Green&39;s and spectral functions of the small Frolich polaron

According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupl

Green’sandspectralfunctionsofthesmallFr¨ohlichpolaron

A.S.AlexandrovandC.Sricheewin

DepartmentofPhysics,LoughboroughUniversity,LoughboroughLE113TU,U.K.

arXiv:cond-mat/9911376v2 [cond-mat.supr-con] 26 Nov 1999AccordingtorecentQuantumMonteCarlosimulationsthesmallpolarontheoryispracticallyexactinawiderangeofthelong-range(Fr¨ohlich)electron-phononcouplingandadi-abaticratio.WeapplytheLang-Firsovtransformationtoconvertthestrong-couplingtermintheHamiltonianintotheformofane ectivehoppingintegralandderivethesingle-particleGreen’sfunctiondescribingpropagationofthesmallFr¨ohlichpolaron.Oneandtwodimensionalspectralfunc-tionsarestudiedbyexpandingtheGreen’sfunctionpertur-batively.Numericalcalculationsofthespectralfunctionsareproduced.Remarkably,thecoherentspectralweight(Z)ande ectivemass(Z′)renormalisationexponentsarefoundtobedi erentwithZ′>>Z,whichcanexplainasmallcoherentspectralweightandarelativelymoderatemassenhancementinoxides.PACSnumbers:74.20.Mn,74.20.-z,74.25.Jb1.IntroductionTheproblemofafermiononalatticecoupledwiththebosonic eldoflatticevibrationshasanexactsolutionintermsofthecoherent(Glauber)statesintheextremestrong-couplinglimit,λ=∞foranytypeofelectron-phononinteractionconservingtheon-siteoccupationnumbersoffermions1,2.Fortheintermediatecouplingthe1/λperturbationtechniquehasdevelopedbothforasingle3andsystems4.Theexpansionparameteractuallyis1/2zλ23,5,6,4,sotheanalyticalperturbationtheorymighthaveawiderregionofapplicabilitythanonecanexpectfromanaivevari-ationalestimate(zisthelatticecoordinationnumber).However,itisnotclearhowfasttheexpansionconverges.Theexactnumericalofvibratingclus-ters,variationalcalculations717dynamicalmean- eldapproachin18,andQuantum-Monte-Carlosimulations19–22revealedthatthegroundstateen-ergy(thepolaronbindingenergyEp)isnotverysensitivetotheparameters.Onthecontrary,thee ectivemass,thebandwidthandtheshapeofpolarondensityofstatesstronglydependonpolaronsizeandadiabaticratioincaseofashort-range(Holstein)interaction.Inparticu-lar,numericaldiagonalisationofthetwo-site-one-electron

Holsteinmodelintheadiabaticω0/t<1aswellasinthe

nonadiabaticω0/t>1regimesshowsthatperturbation

theoryisalmostexactinthenonadiabaticregimefor

allvaluesofthecouplingconstant.However,thereisno

agreementintheadiabaticregion,wherethe rstorder

perturbationexpressionoverestimatesthepolaronmass

byafewordersofmagnitudeintheintermediatecoupling

regime8.Hereω0isthecharacteristicphononfrequency

andtisthenearest-neighbourhoppingintegralsothat

1thedimensionlesscouplingconstantisλ=Ep/(zt).Amuchlowere ectivemassoftheadiabaticsmallpolaronintheintermediatecouplingregimecomparedwiththatestimatedby rstorderperturbationtheoryisaresultofpoorconvergenceoftheperturbationexpansiontotheappearanceofthefamiliardouble-wellpotential23intheadiabaticlimit.Thetunnellingprobabilityisex-tremelysensitivetotheshapethispotential.Ithasalsobeenunderstood21therangeofappli-cabilityoftheanalyticaltheory3,4stronglydependsontheradiusofinteraction.Whiletheanalyticalapproachisapplicableonlyifω0≥tforshort-rangeinteraction,thetheoryappearsalmostexactinasubstantiallywiderregionoftheparametersforalong-range(Fr¨ohlich)in-teraction.Theexacte ectivemassofbothsmallandlargeFr¨ohlichpolarons,calculatedwiththecontinuous-timepath-integralQuantumMonteCarlo(QMC)algo-rithm,m (λ)iswell ttedbyasingleexponent,22.Asanexample,e0.73λforω0=tande1.4λforω0=0.5tdescribem (λ)inaone-dimensionallattice.Thenumer-icalexponentsareremarkablyclosetothoseobtainedfromtheLang-Firsovtransformation,e0.78λande1.56λ,respectively.Hence,inthecaseoftheFr¨ohlichinterac-tionthetransformationisperfectlyaccuratealreadyinthe rstorderof1/λexpansionevenintheadiabaticregime,ω0/t≤1foranycouplingstrength.Inthispaperweusethisresulttocalculatethesingle-particleGreen’sfunctionofafermiononalatticecoupledwiththebosonic eldviathelong-rangeFr¨ohlichinter-action.2.Green’sfunctionsofthesmallFr¨ohlichpolaronTheclassicalapproachtothesmallpolaronproblemisbasedonthecanonicaldisplacement(Lang-Firsov)trans-formationoftheelectron-phononHamiltonian3allowingforthesummationofalldiagramsincludingthevertexcorrections, ωqn i[ui(q)dq+H.c.]c+t(m n)δs,s′c H=ij+i,j qq,iωq(d qdq+1/2)(1)withthebarehoppingintegralt(m)andthematrixele-mentoftheelectron-phononinteractionui(q)=1γ(q)eiq·m.2N(2)Herei=(m,s),j=(n,s′),includesitemandspins,n i=c ici,andci,dqareelectron(hole)andphononoperators,respectively.

According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupl

ThecanonicaltransformationeSdiagonalisingtheHamiltonianintheλ=∞limitis

H =eSHe S,(3)

where

S= n i[ui(q)dq

q,i H.c.].(4)

Theelectronoperatortransformsas

c i=ciexp ui(q)dq H.c. (5)

q

andthephonononeas:

d q=dq+ n iu i(q)(6)

i

ItfollowsfromEq.(6)thattheLang-Firsovtransforma-tionshiftsionstonewequilibriumpositions.Inamoregeneralsenseitchangesthebosonvacuum.Asaresult,

H = σ ijc

icj Ep

i,j n ii

+ ωq(d qdq+1/2)+1

q

2N q|γ(q)|2ωq(9)

thepolaronlevelshift(bindingenergy),and

vij= 1 mt(m)exp( ik·m)(14)withg2(m)= q|γ(q)|2[1 cos(q·a)].(15)Quitegenerallyone ndsZ′=exp( γEp/ω),wherethenumericalcoe cientγ= |γ(q)|2[1 cos(q·a)]/ |γ(q)|2,(16)qqmightbeassmallas0.421andevensmallerinthecuprateswithnearestneighbouroxygen-oxygendistancelessthanthelatticeconstant,γ 0.225,26.ApplyingtheLang-FirsovcanonicaltransformationtheFouriercomponentoftheretardedGF, ∞GR(k,ω)= ie i(k·m ωt) 0|c0(t)c m(0)|0 dtm0(17)isexpressedasaconvolutionoftheFouriercomponentsofthecoherentretardedpolaronGF,Gp(n,t),andthe

multiphononcorrelationfunctionσ(n,t):

GR(k,ω)=1

According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupl

withXm=exp

stateofH .

Straightforward qui(q)dq H.c. ,and| 0 thegroundcalculations1,2yield

G1p(m,ω)=ω ξ(k′)+iδ,(21)and

σ(m,ω)=iZ1

∞

l=0

|γ(q)|2exp(iq·m).(22)

q l2N

Here

Z=exp( q|γ(q)|2).(23)

Inthefollowingweconsiderdispersionlessphonons,ωq=ω0,andtheFr¨ohlichinteractionwithγ(q)～1/q.TheconvolutionofEq.(19)andEq.(20)yields

GR(k,ω)= ∞

G(l)

R(k,ω),(24)

l=0

where

G(l)

R (k,ω)=

|γ(q1)|2×|γ(q2)|2×...×|γ(ql)|2

Z

q1,...ql

ω ξ(k)+iδ.(26)

Thespectralweightofthecoherentpartisstrongly(ex-ponentially)suppressedasZ=exp( Ep/ω0)whilethee ective′massmightonlybeslightlyenhanced,ξk=ZEk µ,becauseZ<<Z′<1inthecaseoftheFr¨ohlichinteraction.

Thesecondincoherentphonon-assistedcontributionwithl≥1describestheexcitationsaccompaniedbyemissionofphonons.Webelievethatthistermisrespon-sibleforthebackgroundinopticalconductivityandinphotoemissionspectraofcupratesandmanganites.Wenoticethatitsspectraldensityspreadsoverawideen-ergyrangeofabouttwicethepolaronlevelshiftEp.Onthecontrarythecoherenttermshowsanangulardepen-denceintheenergy′windowoftheorderofthepolaronbandwidth,2Zzt.Interestingly,thereissomekdepen-denceoftheincoherentbackgroundaswell,ifthema-trixelementofelectron-phononinteractiondependsonq(seealsoRef.28).Toillustratethispointwecalculatethesingle-phononcontribution(l=1)tothespectralfunctionA(k,ω)≡ (1/π) GR(k,ω)= ∞A(l)(k,ω).(27)l=0Thecoherentpart(l=0)ofthespectralfunctionisaδ-functioninagreementwiththewell-establishedfact(see2,1andreferencestherein)thatsmallpolaronsex-istintheBlochstatesatzerotemperaturenomatterwhichvaluestheparametersofthesystemtake.Thesingle-phononcontributiontotheincoherentbackgroundisgivenbyA(1)(k,ω)～ dqq 2δ(ω ω0BZ ξ(k+q),(28)wheretheintervalofintegration(theBrillouinzone,BZ)isdeterminedbylatticeconstantsa,b,c.Wethisintegralforone-dimensional(1D),ξ(k)= calculateand 2tcos(kxa)2 t′two-dimensional(2D),ξ(k)=2tcos(kxa)+cos(kyb)polaronbandsinthetight-bindingapprox-imationwiththepingintegrals, (renormalised)nearestneighbourhop-t=Z′t,t ′=Z′t′.The1Dsingle-phononspectralfunctionisreducedtoA(1)(k,ω)～(1 ω 2) 1/2 π/3dz[f(z,ω,kx)+f(z,ω, kx)],(29)0wheref(z,ω,kx)=[z2+(kx cos 1ω )2] 1/2tan 1(π[z2+(kx cos 1ω )2] 1/2).(30)Hereandfurtherwetakea=b=1andc=3,t and′=t/4.ItsspectralandangulardependenceareshowninFig.1.Apartfromtwonondispersivesingularities(vHs)atω ≡(ω ω0)/2 1Dvan-Hovet=±1thereisaninterestingdispersivepeakatω =coskx,whichisdueentirelytothelong-rangecharacteroftheFr¨ohlichinteraction.Indeed,approximatingtheBrillouinzonebyacylinderalongx,onereadelyobtainsalogarithmicsingularityinthespectralfunction,A(1)(k,ω)～(1 ω 2) 1/2ln (kx cos 1ω )2+q2D

According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupl

HereqDistheradiusofthecylinder.Forω=ω0+2tcoskx,thex-componentofthephononmomentumiszero,andthesingularmatrixelementsquared(～1/q2)integratedoverqyandqzyieldsasingularity.

The’long-range’dispersivefeaturesappearinthe2Dsingle-phononspectralfuctionaswell.Thisfunctionisreducedto

A(1)(k,ω)～ b(ω)

dz[1 ( ω 1

a(ω)

3[(x kx)2+(z ky)2] 1/2

andx=cos 1( ω 1 (33)

According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupl

34

A.S. Alexandrov and C.J. Dent, Phys. Rev. B 60, (1999).

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1

Figure Captures Fig.1. 1D single-phonon contribution to the polaron spectral function. Fig.2. 2D single-phonon contribution to the polaron spectral function along theΓ Y direction.

According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupl

Relative Energy

According to recent Quantum Monte Carlo simulations the small polaron theory is practically exact in a wide range of the long-range (Frohlich) electron-phonon coupling and adiabatic ratio. We apply the Lang-Firsov transformation to convert the strong-coupl

Relative Energy

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