Dynamical properties of the two-dimensional Holstein-Hubbard

Dynamical properties of the two-dimensional Holstein-Hubbard model in the normal state at zero temperature:A?uctuation-based effective cumulant approach

T.Hakiog?lu

Physics Department,Bilkent University,06533Ankara,Turkey

M.Ye.Zhuravlev

N.S.Kurnakov Institute of General and Inorganic Chemistry of the Russian Academy of Sciences,

31Leninsky Prospect,GSP-1,117907Moscow,Russia

?Received23February1998?

The two-dimensional many-body Holstein-Hubbard model in the T?0normal state is examined within the

framework of the self-consistent coupling of charge?uctuation correlations to the vibrational ones.The pa-

rameters of our model are the adiabaticity,the electron concentration,as well as the electron-phonon and the

Coulomb interaction strengths.A?uctuation-based effective cumulant approach is introduced to examine the

T?0normal-state?uctuations and an analytic approximation to the true dynamical entangled ground state is

suggested.Our results for the effective charge-transfer amplitude,the ground state energy,the?uctuations in

the phonon population,the phonon softening as well as the coupling constant renormalizations suggest that,the

recent numerical calculations of de Mello and Ranninger?Ref.5?,Berger,Vala′sˇek,and von der Linden?Ref.

2?,and Marsiglio?Refs.4and8?on systems with?nite degrees of freedom can be qualitatively extended to the

systems with large degrees of freedom.?S0163-1829?98?03728-X?

I.INTRODUCTION

In this work we focus on the dynamical properties of the

polaronic ground state in the Holstein-Hubbard model from

the perspective of what we call as the charge-density wave ?CDW??uctuation-based effective cumulant approach.In this many-body model,the qualitative aspects of the transi-

tion from large to small polarons as the electron-phonon ?e-ph?adiabaticity and the Coulomb interaction strengths are varied,with the full assessment of these interactions,is still an unresolved problem since the celebrated work of Holstein.1Recently quantum Monte Carlo?QMC?calculations,2–4semianalytic direct diagonalization5–8using ?nite lattice and electronic degrees of freedom,and varia-tional ground-state techniques9–11have revealed evidence of a smooth transition of the ground state from the large ex-tended to the small localized polaronic one as the interaction parameters are varied from the weak-coupling adiabatic to strong-coupling antiadiabatic ranges.The ground-state dy-namics of the Holstein-Hubbard model is determined by the three dimensionless scales;viz.,the adiabaticity??t/?0, the e-ph mediated coupling??(g/?0)2,and the repulsive Coulomb interaction strength V c e-e/?0where?0is the fre-quency of Einstein phonons,t is the charge transfer ampli-tude and g is the linear e-ph coupling strength.In the weak-coupling adiabatic regime?i.e.,??0.5,1??and V c e-e suf?ciently small?,the Migdal random-phase approximation ?RPA?is quite accurate in describing the quasiparticle renor-malization.However,the extension of Migdal RPA beyond ??0.5encounters super?cial instabilities in the phonon vacuum.This point has been critically questioned for in-stance,in Refs.2,5,and6when it is no longer possible to assign independent degrees of freedom to phonon and elec-tron systems beyond the weak-coupling strongly adiabatic ranges,and one has to self-consistently deal with an en-tangled dynamical picture by abandoning the simpler quasi-particle one.On the semianalytic side progress has been made in the diagrammatic approaches by extending the Migdal RPA to the Migdal-Eliashberg?ME?formalism with self-consistent handling of the phonon and electron renor-malizations within the RPA,where compatible results to more reliable QMC simulations2,4have been obtained.At the other extreme,the crucial role played by the adiabaticity pa-rameter was clearly shown in Ref.5such that the strong-coupling Lang-Firsov?LF?approximation is strictly appli-cable only in the strongly antiadiabatic range??1and contrary to the common belief,the convergence to LF behav-ior can be considerably weakened in transition from strongly antiadiabatic??1to weakly nonadiabatic ranges??1.In our opinion,although these results do not contradict the con-ditions of applicability of the LF approach or strong-coupling1/?expansion,12they con?ne their validity to the strongly antiadiabatic ranges.

The crucial point needed for a global perspective of the ground-state properties in the Holstein-Hubbard problem for a large range of coupling constants and adiabaticities is in the understanding of the nonlinear,self-consistent coupling of the charge?uctuations to the?uctuations in the vibra-tional degrees of freedom.In this respect,the main motiva-tion for our?uctuation-based approach was provided by the numerical direct-diagonalization results in Ref.5regarding the correlated charge-deformation dynamics,as well as the CDW susceptibility based QMC and self-consistent ME cal-culations of Refs.2,4,and7.

It is desirable that these numerical calculations,despite the limitations in the consideration of?nite degrees of freedom—such as the?nite lattice size,truncated Hilbert space,small number of electrons etc.,which are necessary from the feasibility point of view of the numerical

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methods—can be qualitatively extended to reach at conclu-sive results on the nature of the polaronic transition for more generalized systems with large degrees of freedom.In fact, this point has been raised a long time ago by Shore and Sander9and also stressed by the authors of Refs.5,7,and8. However there has not been conclusive evidence,in particu-lar at the intermediate ranges??1,??1,on whether the theoretical results obtained using models with?nite degree of freedom could be extended to systems with realistic sizes. Moreover,apart from the variational ground state calculations,9–11direct attempts to tackle the many body dy-namical?uctuations,in particular in the intermediate ranges, have not been possible on practical grounds.On the other hand,although the self-consistent ME RPA as well as QMC calculations provide an improved understanding of the prob-lem,a clear self-consistent picture of?uctuations in the ground state?and,perhaps an approximate analytic form?still remains to be established.

In this work we approach the many-body problem in the normal state and at zero temperature by improving the CDW ?uctuation-based effective cumulant approach that was re-cently introduced in Ref.13and applied to the superconducting-state solution to examine the low-temperature T c-dependent phonon anomalies in certain high-temperature superconductors.

In Sec.II the Holstein-Hubbard model is introduced and studied in the momentum space.The nature of the interacting ground state is examined in Sec.II A where an approximate analytic form is suggested in the direct product form,decom-posing the entangled nonlinear polaronic wave function in the coherent and two-particle correlated subspaces.The pa-rameters of this effective wave function are calculated using the CDW?uctuation-based effective cumulant approach,re-producing the all?rst and second-order phonon cumulants of the entangled polaronic wave function.The effective wave function is an analytic and continuous function of?,?,and V c e-e,which ensures the same properties for the ground-state energy as well as other physical parameters induced from the model.The solution of the wave-function parameters as well as the calculation of the approximate ground-state energy is presented in Sec.II B.Section III is devoted to the renormal-ization of the charge-transfer amplitude.In Sec.IV,the renormalization of the effective e-e interaction is examined. The statistics of the?uctuations in the ground state of the renormalized phonon subsystem and the renormalization of the vibrational frequency are examined in Secs.V A and V B,respectively.

II.MODEL

We investigate the Holstein-Hubbard problem via the Hamiltonian,

H?H e?H ph??k,m,?g?k?e i km c m??c m??a k?a?k??

?1

2

?

m,n,?,??

V m,n c m??c

n??

?c

n??c m?,?1?

where c m??,c m?create and annihilate electrons at site m with spin?on a two-dimensional?2D?lattice,a k?,a k create and annihilate phonons at momentum k with g(k),and V m,n de-scribing the linear e-ph and electron-electron Coulomb inter-actions,respectively.The?rst two terms in the Hamiltonian describe the electron charge transfer and the harmonic pho-non contributions as

H e???mn??t mn c m??c n?,and

H ph??k?k2?a k?a k?a k a k??,?2?where t m,n is the translationally invariant charge-transfer am-plitude between neighboring sites m,n and?k is the har-monic phonon frequency.

The central theme of this work is to calculate the?uctua-tions in the vibrational degrees of freedom in a self-consistent frame together with the charge-density?uctua-tions in the correlated electron subsystem.The charge-density?uctuations are de?ned by the expressions

c m??c m???c m??c m?????c m??c m??or,equivalently,

?k?2nˉk???k,?3?where?k??k?,?c k??k,?

?c

k?,?with c k,?

?,c

k,?

describing the electron operators in the momentum representation,and 2nˉk???k?describing the CDW order parameter.The factor of2in Eq.?3?arises from the spin 8e50d6addd3383c4bb4cd230ing Eq.?3?,Eq.?1?is separated into H?H0?H I such that

H0?H e??k??k2?a k?a k?a k a k???2g?k?nˉk?a k?a?k???, H I??k g?k???k?a k?a?k???12?k V c?k??k??k,?4?

where V c(k)?1/N?k e i k.…m?n…V m,n,and H0corresponds to the exactly solvable part associated with the eigen-wave-function,

??0????c? ??e?,

??c??U c?0ph??exp?2?k g?k??k nˉk?a k?a?k????0ph?.?5?

Here??c?describes the pure coherent part of the ground-state wave function in the phonon subsystem,and?0ph?is the phonon vacuum state.At the exactly solvable level the prod-uct form of the wave function remains to be valid with??e?representing the wave function of the electron subsystem. The coherent part??c?describes the coupling of the phonons to the static charge-density wave described by the CDW or-der parameter nˉk?1/2?k???c k?k?

?c

k???.To examine the dy-namical contributions to the interacting ground-state wave function we eliminate this part from the Hamiltonian by the unitary Lang-Firsov transformation U c in Eq.?5?as

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T.HAKIOG?LU AND M.YE.ZHURAVLEV

H??U c??H0?H I?U c

?H e??k?k2?a k?a k?a k a k??

??k?g?k??2?k?nˉk???k?nˉ?k?H I,?6?where the coherent part of the wave function in Eq.?5?is now shifted to??0???U c???0???0ph? ??e?.Now the inter-action term H I in Eq.?4?is given purely in terms of the coupling of phonons to the?uctuations in the CDW that contribute to the dynamical part of the interacting ground-state wave function.This interaction term is also convention-ally transformed away by another unitary transformation U?,

U??exp??k g?k??k??k?a k?a?k???,?7?for which the transformed Hamiltonian reads

H??U??H?U?

???mn??t mn??m,n?c m??c n?

??k?k2?a k?a k?a k a k????k?g?k??2?k??k???k

??k?g?k??2?k?nˉk???k?nˉ?k?12?k V c?k??k??k.

?8?The expense paid by this transformation is the introduction of the multiphonon operator,

??m,n??exp?12?k g?k??k?e ik?m?e ik?n??a k?a?k???.?9?

Combining the transformations in Eq.?5?and Eq.?7?we obtain a highly entangled dynamical wave function?????U????0??.Although the rest of the Hamiltonian in Eq.?8?is decoupled in electron and phonon degrees of freedom,a major dif?culty is introduced by the multiphonon-electron scattering in the?rst term in Eq.?8?.In the conventional Lang-Firsov approach this term is replaced by its average in the coherent part??c?of the wave function by?(m,n)→??c??(m,n)??c?,which completely decouples the Hamil-tonian.On the other hand,a re?ned treatment of the residual interactions induced by?(m,n)???c??(m,n)??c?has to in-corporate the highly nonlinear phonon correlator,12which,in our opinion,can obscure the physical picture of the dynami-cal properties of the wave function.

In fact,the dif?culties in the solution of the many-body problem are,at least,twofold.At one end,there is the im-practicality of a formal diagrammatical approach to the re-sidual interactions.12At the other end,even if one can get away with neglecting the residual interactions by using a Lang-Firsov-like formalism,a full understanding of the dy-namical wave function????is still not promised due to its highly entangled nature.In this work,we will approach the problem within a Lang-Firsov-like approach?namely,by re-placing?m,n by?????m,n????where????is,in contrast to the LF approach where the coherent part is used,the dynami-cal?uctuating part of the polaron wave function?by demon-strating that it is,in principle,possible to construct an effec-tive wave function???eff?as an approximation to the dynamical part????,which adopts a special product form at the cumulant-generating-operator level in the n-phonon cu-mulant correlation space.Then,in an approximation scheme, an analytic form???eff?will be constructed by reproducing all ?rst-and second-order cumulants of the phonon operators in ????.

A.Nature of the interacting ground state

Our purpose in this subsection is to understand the nature of the dynamical strongly entangled wave function????.In the static CDW limit?i.e.,nˉk 0?,the?uctuations in the charge density are negligible.It is known that the static CDW limit corresponds to strongly antiadiabatic regimes when the e-ph coupling constant is in the extreme weak-or strong-coupling limits.This is the limit where??c?can ac-curately approximate the exact polaron ground state of H in Eq.?1?.In the weak-coupling antiadiabatic limit??1,??1,a perturbative scheme based on charge?uctuations is adequate where the magnitude of?uctuations in the residual interactions is limited?i.e.,??(m,n)???(m,n)???1?;since ?(m,n)is a positive and bounded operator by unity from above and??(m,n)??1.In the strong-coupling antiadiabatic regime,the small polaronic bandwidth is strongly reduced where we also have negligible contribution of the residual interactions.There,?(m,n)is bounded from below by zero since??(m,n)??1.It is clear that the corrections to??c?as well as the importance of the residual interactions arise from the nonnegligible presence of the dynamical?uctuations in the intermediate ranges between these limits.

We will examine????by calculating the characteristic cumulants of the phonon coordinates Q k?1/&(a k?a?k?) and P k??i/&(a?k?a k?).In order to study the dynamical ?uctuations in the ground state we shift the phonon coordi-nates in the Hamiltonian?1?to the origin by Q k→Q k ??Q k?and P k→P k??P k?where?Q k?and?P k?are deter-mined in the coherently shifted component??c?as?Q k??2?g(k)/?k?nˉk and?P k??0.This is equivalent to a unitary transformation by U c of the initial Hamiltonian yielding Eq.?6?.Note that from here on all expressions involving factors of Q k and P k will be expressed in the shifted coordinates.

We start with calculating?ve distinct types of the phonon moments de?ned by

R s

1

??????Q k?s1????,

P s

2

??????P k?s2????,

K s

3

??????P k P?k?s3????,

F s

4

??????Q k Q?k?s4????,

G s

5

??????Q k?s5?P k?s5????.?10?After a tedious but straightforward calculation using?????U???0ph? ??e?,these are explicitly given by

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DYNAMICAL PROPERTIES OF THE TWO-DIMENSIONAL...

R s 1?0,P s 2?0,K s 3?

?1

2

?

s 3

s 3!,

F s 4?s 4!2s

4

?

p ?0

?

2F 1??s 4?p ,0;1;?1?

?

??s 4?p ??2p ?p ?p !?2

?g ?k ??k ?2p

????k ???k ??p ,

G s 5?

?i

2

?

s 5

s 5!,

?11?

where (n )m ?n (n ?1)ˉ(n ?m ?1)and 2F 1(a ,b ;c ;z )is the Gauss hypergeometric function and we assumed Gauss-ian density ?uctuation correlations.In principle,an effective wave function ???eff ?that is expected to be equivalent to ????in the phonon sector should consistently reproduce the entire set of an in?nite number of cumulants in Eqs.?11?with 1рs i ??,(i ?1,...,5).Hence,the effective wave function also comprises an in?nitely large set of correlation subspaces where the correlations in each subspace is produced by the unitary n -phonon cumulant correlation generator U (n )as

???eff

??

?n ?1

?

U ?n ??0ph ? ??e ?,where

?n ?1?

U ?n ???n ?m

?

U ?n ?U ?m ?1?ˉU ?2?U ?1??12?

with U (1),U (2),...,etc.describing the one-particle coher-ent,the two-particle coherent correlations,etc.,respectively.

In fact,in this decomposition in terms of correlation sub-spaces,U (1)corresponds to the coherent shift U c ?in Eq.?5?and U (2),U (3),etc.describe the two-particle and three-particle correlated sectors of U ??in Eq.?7?,etc.In this case the projection of the effective wave function ???eff ?on the m -dimensional correlation subspace is ????m ,which is de-termined by the projection operator,

T m ?

??n ?m ?1

?U ?n ?

?

?

as

????m ?T m ????.

?13?

In order for the product form in Eq.?12?to be a sensible expansion of the wave function in terms of its independent sectors in the correlation space,each unitary n -phonon cor-relation generator U

?(n )must reproduce the n th-order phonon cumulants obtained from the moments in Eqs.?10?but not the moments themselves.This is indeed the reason why we shifted the phonon coordinates in order to eliminate the in-?uence of the coherent one-particle sector on the second-order and higher dynamic correlations in the wave function.This is equivalent to subtracting the coherent one-particle contributions by performing the shift Q k →Q k

??0ph ?U ?c ?Q k U ?c ?0ph ?.For those of the m th-order ones,this

procedure de?nes generalized shifts

?0ph ?(?n ?1m ?1U ?(n ))?Q k (?n ??1m ?1U

?(n ?))?0ph ?and similarly for P k .In result,it is technically possible to decompose the wave function in direct product form in the cumulant corre-lation space.Despite the fact that the technical principles of such a decomposition prescribed in Eq.?12?can be exam-ined,it is not practically possible to go beyond the second-order correlations,because of the fact that a possible general analytic form for the third-and higher-order cumulant gen-erators U

?(m ),(3рm )have not been studied in the literature from the mathematical point of view.The ?rst-and the second-order cumulant correlations,on the other hand,are well known in quantum optics as the one-particle coherent 14and the two-particle coherent states,15–17respectively,and have been extensively applied to the polaron problem in the context of the dynamical 13and the variational ?see,for in-stance,Refs.18and 19?approaches.

Under these practical limitations arising for 3рm ,we consider a subset of Eqs.?10?comprising the entire ?rst-and second-order cumulants,which correspond to s 1,s 2?1,2,s 3,s 4,s 5?1.Hence,it is implied that the polaron ground-state wave function will be approximated in the cumulant correlation space using only the ?rst-and the second-order cumulants.From Eq.?11?these seven cumulants are explic-itly given by

????Q k ?????????P k ?????????Q k Q k ????

?????P k P k ?????0,

????Q k Q ?k ?????1/2?1?4

??g ?k ?

?k

??

2

???k ???k ??

,

????P k P ?k ?????1/2,????Q k P k ????i /2.

?14?

In order to reproduce Eqs.?14?,we propose the effective

wave function,???eff ??S ?????

?k

??k ??k a k ?a ?k ???k ?a k ??2?a ?k ??2

??0ph ? ??e ?,?15?

where the phonon coordinates are coherently shifted for the

calculation of second-order correlations according to the pro-cedure outlined above.The wave function is normalized as ??k ?2???k ?2???k ?2?1,where we neglect the overall phase

of ???eff ?by considering ?k ??ˉk ,and

S ??????exp ?

?

?k

??k a k a ?k ??ˉk a k ?a ?k ?

?

?

,

?k ???k ?e i 2?k ,S ??S ?1

?16?

describes the two-particle coherent,translationally invariant unitary operator ?squeezing operator in quantum optics 15,16?.The unitary transformation de?ned by S (???)on the phonon coordinates is given by

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T.HAKIOG

?LU AND M.YE.ZHURAVLEV

S??????Q k S????????k?Re??k??Q k?Im??k?P?k,

S??????P k S????????k?Re??k??P k?Im??k?P?k,

?17?where?k?cosh(2??k?)and?k?e?i2?k sinh(2??k?)such that ??k?2???k?2?1as imposed by the unitarity of S(???).

Using the wave function in Eq.?15?and the properties in Eqs.?17?we

obtain

???eff?Q k???eff??0,

???eff?P k???eff??0,

???eff?Q k Q k???eff??0,

???eff?P k P k???eff??0,

???eff?Q k Q?k???eff??Re??k?k??k??k?2?2?k?ˉk??k??k?2??1

2

??k2?3??k?2?5??k?2???k??k?2,

???eff?P k P?k???eff???Re??k?k??k??k?2?2?k?ˉk??k??k?2??1

2

??k2?3??k?2?5??k?2???k??k?2,

???eff?Q k P k???eff??i

2

?1???ˉk?k??k?ˉk???k2?3??k?2?5??k?2???Im???k?k?2?k?ˉk???ˉk2??ˉk2??.?18?

The parameters?k,?k,?k,?k,?k are determined by de-

manding the equality of Eqs.?18?and Eqs.?14?.In fact,

independently from speci?c values of?k,?k,?k,and?k,

the effective wave function???eff?satis?es a larger set of

cumulants than given by the subset in Eqs.?18?.First of all,

the?rst two conditions on R s

1and P s

2

in Eq.?11?are very

strict,corresponding to the translational invariance of????. These are also respected for all s1,s2by???eff?independently from?k,?k,?k and?k.Furthermore,we also have ????(Q k)s5(P k)s5????????eff?(Q k)s5(P k)s5)???eff??(i/2)s5s5! for all s5and for all arbitrary but real?k,?k,?k,and?k. Hence,we are motivated to?nd a solution where the param-eters are all real.Here,we switch to the polar coordinates ?k???k?exp(i??),and similarly for the other parameters. From the last equations in Eqs.?18?and?14?,we infer that ??????m?with m?0,1and Im??k?k?2?k?ˉk??0.For real parameters this trivially implies??k??k?sin????2??k???k?sin(?????)?0,hence,???r?(r?0,1),and???n?(n?0,1).With these conditions,there are?ve real equalities in the simultaneous solution of Eqs.?18?and?14?and four conditions?including two normalization conditions?to be satis?ed.We consider the?fth condition as the mini-mization of the ground-state energy.Since all parameters are now real,we drop the absolute value signs,i.e.,??k?→?k and similarly for the others.We now have an effective wave function that respects the strict conditions imposed by the

translational invariance indicated by R s

1and P s

2

as well as

the last condition indicated by G s

5in Eqs.?10?at all orders.

Consistency between Eqs.?18?and?14?now implies

?k2??k2?1,

?k2??k2??k2?1,

???1?r?k??k?2??1?n?k??12??k2?3?k2?5?k2?????k???1?m?k?2

?1/2?1?4??g?k??k??2???k???k??,

???1?r?1?k??k?2??1?n?k??12??k2?3?k2?5?k2??

???k???1?m?k?2?1/2.?19?

This set of four equations will be closed by one additional constraint from the ground-state minimization,which we ad-dress in the following section.

B.Solution of the parameters and approximations

to the true ground-state energy

We now de?ne the ground-state energy of the Hamil-tonian in Eq.?8?by

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DYNAMICAL PROPERTIES OF THE TWO-DIMENSIONAL...

E0????eff?H????eff?

??m,n t m,n??m,n??c m?c n???k V0?k????k???k???k V0?k?nˉk nˉ?k??k?k2????k?2???k?2?

??1?3??k?2?8??k?2??4?k??k???k???k?

?Re?e i????????????8??k???k???k???k?

?Re?e i???????????????,?20?

where V0(k)?1

2V c(k)???g(k)?2/?k?,is the bare e-e inter-

action,and,the last sum in Eq.?20?is the result of ?k?k/2???eff?(a k?a k?a k a k????eff?.The contribution from the multiphonon operator is more tedious to calculate,for which we obtain

???eff??m,n???eff???k exp??A k???k2???k?2?1?A k?2

???k?2/4?4?12A k2?A k4??2A k Re??k?ˉk?

?A k2Re??k?ˉk??A k?2?A k?2Re??k?ˉk??,

?21?with A k?12?g(k/?k?2e?4?k(1?cos k x?a?cos k y?a)where a describes the lattice constant,which we take to be unity. For the lowest possible energy we must satisfy in Eq.?20?

??????????,

?????????????,

??k??1

2

??k???k???k?????k?2???k?2?,?22?

where the last one in Eqs.?22?is obtained by minimizing the phonon part in Eq.?20?with respect to??k?.The?rst two yield???2???0,thus???8e50d6addd3383c4bb4cd230ing this as well as???r?obtained previously we?nd two possible solutions

???0,???0,???0,????,and

????,???0,???0,???0.?23?Since the phases are all?xed,we turn to the calculation of the density?uctuation correlations.The ground-state energy in Eq.?20?,as well as the parameters of the wave function in Eqs.?19?and?22?are functions of???k???k?,which we determine using the dielectric function?(k,?)formalism as

V0?k????k???k????0?d??Im?1?k,??.?24?

In the RPA,?(k,?)is given by

??k,???1?

V0?k?P?k,??

1?V0?k?P?k,??.?25?

The electron polarization P(k,?)is obtained in the standard formulation by

P?k,???2?p????p??p?????p?k??p?k?

???p??p?k??p??p?k?i?,?26?

where?k?t eff(k)??,t eff?2t???(1?cos k x?cos k y).Since ?k,?k,?k,and?k are not determined at this level,we consider in t eff,the zeroth-order approximation where we replace???by its LF limit???LF?exp??1/2?g(k)?2/?k2?. The chemical potential?is?xed self-consistently by the zero-temperature constraint,

n0??k????k???k??,?27?

with?(k)???k?V(k??k)????k???(k?)?describing the exchange contribution to one particle energy renormaliza-tion.Since we are con?ned here to zero-temperature formal-ism,??k?is independent from k and just renormalizes the chemical potential.Hence the exchange contribution is inef-fective in the denominator of Eq.?26?.

1.Density?uctuation correlations

We obtain the solution Eqs.?24–27?numerically in two dimensions using Einstein phonons?k??0and k indepen-dent dimensionless bare e-ph coupling??(g/?0)2.All en-ergies are normalized by?0.The dependence of???k???k?on the dimensionless parameters?,?,and V c(?,?)/?0is shown in Figs.1?a–c?at k?(?,?)and at half-?lling,for the values V c(?,?)/?0?0,1,2,3,4,and??0.05,0.1,1with 0р?р2.In each curve the solid line,open circles,open triangles,solid circles,and solid triangles represent values of V c(?,?)/?0as,respectively,indicated above.A quantita-tive comparison of the?gures for a?xed Coulomb interac-tion strength indicates that,as the adiabaticity?is decreased, there is an overall suppression in the magnitude of the?uc-tuation correlations.This effect is also enhanced further by strong e-ph coupling particularly in the strongly antiadiabatic ?i.e.,??1?ranges.On the other hand,as?increases towards the adiabatic range,correlations gradually increase for stron-ger e-ph coupling.This picture qualitatively agrees with the results obtained by direct-diagonalization calculations on?-nite systems where a cooperation is observed in the antiadia-batic range between the decreasing adiabaticity and the in-creasing coupling constant to suppress the quantum ?uctuations.The overall effect of the increasing repulsive Coulomb interaction is to overcome the phonon-induced po-laron attraction,which amounts to suppressing the?uctua-tions for small couplings and enhancing them in the strong-coupling ranges.At this level,we solve Eqs.?19?and?22?for the parameters of the effective wave function before we calculate the ground-state energy.

2.Parameters of the effective wave function

Once?uctuation correlations are determined,the phonon effective ground-state parameters?k,?k,?k,and?k can be calculated from Eqs.?19,22?for two branches as character-ized by Eqs.?23?.The solutions corresponding to these two branches are identical for?k,?k,and?k and only differ

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very slightly for ?k and ?k .In this subsection,we only present the results for the ?rst branch,whereas,both solu-tions will be explicitly used in the calculation of the approxi-mate ground-state energy.In Figs.2?a–d ?the parameters of the effective wave function are plotted for k ?(?,?)at half-?lling in the same ?range as in Fig.1.As the e -ph interac-tion is increased,a strong competition is observed between

the strengths of the pure two-particle coherent component given by ??,?and the pair excitations on this state given by ??,?.In the intermediate ranges of the e -ph coupling ?i.e.,??1?,the pair excitation strength becomes comparable to the strength of the underlying two particle coherent compo-nent.The four particle excitation given by ??,?is limited in strength in the whole ?range.On the other hand,Fig.2?d ?represents the parameters within the two-particle coherent component.For increasing e -ph interaction a rapid reduction is observed in exp(?2???,??).We observe that,because of the non-negligible strength of ??,?,the whole picture here is quite contrary to the common practice of replacing the effec-tive phonon ground state by a variational pure two-particle coherent ?squeezed ?component ?in which case we would have ?k ?1,?k ??k ?0for all k ?in the intermediate and strong coupling regimes.In Figs.3?a–d ?the same param-eters are calculated for ??0.05.As the system is shifted to increasingly antiadiabatic ranges ?i.e.,??1?,the relative strength ??,?of the pure two-particle coherent component is approximately maintained in the entire coupling range with respect to the two-and four-particle correlated excitations represented by ??,?and ??,?,respectively.Hence,in this range of the interaction parameters,the two-particle coherent component ??,?dominates the wave function where the two-and four-particle correlated excitations ??,?and ??,?compete only with each other.Within the two-particle coher-ent component ?as indicated in Fig.3?d ??there is a also an increasing tendency to overlap with the conventional phonon vacuum.Nevertheless,we observe that exp(?2???,??)satu-rates around 70%,implying that the overlap with the vacuum does not exceed 30%?see Fig.3?d ??even for such a strong antiadiabaticity as ??0.05.Note that,a strong overlap of the dynamical part ???eff ?with the vacuum would indicate that the coherent part ??c ?is dominating the ground-state wave function.These results are in qualitative agreement with the direct diagonalization results of Ref.5where the

observed

FIG.1.The solution of the density ?uctuation correlations at k ?(?,?)and at half-?lling as a function of the e -ph coupling for V c /?0?0,1,2,3,4as represented by solid line,open circles,open triangles,solid circles,and solid triangles,

respectively.

FIG.2.The parameters of the effective wave function at k ?(?,?)for ??1and at half-?lling n 0?1as a function of the e -ph coupling for various Coulomb strengths as V c /?0?0,1,2,3,4represented by solid,dotted,dashed,long-dashed,and dotted-dashed lines,respectively.

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convergence of the true ground state to the Lang-Firsov

small polaron limit ?indicated by the pure coherent part ??c ??is weaker than expected and strongly adiabaticity dependent.As the system is driven to even more antiadiabatic ranges,the charge ?uctuations reduce their overall amplitude as the ?uctuating component ???eff ?of the polaron wave function develops an ever increasing overlap with the conventional vacuum ?i.e.,as implied by the saturation in ??,?at approxi-mately 90%with ??,?,??,?saturating at limited strengths as well as the tendency of exp(?2???,??)to stay closer to unity in Fig.3?d ??.Hence the ground-state polaron wave function gradually becomes more coherent and localized;nevertheless,we also observe that the convergence to this limit is weaker than conventionally expected.

As the dependence of this overall picture on the electron concentration is concerned,the ?rst observation we make is

that,when n ˉk ?n 0is shifted away from half-?lling the in?u-ence of the Coulomb interaction becomes weaker on all pa-rameters.In addition,the relative strength of the correlated pair excitations ?i.e.,??,??with respect to the two-particle coherent component ?i.e.,??,??becomes weaker as shown in Figs.4?a,b ?for n 0?0.6.The four-particle correlations as given by ??,?in Fig.4?c ?,maintain their negligible strength.We also observe in the same result that the parameters of the two-particle coherent component as indicated in Fig.4?d ?are not too sensitive to changes in the electron concentration in this range.

3.Approximate ground-state energy

In Figs.5?a,b ?the ground-state energy difference calcu-lated in reference to the noninteracting limit ?i.e.,??0?

and

FIG.3.The same as in Fig.2for ??0.05at half-?lling n 0?

1.

FIG.4.The same as in Fig.2for ??1at electron concentration n 0?0.6.

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corresponding to each phonon branch as a function of ?is plotted for the same parameter values as the previous ?gures at half-?lling.Note that in this section,we intentionally in-clude the results of both branches in Eqs.?23?.To clearly demonstrate the in?uence of the charge ?uctuation correla-tions,the ground-state energy of the background uniform distribution ?i.e.,V 0(k )n ˉk n ˉ?k ?is subtracted in both Figs.5?a ?,and 5?b ?.The ?rst solution obtained for the parameters is identi?ed for each Coulomb strength,by a solid line (V c /?0?0),an open circle (V c /?0?1),an open triangle (V c /?0?2),a solid circle (V c /?0?3),and a solid triangle (V c /?0?4),respectively,in accordance with symbols used in Fig.1.The second solution is represented by dotted lines,for all Coulomb strengths.At weak e -ph coupling strength,a ?nite positive contribution to the energy is present from Cou-lombic charge ?uctuations.A common feature of all ground-state energy solutions in Fig.5?a ?is that at a ?xed Coulomb interaction strength,a slightly lower ground-state energy is obtained with the second branch for coupling strengths ??1than with the ?rst branch.In the approximate range 1??the ?rst branch yields a lower ground-state energy than the second one.In the transition from one branch to the other no discontinuity is present.In addition to the continuous na-ture of the transition,a kinklike feature is also present near ??1,where the transition is observed.The continuity of the ground-state energy is widely accepted on grounds of direct-diagonalization studies on ?nite systems 5–8as well as varia-tional calculations.9–11The kinklike feature has also been reported in one-dimensional calculations but it was attributed to the ?nite-size effects.8We also observe,in accordance with Ref.8that,as the system parameters are driven into antiadiabatic ranges ?i.e.,??1?the kinklike feature disap-pears as shown in Fig.5?b ?,and the ?uctuations calculated at

distinct Coulomb interaction strengths become less viable for the ground-state energy due to the suppression of the dy-namical ?uctuations.

III.EFFECTIVE CHARGE-TRANSFER AMPLITUDE

It has been shown in the direct-diagonalization calcula-tions on ?nite systems 5that the convergence of the intersite charge-transfer amplitude to the conventional Lang-Firsov ?LF ?limit is weak particularly in the intermediate coupling weakly antiadiabatic regimes.In the conventional LF ap-proach the adibaticity does not play a role in the renormal-ization of the t eff .The reason behind the independence of t eff from ?is that the standard LF polarons are renormalized only with respect to the lattice site on which the polaron is located;whereas,this approximation is only expected to be manifest in the extreme antiadiabatic strong-coupling limit.On the other hand,the response time scale for the phonon cloud to follow the charge is expected to be a monotonously increasing function of adiabaticity.This implies that in the strongly adiabatic ranges the renormalization of the effective charge-transfer amplitude by the following phonon cloud is expected to be weaker than it is for weakly adiabatic and nonadiabatic ranges.Hence,the localizing effect of the strong e -ph coupling should be a function of adiabaticity.This means that t eff /t ,as a measure of the kinetic-energy renormalization scale for electrons,is expected to be a mo-notonously decreasing function when ?decreases,which was indeed observed in the numerical calculations of Ref.5,7,and 8.In another way of saying it,the expected renormal-ization of t eff with respect to ?is itself a strong result against the use of the LF approach in the large and intermediate adiabatic ranges and the generality of the argument

requires

FIG.5.?a ?The ground-state energy difference ?E 0?E 0(?)?E (??0)as calculated by Eq.?20?for ??1at half-?lling and for the two solutions of the wave-function parameters as determined by the values of the phases in Eq.?23?.Here,for the second solution the same symbols are used as in Fig.1for the same parameter values.Since the ?rst solution and the second one meet on the vertical scale at a value corresponding to a particular value of V c the second solution for each V c can be identi?ed easily.For the sake of clarity we thus represent all second solutions with dotted lines.?b ?Same as in part ?a ?for ??0.05at half-?lling.

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that a similar scenario is expected to hold for the many-body case.

We de?ne the effective charge transfer amplitude t eff us-ing Eq.?21?as

t eff ?t ???eff ???m ,n ????eff

?.

?28?

Note that the coherent CDW sector ??c ?would have no con-tribution in Eq.?28?if it was included in the wave function.In Eq.?28?,or in its explicit form in Eq.?21?,the Lang-Firsov limit would only correspond to ?k ?1,?k ??k ??k ?0,yielding the standard Holstein band reduction t eff ?t exp(??/2).It can be seen that this limit is unphysical in our dynamical approach here.The reason is that,since all parameters are de?nite functions of ?,the limit ?k ?1,?k ??k ??k ?0would only be obtained if no e -ph coupling was present.Hence,deviations from the standard LF ap-proach is an inherent feature of the dynamical approach it-self.Since the parameters of ???eff ?are known by Eqs.?19?and ?22?,we can examine Eq.?28?as the e -ph coupling constant and the adibaticity are varied.In Fig.6?a ?,the cou-pling constant dependence of the renormalized charge-transfer amplitude is plotted for ??0.05,0.1,1.Given the general argument discussed above and the previous results obtained for ?nite systems,our results in Fig.6?a ?could be qualitatively anticipated,i.e.,t eff decreases monotonously with decreasing adiabaticity.To indicate that the adiabaticity dependence is a manifestation of charge ?uctuation correla-tions,Eq.?28?as well as the Lang-Firsov-normalized charge-transfer amplitude t eff /(te ??/2)are plotted in Fig.6?b ?as a function of ?for ??0.1,0.5,1.The connected points with solid circles,solid triangles,and solid squares represent the solution of Eq.?28?for ??0.1,0.5and ??1respec-tively.The LF-normalized solutions are indicated with the same type of unconnected points for the same ?values.The difference between the full and LF-normalized solutions is weaker for small couplings as expected.More importantly,the difference is also a function of the adiabaticity,decreas-

ing monotonously for decreasing ?.Hence,the qualitative

features of Figs.6?a ?and 6?b ?reasonably agree with those in Refs.5,7,and 8.

IV.EFFECTIVE ELECTRON-ELECTRON INTERACTION

The effective electron-electron interaction will be calcu-lated from

V eff e -e ?k ,???

V 0?k ?

??k ,??

,

?29?

where ?(k ,?)is given by Eq.?25?.At half-?lling,the cal-culations are shown for the Coulomb dominated bare inter-action in Figs.7?a,b ?for the real and imaginary parts of the inverse dielectric function,Since Re ?1/??is even and Im ?1/??is odd in ?,we only include the positive excitation energies.In the Coulomb dominated region,high-energy ex-citations across the Fermi surface ?i.e.,??2?and k ?(?,?)?are strongly susceptible to a sharp singularity in the electron density of states where a strong enhancement in the effective e -e coupling is observed.In the same limit Im ?1/??has a coherent peak for excitations across the Fermi energy,which is consistent with the known presence of high-energy dynamical CDW ?uctuations.In this regime,the qua-siparticle screening is inactive and the charge ?uctuations are dominated by high-energy processes.We observe that,for weaker bare Coulomb interaction strength the enhancement is also weaker ?not shown in Fig.7?.As the bare e -ph cou-pling is increased,the peak position shifts to lower energies due to the quasiparticle band narrowing and the CDW peak amplitude is much less pronounced.In contrast,in the low-energy excitation range ?i.e.,????,one enters the particle-hole continuum where the screening is active.In this regime,Re ?1/???1,which suppresses the effective e -e coupling be-low its bare strength.

At the other limit,where the net bare e -e coupling is phonon dominated,as shown in ?Figs.8?a,b ??,the high

en-

FIG.6.?a ?The effective charge transfer amplitude t eff /t as a function of ?for the indicated value of interaction parameters.?b ?The

adiabaticity dependence of t eff /t .

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ergy excitations become incoherent and the coherent CDW instability disappears.Note the presence of a minus sign on the vertical scale in Fig.8?a ?to indicate that the effective e -e coupling is attractive (0?Re ?1/??).In this regime,the particle-hole continuum is narrowed from below to interme-diate excitation energies where the screening is effective,resulting in a net suppression of the attractive coupling.The limitation of the particle-hole continuum at the low-energy end is dictated by the small polaron formation where a strong enhancement of the attractive coupling is observed.As the bare e -ph coupling is increased,the effective polaron mass is strongly enhanced within a low-energy window and the in-teractions are dominated by low-energy exchange processes.With increasing bare attractive coupling,the low-energy window is compressed to even lower energies,apparently approaching to a ?-like peak at ??0for 1??.For an in-creasing bare e -ph coupling constant,the pergence in the behavior of Re ?1/??is also consistent with the gradual de-velopment of the sharp low-energy peak in Im ?1/??in Fig.8?b ?.We believe that this is an indication of the existence of a very narrow band,itinerant,small ?quasilocalized ?polarons in this low-energy regime.In the ultimate limit of very large e -ph coupling the small polaron band is reduced completely,the effective adiabaticity is strongly decreased and,the effec-tive e -e coupling is strongly renormalized signaling a gradual transition from the itinerant,?uctuating low-energy small polaron picture to self-trapped polarons.Since the cou-pling is strongly attractive,bipolaron bound-state formation is also likely to happen within this range.

Figures 7?a,b ?and 8?a,b ?con?rm the general wisdom 2–4,20that,the electron self-energy as well as vertex corrections are particularly strong across the Fermi surface both in the high-energy Coulombic and low-energy phonon dominating re-gimes.To complete the picture at half-?lling,the k depen-dence of the dielectric function is plotted in Figs.9?a,b ?for ?/?0?8.05,??0,V c (?,?)/?0?4,and Figs.10?a,b ?for ?/?0?0.05,??1.6,V c (?,?)/?0?0.These particular ?values correspond to the vicinity of excitation energies in Figs.7?a,b ?and 8?a,b ?where the peak positions are observed.Hence,Figs.9?a,b ?and 10?a,b ?give representative samplings of the dielectric function in the extreme high-energy Cou-lombic and low-energy phonon dominated regimes and where the strongest ?,k dependence is expected.In the former ?Figs.9?a,b ??a relatively smooth and dispersionless CDW gap is present on the Fermi surface.Across the Fermi surface at k ?(?,?)there is an enhancement both in Re ?1/??and Im ?1/??indicating the dynamical CDW peak in Figs.7?a,b ?.On the other hand,we ?nd in the latter case ?Figs.10?a,b ??that in the presence of a strong attractive coupling the gap ?uctuates at very low energies ?e.g.,?/?0?0.05?,and it is strongly anisotropic on the bare Fermi surface.For instance,at k ?(0,?),and at ??,0?the Re ?1/??it is rather ?at and narrow with no structure in the imaginary part,whereas across the bare Fermi surface towards k ?(?,?)it is strongly k dependent and dynamical with the large dynami-cal small polaron peak at k ?(?,?)?see also Figs.8?a,b ??.An extension of these results to the case away from half ?lling as well as different values of the bare

charge-transfer

FIG.7.?a ?The real part of the vertex renor-malization for the effective e -e coupling as a function of the excitation energy ?in the Cou-lomb dominated regime at indicated values of the interaction parameters.?b ?Same as ?a ?for the the imaginary part.

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amplitude also indicate that the ?,k dependence of the self-energy and vertex corrections maintain their full validity at a qualitative level.Because of the strong ?,k dependence of high-energy excitations in the Coulombic case,the position and the amplitude of the dynamic CDW peak is strongly sensitive to slight changes in the electron concentration.We observed that in the region where low-energy phonon domi-nated excitations are strong,there is an overall suppression in the magnitude of the low-energy excitations on the Fermi surface as well as at k ?(?,?)when the concentration is shifted away from the half-?lling.

The density of states on the Fermi surface is strongly dependent on the strength of the charge-transfer amplitude.For t ?0.7,at half-?lling and in the Coulomb dominated case,we observed an order of magnitude enhancement on the Fermi surface in the effective e -e interaction.The last ex-ample is the extreme phonon dominated region at t ?0.7at low energies.There,the previously observed low energy small polaron peak is enhanced and broadened in the vicinity of k ?(?,?).In addition to that,two dynamical peaks ap-pear in symmetric position at k ?(0,?)and ??,0?.In all ex-amples we examined,relatively more structure is observed in the k space in the phonon dominated regions than in the Coulombic ones.

The strong sensitivity of the vertex corrections as func-tions of ?,k on the bare interaction parameters and the elec-tron concentration renders the analysis delicate particularly near the instabilities.It has been argued that,in the presence of strong short-range Coulombic or magnetic correlations,the strong enhancement in the phonon-mediated effective at-

traction can drive the system into superconductivity near the dynamical CDW instability.20We believe that this mecha-nism might be more likely to happen ?if it does ?in the strongly antiadiabatic ranges in otherwise the same regime where the phonon excitation energies are more compatible with the electronic ones.On the other hand,Coulomb domi-nated strong coupling antiadiabatic ranges,where the excita-tions are on the order of bare phonon frequency or smaller with exchange momenta on the order of k ?(?,?),are also favored by the small polaron formation.Hence the competi-tion in this regime between the superconductivity and quasilocalized polarons,must be decided by the effective adiabaticity as well as the coupling constants.This renders the analysis of the competing effects of the vertex (?eff )and phonon (?k )self-energy against the electron self-energy (t eff )renormalizations to be particularly critical near these instabilities.

V.RENORMALIZED PHONON SUBSYSTEM

A.Phonon number distribution

We now examine the distribution of the number of phonons p (n k )in the approximate ground state ???eff ?by

p ?n k ????n k n ?k ???eff ??2

.

?30?

Since ???eff ?is de?ned in terms of pair excitations we con-sider n k ?n ?k ,which allows us to use Yuen’s formula,

15,16

FIG.8.?a ?The real part of the vertex renor-malization for the effective e -e coupling as a

function of the excitation energy ?for the pho-non dominated regime at the indicated interaction parameter values.?b ?Same as ?a ?for the imagi-nary part.

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?n k,n k?S??????0????2n

k

?!

n k!

?tanh?2??k???n k

?cosh?2??k???1/2,?31?

in the calculation of Eq.?30?.We?nd that

?n k n?k???eff???n k,n k?S??????0???k??k?k?k?2n k?1?

??k?k2?k2?3n k2?3n k?1??

??n k?1,n k?1?S??????0??k?k2n k

??n k?1,n k?1?S??????0??k?k2?1?n k?

??n k?2,n k?2?S??????0??k4n k?n k?1?

??n k?2,n k?2?S??????0??k4?n k?1?

??n k?2?.?32?Using Eq.?32?and?31?,the phonon number distribution in

Eq.?30?is plotted for different values of?and?,n0and V c/?0at k?(?,?)in Figs.11?a–d?.The values of the cou-pling constants are chosen suf?ciently below and suf?ciently

above the critical crossover of the two solutions near??1in

Fig.5?a?so that p(n k)is calculated using the?rst solution

for?1and?2and the second one for?3.A common feature

of Figs.11?a–d?is that,for suf?ciently small?i.e.,???1?, the phonon probability distribution is always the largest at

n k?0.As?increases,the maximum value is smoothly shifted towards?nite number of phonons and the overlap with the vacuum state decreases.As the system is driven into antiadiabatic ranges,as shown in Fig.11?b?,there is an over-all decrease in the dynamical charge?uctuation correlations where the phonon distribution is narrower and the overlap with the vacuum is strongly increased.A comparison be-tween Figs.11?a?and11?b?indicates that there is a delicate competition between?and?to determine the shape of the probability distribution.The decreasing?tends to compress the distribution towards n k?0by increasing the vacuum component.On the other hand,a weak?i.e.,???1,?2?but

increasing?broadens the distribution and attempts to shift it away from the vacuum,where it?ghts against the stabilizing effect of the decreasing?.Whereas,if?is strong?i.e.,???3?,the increasing?cooperates with the decreasing?to

stabilize the coherent polaron formation as indicated by the increasing n k?0component in p(n k).We identify the co-operation of increasing?and decreasing?as the correct route to the Lang-Firsov limit in which the dynamical com-ponent of the probability distribution very strongly overlaps with the vacuum where the phonon statistics is driven by the dominating coherent part.

A similar competition is observed in Figs.11?a?and11?c?between the e-ph and the Coulomb interactions,as well as in Figs.11?a?and11?d?for different electron concentrations. When?is weak,increasing?competes with the stabilizing effects of Coulomb interaction or reduced electron concen-tration.When?is strong,it cooperates with them to stabilize the coherent polaron formation.We observe that the overall picture here is also consistent with the results of de Mello and Ranninger in Ref.5.

It should be noted that the nonclassical structure of p(n k) is entirely a manifestation of the dynamical?uctuations.The ?uctuating part given by???eff?in Eq.?15?of the true ground-state wave function does not support any structural changes ?i.e.,???eff?Q k???eff?????eff?P k???eff??0as also enforced by Eqs.?18??.Hence,the decomposition of the wave function in the correlation space also enables one to examine the dynamical and static parts of the distribution function independently. The true probability distribution is obtained by a convolution between the dynamical and static coherent sectors of the wave function.The static coherent sector yields the non?uc-tuating Poisson distribution,which is not addressed in this paper.

B.Renormalized frequency of vibrations

In principle,the phonon frequency renormalization should be calculated by?nding the corresponding effective phonon Hamiltonian for which the dynamical polaron wave function in Eq.?15?is the lowest eigenstate.This would be a tedious, but relatively straightforward inverse eigenproblem if we could write the operator in Eq.?15?in the form of an invert-ible unitary operator acting on the phonon vacuum state.In the following,we will present our results instead,using the RPA where the phonon self-energy?(k,?)is calculated

by FIG.9.?a?The real part of the vertex renormalization for the effective e-e coupling in the Coulomb dominated regime at the peak value?/?0??2?/?0?8in k space for V c/?0?4,??0,??1and n0?1.?b?Same as?a?for the imaginary part?note the negative sign on the vertical scale?.

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??k ,???V 0?k ?P ?k ,??,?33?

where P (k ,?)is the electron polarization given by Eq.?26?.We will present our results for the phonon dominated regime without Coulomb interaction.Hence V 0(k )???.The RPA is known to yield compatible results to the self-consistent ME calculations 2,4in the relatively weak-coupling constant ranges ??0.5whereas it strongly overestimates the dynami-cal phonon softening for 0.5??as compared to more reli-able QMC simulations.2In the conventional RPA the renor-malized phonon frequency is given by

?k ???k 2?2?k ??k ,?k ?,

?34?

where bare electron Green’s functions are normally used in

the calculation of P (k ,?).Using Eq.?34?,we plot in Fig.12the renormalized phonon frequency ?k in the RPA ?thin solid lines ?as a function of ?for ??0.3,0.4,1and for no Coulomb repulsion.The ME calculations ?dotted lines ?and QMC results ?with error bars ?of Ref.2for ??1are also included for comparison.It is known that the conventional RPA overestimates the charge ?uctuation correlations due to neglected corrections of the self-consistent renormalizations

in the electron self-energy and the coupling constant.2–4,20

This is re?ected in an unbounded negative increase of the phonon self-energy,which in turn derives the renormalized phonon frequency into an instability for the intermediate and strong-coupling ranges 1??.

If the vertex corrections are properly included,in the at-tractive case,the effective e -ph coupling constant ?eff ??Re ?1/?(k ,?)?is suppressed for high frequency excita-tions due to the charge screening effect and is enhanced in the low frequency range due to the small polaron formation ?see the coupling constant renormalization in Sec.IV Fig.8?a,b ??.On the other hand the electron self energy is also re?ected upon the renormalization of the charge transfer am-plitude t eff of which the band narrowing effect,according to Fig.6?a ?,is to derive the system into an effectively nonadia-batic range.Hence a physically more relevant calculation should properly include both corrections which is suggested by replacing ?(k ,?k )→?eff (k ,?k )in Eq.?34?where the latter is calculated with t ???LF →t eff where t eff is now given by Fig.6?a ?,and,with ?→?eff where ?eff ??Re ?1/?(k ,?k )?is calculated in Fig.8?a ?.The self-consistent solution of

?k ???k 2?2?k ?eff ?k ,?k ?,

?35?

which we term as the corrected RPA ?CRPA ?,is technically different from those calculations using ?nite lattice and elec-tron degrees of freedom where it is numerically feasible to maintain the self-consistency from the beginning.4The solu-tion of the CRPA is depicted in Fig.12with the thick solid line as a function of the bare coupling constant ?.In the solution of CRPA,we were not able to beyond ??1.6due to an unstability in the numerical calculations in Eq.?35?.Nev-ertheless,the agreement with the QMC results for a reason-ably large range of e -ph coupling clearly indicates the im-portance of the vertex as well as the self-energy corrections in the antiadiabatic strong-coupling case.The picture can be made more transparent if one pides the ?range in Fig.12by imaginary lines into the weak-coupling ??0.5,intermediate-coupling 0.5???1.2,and strong-coupling 1.2??sectors and compare the ??1RPA solution where such renormalizations are not present with the ??1CRPA solu-tion where they are included.In the weak sector,the phonon softening is weak and typical excitation energies are on the order of bare phonon frequency where the charge screening effects weakly suppress the coupling constant ?i.e.,Re ?1/???1?.By the weak screening in this sector,further softening of phonons is slightly delayed to the larger coupling strengths.In the intermediate range,the charge ?uctuations become important where the electron self-energy and vertex corrections compete to determine the phonon softening.This can be qualitatively understood by the following argument.As ?is increased in the intermediate range,the band narrow-ing effect of the electron self-energy corrections tend to op-pose further softening,but in the intermediate sector the pho-non frequency is already suf?ciently softened and the low-energy excitations slowly start dominating as a precursor of the ?uctuating polaronic regime where the large low-energy vertex corrections enhance the effective coupling constant 1?Re ?1/??.Hence,more softening is observed.On the other hand,in the third sector at relatively large coupling

con-

FIG.10.?a ?The real part of the vertex renormalization for the

effective e -e coupling in the phonon dominated regime at the peak value ?/?0?0.05in k space for V c /?0?0,??1.6,??1and n 0?1.?b ?same as ?a ?for the imaginary part.

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stants,the outcome of the competition between the electron self-energy and vertex corrections is decided by the bare adiabaticity parameter ?.At this point,it is necessary to go back and examine the renormalization of the charge-transfer amplitude in Fig.6?a ?for various values of ?.For interme-diate and large values of ?,the band reduction is opposed by the suppression factor exp(?4?k )in Eq.?21?arising from the strong presence of the two-particle coherent ?i.e.,0???,?,??,??1?and,the two-particle pair excitations ?i.e.,0???,??in the ground state.The net effect of the coherent two-particle pair excitations is to slow down the rapid reduc-tion of the electron band as ?increases.The in?uence of this factor has also been noticed in the variational calculations in the intermediate and strong couplings as well as intermediate and low excitation energies in the phonon spectrum.21,22We observe in Fig.6?a ?that,this effect is visible for ??1by the presence of a bulge near ??0.6and the decrease of t eff /t for increasing ?is much slower for the larger values of ?.This implies that,a smaller ?yields a more rapid band reduction,resulting in a stronger suppression of the charge ?uctuations.In the strongly antiadiabatic regime,the increasing e -ph cou-pling cooperates with the strong nonadiabaticity ?as also ob-served in Fig.11?b ??and the phonon softening is completely destroyed.This is indicated in Fig.12by the thin solid lines corresponding to ??0.4and 0.3.On the other hand,for larger ?,the phonon softening can continue in the presence of marginal charge ?uctuations.For instance,for ??1and for the CRPA solution,as ?is increased further,the charge ?uctuations decrease,leading into a ?nite saturation regime where the phonon softening is relatively unchanged with ?.

VI.CONCLUSIONS

In this work,we improved and extended the dynamical charge ?uctuation based effective wave-function scheme of

our previous work in Ref.13to the normal state in the two-dimensional Holstein-Hubbard model in the intermediate in-teraction ranges.In particular,the possibility of representing the effective wave function in the decoupled subspaces of n -phonon cumulant correlations is exploited and applied to the ?rst two cumulants of the polaron wavefunction.The differences of this approach from the diagrammatic phonon correlator technique of Ref.12as well as the standard Lang-Firsov approaches are emphasized by showing that the nu-merically observed weak convergence to the LF theory in the strong-coupling antiadiabatic limit is inherently built in this model.With the effective cumulant approximation,one is able to construct an effective many-body wave function and compare the results at a qualitative level with the recent nu-merical studies on direct diagonalization,QMC,and varia-tional approaches.The effective wave function provides a clear picture of the dynamical coupling of the correlated pho-non pair ?uctuations to those in the CDW.In this respect,we consider the current work as a possible dynamical many-body extension of these studies.

As far as the general polaron problem is concerned,the decoupled nature of the effective wave function in the cumu-lant correlation space might be a promising tool to under-stand the properties of the polaron ground state at a deeper level.This procedure also decouples the static coherent sec-tor from the dynamical ?uctuating part of the wave function.In this article we took this as an advantage to study the dynamical sector independently.The authors believe that the possible improvements of this extended LF-like approach can be done in two directions.At ?rst one can realize that,the true ground state ?as suggested by the multiphonon scat-tering operator ?(m ,n )?has corrections to the coherent part even at the dynamical level,and,the true ground-state wave function includes a dynamically shifted mixture of

coherent

FIG.11.The dynamical phonon distribution in the effective wave function for the indicated values of the parameters.

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states of the phonon coordinate Q k and momentum P k .The static contribution is a Q k coherent state,which is precisely what we called ??c ?in this article.In ??c ?we have neglected these pure dynamical corrections,although a more rigorous treatment should also embody those effects self-consistently.A second means of improvement is in the understanding of ????itself.At this point,some formal dif?culties arising from the formulation of the unitary generators of the m -phonon cumulant correlations for 3рm have to be over-come.The cumulant correlation corrections for 3рm also depend heavily on the corrections to the Landau-Fermi liquid picture.The reason behind this is that higher cumulants in

????are more susceptible to deviations from the standard assumption of Gaussian density ?uctuation correlations in the Landau-Fermi liquid.This assumption was indeed used in the calculation of Eqs.?11?.In this respect,these two corrections to phonon as well as fermion statistics should be attacked simultaneously in a more re?ned self-consistent frame.Possible advances made in this direction might reveal the importance of these deviations and might also shed light on the likely presence of the not-completely-understood strongly nonlinear self-trapping regime both in the Coulom-bic high-energy and phonon dominated low-energy sectors.Although the revival of the Holstein-Hubbard model in the past 15–20years was heavily stimulated by the progress in high-temperature superconductivity,we did not enter into such discussions in this 8e50d6addd3383c4bb4cd230ing an oversimpli?ed model,it was suggested in Ref.13that the low-temperature T c -dependent phonon anomalies observed in certain Cu-O–based compounds might be connected with the dynamical vibrational ?uctuations self-consistently coupling to the po-laronic charge ?uctuations in the superconducting phase.It should be noted that a more realistic model for high-temperature superconductors is suggested by the Holstein–t -J model in the presence of strong Coulomb correlations with the electron concentration being slightly shifted away from half-?lling where the vibrational ?uctuations strongly couple in a self-consistent frame to charge but also spin ?uc-tuations in the Cu-O planes.One then has to incorporate all self-energy and vertex corrections in the Coulomb dominated regime,both for the ?uctuations in the charge and spin de-grees.Hence,one possible direction to take in the supercon-ducting phase is to examine the Holstein–t -J model within the ?charge and spin ??uctuation-based effective cumulant approach presented here.

ACKNOWLEDGMENTS

T.H.is grateful M.Arai,C.H.Booth and,in particular,to N.Bulut for helpful and stimulating discussions.Both au-thors are indebted to V.A.Ivanov for the exchange of cru-cial ideas.M.Y.Z.is grateful to TU

¨BI ˙TAK ?Scienti?c and Technical Research Council of Turkey ?and Bilkent Univer-sity for support and hospitality.

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