|Sz

tionprogresses,thetruncationj|ψ(errort) .Asaccumulates.thetimeevolu-Theintegratedtruncationerrorprovidesausefulestimateoftheerror,andsolongertimesrequiresmallertrunca-tionerrorsateachstep,attainedbyincreasingthenum-berofstateskeptm.Thetruncationerrorgrowswithtimefor xedm,andislargestnearthecenterwherethespinoperatorwasapplied.Wespecifythedesiredtruncationerrorateachstepandchoosemtoachieveit,withinaspeci edrange.Typicallyforlatertimeswehavem≈1000.Finitesizee ectsaresmallfortimeslessthan(L/2)/v.WeareabletoobtainveryaccurateresultsforG(t,j),witherrorsbetween10 4and10 5,fortimesuptoJt～30-60.

ForJt>10 20,we ndthebehaviorofSzz(q,t)andG(t)iswellapproximatedbyasymptoticexpressions,de-terminedbythesingularfeaturesofSzz(q,ω)andG(ω).Byutilizingtheleadingandsubleadingtermsforeachsingularity,wehavebeenableto twithatypicalerrorinSzz(q,t)orG(t)forJt～20-30between10 4and10 5.Wecan twiththedecayexponentsdeterminedanalyt-icallyorasfreeparameterstochecktheanalyticexpres-sions.TableIshowsthecomparisonbetweentheexpo-nentsforG(t)extractedindependentlyfromtheDMRGdataandtheFTpredictions.Inallcasestheagreementisverygood.BysmoothlytransitioningfromthetDMRGdatatothe tastincreases,weobtainaccurateresultsforalltimes.AstraightforwardtimeFouriertransform

Exact formulas for the singularities of the dynamical structure factor, S^{zz}(q,omega), of the S=1/2 xxz spin chain at all q and any anisotropy and magnetic field in the critical regime are derived, expressing the exponents in terms of the phase shifts wh

TABLEI:Exponentsforthespinself-correlationfunctionG(t)forh=0.TheparametersW,η,η2andσwereob-tainednumericallyby ttingthe

DMRGdataaccordingtoEq.(15).ThesearecomparedwiththecorrespondingFTpredictions(withvandKtakenfromtheBetheansatz).

η1

σ2K

01.51.5110.1251.4511.4261.76120.251.3661.3612.03420.3751.3131.3032.00020.51.2871.252.12020.75

1.102

1.149

2.226

2

withaverylongtimewindowyieldsveryaccuratehighresolutionspectra.ExamplesoflineshapesobtainedthiswayareshowninFig.1.WealsodidDMRGfortheholeGreen’sfunctionforthefermionicmodelcorrespondingtoEq.(1),obtaininggoodagreementwiththepredictedsingularitiesfromthex-rayedgepicture.

Wehavenotseenanyexponentialdampingoftheη2terminG(t)for >0.Thissuggeststhatthesingularityattheupperedgeisnotsmoothedoutintheintegrablexxzmodel,evenwhenthestabilityoftheexcitationisnotguaranteedbykinematicconstraints[25].Integrabil-ityalsoprotectsthesingularityatωUfor nite eld,asimpliedbytheCFTformofthespectruminEq.(6).Inconclusion,wepresentedamethodtocalculatethesingularitiesofSzz(q,ω)forthexxzmodel.Theex-ponentsforgeneralanisotropy,magnetic eldandmo-mentumcanbeobtainedbysolvingtheBetheansatzThewidthofthepeakisverysmallforsmall| |.

4

theparticle-holesymmetriczero eldcase,weshowedthattheloweredgeexponentisq-independentandthe(“exciton-like”)upperedgehasauniversalsquarerootsingularity.ThecombinationofanalyticmethodswiththetDMRGovercomesthe nitetlimitationontheres-olutionofthetDMRGandcanbeusedtostudydynamicsofotherone-dimensionalsystems(integrableornot).WethankL.Balents,J.-S.Caux,V.Cheianov,L.I.Glazman,N.Kawakami,M.PustilnikandJ.Sirkerfordiscussions.WeacknowledgethesupportoftheCNPqgrant200612/2004-2(RGP),NSERC(RGP,IA),NSFundergrantDMR-0605444(SRW)andCIfAR(IA).

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