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Gutzwiller scheme for electrons and phonons: the half-filled Hubbard-Holstein model

P. Barone1, R. Raimondi2, M. Capone3,4, C. Castellani3, and M. Fabrizio1

1International School for Advanced Studies (SISSA) and CNR-INFM-DemocritosNational Simulation Centre, Via Beirut 2-4 34014 Trieste Italy

2CNISM and Dipartimento di Fisica “E. Amaldi” Universita di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy3SMC, CNR-INFM and Dipartimento di Fisica, Universita di Roma “La Sapienza”, P.le Aldo Moro 2, 00185 Roma, Italy and

4 ISC-CNR, Via dei Taurini 19, 00185 Roma, Italy(Dated: July 29, 2008)

We analyze the ground-state properties of strongly-correlated electrons coupled with phonons bymeans of a generalized Gutzwiller wavefunction which includes phononic degrees of freedom. Westudy in detail the paramagnetic half-filled Hubbard-Holstein model, where the electron-electron in-teraction can lead to a Mott transition, and the electron-phonon coupling to a bipolaronic transition.We critically discuss the quality of the proposed wavefunction in describing the various transitionsand crossovers that occur as a function of the parameters. Previous variational attempts are recov-ered as particular choices of the wavefunction, while keeping all the variational freedom allows toaccess regions of the phase diagram otherwise inaccessible within previous variational approaches.

I. INTRODUCTION

Understanding the interplay between electron-phonon(e-ph) and strong electron-electron (e-e) interactions canbe important to understand the properties of many com-pounds, such as manganites1 or fullerides2. This problemhas recently received a renewed attention, after a numberof experiments on high-Tc superconducting cuprates tes-tified for a non trivial role of the e-ph coupling in the un-derdoped region, where correlation effects are importantand standard e-ph theories are hardly reliable3,4,5,6,7. In-deed a full theoretical understanding of lattice effects instrongly-correlated systems is still lacking when the twointeractions have intermediate or large strength.

The essential features of a strongly-correlated systemare captured in the celebrated Hubbard model8 and inits extensions. On the other hand, polaron formationdue to e-ph coupling has been extensively studied in theabsence of e-e repulsion, mainly in the limit of low elec-tron density within the molecular-crystal model first in-troduced by Holstein9. Nonperturbative approaches suchas Quantum Monte Carlo, Density Matrix Renormaliza-tion Group and Dynamical Mean-Field Theory (DMFT)have enormously contributed to the understanding ofthese models and have been successfully applied tothe Hubbard-Holstein model10,11,12,13,14,15. Nonethelessthese methods require a considerable numerical effort,and it is not unfair to say that approximate analytic re-sults, even if less accurate, are still highly desirable.

From this point of view, variational approaches area very natural choice as they provide immediate in-formation on the ground-state properties. In the fieldof strongly-correlated systems, the trial wavefunctionintroduced by Gutzwiller16 and then generalized byBunemann and coworkers17 has proved quite accuratefor electronic systems with local interactions. The un-derlying recipe is to construct the correlated wavefunc-tion starting from an adequate uncorrelated one andprojecting out the energetically unfavored states. Theequivalence with the Kotliar-Ruckenstein (KR) slave-

boson saddle-point solution18 allows also to go beyondthe mean-field approximation in a controlled way. Forthe Holstein model several trial wavefunctions have beenproposed generalizing the Lang-Firsov transformation19,which yields the exact solution in the atomic limit,and it is believed to capture the essence of polaronicphysics, i.e., the entanglement between electronic andphonic degrees of freedom. These kind of approacheshave been successfully applied to the Holstein model,mainly in the limit of low carrier density or for spin-less electrons20,21,22,23,24,25. In fact, a further complica-tion arises at finite density, since the e-ph coupling in-duces an effective e-e attractive interaction that cannotbe handled in an independent particle picture. When theHubbard repulsion is included, it directly competes withsuch phonon-mediated attraction.

The typical strategy to treat correlated fermions cou-pled with phonons consists in deriving an effective modelfor electrons by averaging the full Hamiltonian overa suitable trial phonon wavefunction (which also im-plies the neglect of any electron-multiphonon residualinteraction)26, and then to resort to some appropriatetechnique in order to tackle the electronic effective in-teraction. This can be done for instance by means of aHartree-Fock decoupling scheme23 or exploiting the moreaccurate KR slave-boson mean field27,28,29 suitable forlocal e-e interactions. As discussed in details in Refs.24,30 the assumption of a factorized wavefunction be-comes questionable in the regime where retardation ef-fects between the motion of electrons and lattice defor-mations imply a strong exchange of momentum.

In order to better describe the interplay between latticevibrations and local electronic configurations, we intro-duced in a previous paper31 a Gutzwiller Phonon Wave-function (GPW) to account for the e-ph coupling be-sides strong electronic correlation and benchmarked itin the infinite-U limit of the Hubbard-Holstein modelby comparing with exact DMFT calculations. In thislimit a finite phonon-driven attraction is obviously inef-fective against the infinite repulsion, and the only effect

2

of phonons is the formation of polarons.

In this paper we turn our attention to the finite-Uregime, analyzing in details the paramagnetic solution(i.e., neglecting broken-symmetry phases). Relaxing theinfinite-U assumption, the competition between repul-sion and attraction becomes effective, leading to a richphase diagram. We focus on the half-filling case, in whichthe e-e and e-ph interactions can drive, respectively, theMott and bipolaronic metal-insulator transition, so thattheir interplay is both more transparent and more qual-itative than for arbitrary fillings. Out of half-filling thesystem is indeed always metallic in the absence of sym-metry breaking, and the effect of the competition is es-sentially quantitative and therefore less clear32.

The key property of our GPW wavefunction is to treatsimultaneously electrons and phonons from the onsetby introducing a generalized Gutzwiller projector wherephonon quantum states are determined by the local elec-tronic configurations rather than by the average electrondensity. This allows to address the modification inducedby the electronic correlation onto the lattice groundstate.Therefore, even if the multiphonon adiabatic regime ofthe weakly-correlated system is probably still misrepre-sented, we expect to capture the essential features ofthe model in the presence of a sizeable repulsive inter-action, when the relevant physics is essentially local be-cause the electron motion is slowed down by correlations.We derive a set of variational equations for the phononwavefunctions and show that standard approaches canbe obtained as particular choices for the solution of suchequations. We obtain a phase diagram which strongly de-pends on the adiabaticity regime, and critically discussour results and the quality of the proposed wavefunction.

The paper is organized as follows. In Section II weintroduce the model Hamiltonian and the relevant equa-tions for the general case of arbitrary electron filling, andthen we specialize them for the half-filling case we are in-terested in. At the end of this section we derive standardvariational approaches for the Holstein problem as partic-ular variational ansatzs on the Gutzwiller wavefunction.In Section III we present the general solution provided byour method for the half-filled system, whose propertiesare analyzed in the following Section IV. We then discussin Section V the quality of the proposed wavefunction bycomparing it with other variational wavefunctions.

II. MODEL AND METHOD

The Hamiltonian of the Hubbard-Holstein model athalf-filling reads:

H = −t∑

<i,j>,σ

c†iσcjσ +U

2

∑

i

(ni − 1)2 + ω0

∑

i

a†iai

+αω0

∑

i

(ni − 1)(a†i + ai), (1)

where ciσ (c†iσ) and aiσ (a†iσ) are respectively annihila-tion (creation) operators for tight-binding electrons withspin σ and for optical phonons of frequency ω0 on sitei, t is the nearest-neighbor hopping amplitude, U the lo-cal Hubbard repulsion and α parameterizes the couplingbetween local displacements and electronic density fluc-tuations ni−1. Both interaction terms have been writtenin a particle-hole symmetric form which enforces the half-filling condition. We introduce an adiabaticity parame-ter defined as γ = ω0/D, measuring the ratio betweenthe phonon and electron characteristic energies, and adimensionless parameter λ = 2α2ω0/D = 2α2γ whichmeasures the strength of the e-ph coupling. Here D isthe half bandwidth, which depends on the hopping pa-rameter t once a particular lattice is chosen. We performour calculations in an infinite-coordination Bethe lat-tice with semicircular density of states of half-bandwidthD = 2t.33 For this lattice, as in any infinite-coordinationlattice, the expectation value of the Hamiltonian on theGutzwiller state can be performed without further ap-proximations, because all the configurations with thesame average occupation number equally contribute tothe quantum averages. The same approach can be usedalso for finite-dimensional lattices, where it represents afurther approximation, usually called Gutzwiller approx-imation. Another advantage of considering the infinite-coordination Bethe lattice is the possibility to comparewith the exact results for this lattice, obtained throughDMFT.12,13,14,15

We introduce the GPW as31

|Ψ〉 = ΠiPi(xi)|Ψ0〉, (2)

where |Ψ0〉 is a Slater determinant, that we take as thenon-interacting Fermi sea in order to describe metallicstates. Pi(xi) is a generalized Gutzwiller projection op-erator

Pi(xi) =∑

ν=0,1,2

√

Pν

P(0)ν

φν(xi)|νi〉〈νi|. (3)

Here |νi〉〈νi| are the projectors associated to the differentlocal electronic states, empty site (ν = 0), singly occu-pied site (ν = 1) and doubly occupied site (ν = 2). Ourwavefunction therefore associates to each local electronicstate a normalized first quantization phonon wave func-tion φν(xi) depending on the displacement coordinate

xi = (ai + a†i )/√

2, which has to be determined varia-tionally, exactly like the probabilities of each of the local

states Pν (P(0)ν are the same quantities for the uncorre-

lated system, i.e., P(0)0 = (1−n/2)2, P

(0)1 = n/(1−n/2),

and P(0)2 = (n/2)2, n being the average density per site).

By imposing the standard constraints17,34

∫

dxi 〈Ψ0|Pi|2|Ψ0〉 = 1, (4)

∫

dxi 〈Ψ0|ni|Pi|2|Ψ0〉 = n, (5)

3

and introducing the parameter d = (P0 + P2)/2 andthe doping δ = 1 − n, one gets P0 = d + δ

2 , P1 =

1−2d andP2 = d− δ2 for the correlated occupation prob-

abilities. Setting δ = 0, the three amplitudes are con-trolled by a single variational parameter, d. The varia-tional energy per site is

E =∑

ν=0,1,2

Pν〈h0(x)〉ν +√

2αω0 [P0〈x〉0 − P2〈x〉2]

− 2|ε||S|2 +U

2(P2 + P0) (6)

where h0(x) = (ω0/2)(−∂2x + x2)) and

〈O〉ν ≡∫ ∞

−∞dx φ∗ν(x)Oφν (x) (7)

indicates the average over the phonon wave functionφν(x) of the operator O. The second line of Eq.(6) repre-sents the standard energy of the Gutzwiller approxima-tion for the Hubbard model with

2|ε| = t 〈Ψ0|∑

<i,j>,σ

c†iσcjσ|Ψ0〉 (8)

the free electron kinetic energy and

S =∑

ν=0,1

√

PνPν+1

1 − δ2

∫

dx φ∗ν+1(x)φν (x) (9)

the renormalization factor that accounts for the reducedelectron mobility in the presence of e-ph and e-e interac-tions. Then we have to minimize with respect to d andthe φν(x). The first minimization gives

U + (〈h0(x)〉0 + 〈h0(x)〉2 − 2〈h0(x)〉1)

+√

2αω0(〈x〉0 − 〈x〉2) − 2|ε|∂|S|2

∂d= 0, (10)

while the second yields the following non-linear second-order differential equations

ǫ0P0φ0 = h0(x)φ0 +

√2αω0xφ0 − 2|ε|√

1−δ2S

√

P1

P0

φ1 (11)

ǫ1P1φ1 = h0(x)φ1 − 2|ε|√

1−δ2

(

S∗√

P0

P1

φ0 + S√

P2

P1

φ2

)

(12)

ǫ2P2φ2 = h0(x)φ2 −

√2αω0xφ2 − 2|ε|√

1−δ2S∗

√

P1

P2

φ1(13)

where the ǫν ’s are Lagrange multipliers enforcing the nor-malization conditions on the φν ’s.

Before discussing in some detail the solution ofEqs.(11)-(13), we briefly analyze their structure. In theabsence of e-ph coupling they reduce to three equivalentSchrodinger equations for a free harmonic oscillator cen-tered at each site independent on the electron occupancy.Therefore the phonon wavefunction is the standard gaus-sian, contributing ω0/2 to the on-site energy. Switchingon the e-ph interaction has two major effects. The first

is to induce a shift proportional to ±√

2α to the phononwavefunctions associated to the ν = 0, 2 electron states,which is the expected effect in the atomic limit with thechosen form of the e-ph interaction; the second is a non-local coupling between wavefunctions related to differentcharge configurations, which appears to be proportionalto the average kinetic energy of the electrons. This is notsurprising, since one expects the electron dynamics to af-fect the phononic properties driving them away from theatomic limit scenario. Nonetheless, it is readily seen thatthis coupling term is inversely proportional to the adi-abaticity ratio, that means, as it is well-known, that inthe antiadiabatic regime, i.e. when phonons move fasterthan electrons, the interplay of phonon and electron dy-namics is negligible: the lattice rearranges itself almostinstantaneously with respect to slowly moving electronsand a charge-dependent shift of the phonons is sufficientto capture the ground-state properties of the system. Onthe other hand the coupling term becomes dominant inthe opposite limit, i.e. for γ ≪ 1; in this case each pro-jected wavefunction depends heavily on the other wave-functions and on the electron probability distribution,thus suggesting non trivial dependence on correlation ef-fects.

At half-filling we have P0 = P2 = d and P1 = 1 −2d, that imply φ1(−x) = φ1(x) and φ2(−x) = φ0(x) forthe lowest-energy state. Therefore we are left with twovariational equations for the lattice ground state coupledwith Eq. (10). The latter can be recast in the moretransparent form

4d = 1 − u, (14)

which depends from the important quantity

u =U − 2

√2αω0 〈x〉2 − 2 〈h0(x)〉1 + 2 〈h0(x)〉2

8 |ε0||〈φ2|φ1〉|2(15)

measuring the effective degree of electronic correla-tion as the ratio between renormalized Ueff (nu-merator) and phonon-renormalized kinetic energy εeff

(denominator)29.In spite of these simplifications, an analytical solution

of our equations is still a difficult task. In the next sectionwe present the general solution of Eqs. (11)-(13). Herewe briefly discuss some specific choices for the phononwavefunction that allow for an analytical solution andreproduce popular variational approaches.

a. Gaussian ansatz for the phonon wavefunctions.

As discussed above, in the atomic limit the phonon wave-functions reduce to displaced harmonic oscillators whosedisplacement x0 depends on the local electron occupationaccording to x0(ν) =

√2α(ν − 1). We can assume:

|φν〉 = ei√

2α fν p|0〉 (16)

where |0〉 is the ground state of an undisplaced harmonic

oscillator, p = −i(a − a†)/√

2 is the conjugate momen-tum and fν is a parameter to be variationally deter-mined. Exploiting the particle-hole symmetry we can

4

impose f2 = −f0 = f and f1 = 0 and obtain the varia-tional ground-state energy per-site:

E0 =ω0

2− 8d(1 − 2d) e−α2f2 |ε0|+

d[U − 2α2ω0 f (2 − f)], (17)

that corresponds, within a constant, to the result ofKR slave-bosons supplemented by the variational Lang-Firsov transformation (VLF)28,29. This is not surprising,since the Gutzwiller approach is known to be equivalentto the slave-boson mean-field approach and the VLF de-scribes the phonon wavefunctions as displaced Gaussians.

The mean-field solution is then determined by min-imizing (17), that amounts to solve Eq.(14) with u =exp(α2 f2) [U − 2α2ω0 f (2 − f)]/8|ε0| together with29

f =

[

1 +2|ε0|ω0

(1 + u) e−α2f2

]−1

. (18)

b. Squeezed phonon state ansatz for the phonon

wavefunctions. As widely discussed for the Holsteinmodel23,24,26,27, the harmonic (gaussian) ansatz is notexpected to be accurate except for the so-called light po-laron case, that is realized when anharmonic fluctuationsof the lattice are not so important, i.e. when λ, γ ≪ 1 orwhen γ ≫ 1, |Ueff |26. Such anharmonic fluctuations canbe partially captured by assuming that phonon wavefunc-tions are “squeezed” two-phonon coherent states (Varia-tional SQueezed, VSQ). The simplest choice that is usu-ally made is:

|φν〉 = e−α fν(a†−a) e−F (aa−a†a†)|0〉, (19)

where the e-ph induced displacement of the phonon fieldis still associated to the local charge state whereas thevariational parameter F > 0 controlling the squeezing ofthe wavefunctions is charge-independent, i.e., as noticedfirst in Ref. 24, it describes a kind of mean-field effect onthe lattice due to the electron motion. The ground-stateenergy computed over these phonon wavefunctions reads:

E0 =ω0

4(τ2 + τ−2) − 8d(1 − 2d) e−α2f2τ2 |ε0|+

d[U − 2α2ω0 f (2 − f)], (20)

where we have introduced τ2 = exp(−4F ). This is whatone would obtain by applying the slave-boson mean fieldto the effective polaron model derived by averaging oversqueezed phonon states23,26. The mean-field equationsare modified as follows:

f =

[

1 +2|ε0|ω0

τ2 (1 + u) e−α2f2τ2

]−1

, (21)

τ =

[

1 +4|ε0|ω0

α2 f2 (1 − u2)e−α2f2τ2

]−1/4

, (22)

with u = exp(α2f2τ2) [U−2α2ω0f(2−f)]/8 |ε0|. While fstill accounts for the effective e-ph-induced displacementof the ions, the variational parameter τ can be viewedas a measure of anharmonic fluctuations of the phononwavefunctions.

III. HALF-FILLING PHASE DIAGRAM.

As we briefly discussed in the introduction, the param-agnetic phase of the half-filled Hubbard-Holstein modelis an interesting test-field for our trial wavefunction sincethe interplay of the two local interaction mechanismsis more effective and at the same time more transpar-ent because both the interaction terms are able to drivemetal-insulator transitions of different nature. From atechnical point of view, imposing the particle-hole sym-metry simplifies the calculation, eliminating of the threeself-consistency equations for the phonon wavefunctions.Finally, the half-filled phase diagram has been exten-sively studied by means of DMFT, providing us withan exact benchmark for the infinite coordination Bethelattice.12,13,14,15

In this section we mainly focus on the metal-insulatortransitions driven by the two interaction terms. In theabsence of e-ph coupling a transition from a metal toa Mott insulator (MI) occurs at a critical Uc, whereasthe half-filled Holstein model displays with increasing λa transition to a pair (bipolaron) insulator (BPI), theoccurrence of such instabilities depending on the adia-baticity regime35.

In the present framework, a key parameter to charac-terize the properties of the system is the effective corre-lation parameter u. From Eq. (14), u = 1 correspondsto d = 0, and u = −1 to d = 1/2. These are suffi-cient conditions for, respectively, the Mott and bipola-ronic transitions. In fact, in the present mean-field de-scription, the vanishing of the average double occupa-tion signals the transition to a MI, with one electron persite, whereas d = 1/2 corresponds to a system of elec-tron pairs stuck to lattice sites, i.e., a BPI. Once theexistence of the insulating solution is proved, their ther-modynamic stability must however be checked by com-paring their ground-state energies (i.e., E0(MI) = ω0/2and E0(BPI) = [ω0 + U − 2α2ω0]/2) with that of otherpossible (metallic) solutions. To gain a first insight aboutthe combined role of the two interaction terms, we noticethat the effect of phonons on the correlation parameteru is twofold. On one hand the e-ph interaction reducesthe numerator of u, but it also reduces the denominatornormalizing the kinetic energy. This means that underdifferent circumstances the two interacting mechanismscan either cooperate or compete in localizing the parti-cles.

A full solution of the Gutzwiller equations can be eas-ily achieved numerically by expanding the phonon wavefunctions in the complete basis of the eigenstates |n〉 ofh0, the harmonic oscillator centered around x = 0, witheigenvalues En = ω0(n+ 1/2)

φν(x) =

∞∑

n=0

c(ν)n 〈x|n〉 (23)

This expansion is completely general and it does not

5

introduce any further constraint or approximation. Ex-

ploiting the particle-hole symmetry c(0)n = (−1)nc

(2)n and

plugging (23) in the GPW equations (11)-(13) one gets:

[(

ǫ11 − 2d

− En

)

δnn′ + 8|ε0| d c(2)n c(2)n′

∗]

c(1)n′ = 0

[(ǫ2d

− En

)

δnn′ + 4|ε0| (1 − 2d) c(1)n c(1)n′

∗

−√

2αω0 xn′n

]

c(2)n′ = 0

which can be solved iteratively by keeping an arbitrarynumber of harmonic oscillator levels (in practice with 50-70 levels we get already very accurate results for theground state); of course their solution depends on thevalue of the parameter d, which has to be determinedself-consistently through Eqs. (14),(15) at each iteration.

We show in Fig. 1 the obtained phase diagram in theU − λ plane, displaying three different phases, the twoinsulators MI and BPI and a correlated metal, which, aswe will discuss in the next section, may show bipolaronicfeatures before the metal-insulator transition. Since oneexpects the properties of the system to be strongly de-pendent on γ, as it happens in the absence of correla-tion, the phase diagram has been determined for threedifferent values of the adiabaticity parameter, namelyγ = 0.2, 0.6, 4.

In the antiadiabatic limit, where the phonon wavefunc-tions are simply displaced harmonic oscillators, one findsthat |〈φ1|φ2〉| = exp(−α2/2) ≈ 1 and u ≃ [U−λD]/Uc =Ueff/Uc. The Holstein attraction becomes instantaneousand the system behaves essentially as a repulsive Hub-bard model as long as Ueff > 0 and as an attractiveone for negative Ueff . On the repulsive side we willhave a Mott transition, while on the attractive side thenormal phase displays a metal-insulator transition be-tween a metal and an insulating phase formed by in-coherent pairs, which is the antiadiabatic limit of theBPI36. Therefore transitions to insulating phases are sec-ond order and occurs symmetrically with respect to theUeff = 0 line, i.e. UMI = Uc +λD and λBPID = Uc +U(thin dotted lines in Fig. 1).

Considering finite values of γ we find that the maineffect onto the Mott transition with respect to γ → ∞is to change the slope of the transition line at small λ,which remains second-order and is shifted to lower valuesof U with decreasing γ. Increasing λ the metal-insulatorline bends towards the bisector U = λD when γ is de-creased. We notice that for small λ the Mott transitionline basically coincides with the prediction of the simpleVLF approach. This is easily understood observing that,as long as the transition is of second order, it is definedby the condition u = 1. According to Eq. (22), τ2 → 1as u → 1, which means that squeezing effects do notimprove the VLF variational ansatz. Plugging the VLFexpression for the phonon wavefunctions in Eq. (15) and

imposing u = 1 one gets for small λ the explicit relations:

λMID =Uc

2ω0(U − Uc) γ ≪ 1, (24)

λMID =

[

1 − U

2ω0

]−1

(U − Uc) γ ≫ 1, (25)

in excellent agreement with previous DMFT phasediagrams12,13,14.

Turning to the region of the parameter space domi-nated by the e-ph coupling, a more complex, stronglyγ−dependent picture emerges. As long as γ ≫ 1, themetal-BPI transition is found to be second-order, andthe condition u = −1 is always satisfied at the transi-tion. Since in this limit the φν ’s are essentially displacedharmonic oscillators, one finds:

λBPID = Uc

[

e−λ2γ + U/Uc

]

. (26)

As expected, the presence of U obstacles the stabiliza-tion of the BPI and pushes the pair transition to highervalues of λ. The ability of U to compensate the phonon-mediated attraction is maximum for large γ, so that thesame transition line moves to smaller λ (yet larger thanin the absence of U) as γ is reduced. Decreasing γ thescenario becomes more involved, and the order of thetransition turns out to depend on both γ and U . For0.5 . γ . 1 there is a window of intermediate values ofU in which the transition becomes of first order. There-fore the transition to the BPI is second-order at smallvalues of U , turns to first order at a given U dependingon the considered γ and then, at larger U , the transitionbecomes second-order again (e.g. at γ = 0.6 we obtain afirst-order transition for U/Uc between 0.5 and 1.1). Themetallic region comprised between the MI and the BPIappears to close asymptotically on the line Ueff = 0 withincreasing interactions strength. Eventually, for γ . 0.5the transition to the BPI is first order for all values of U .

Some comments are in order about the evolutionof the order of the transition to the BPI. For zeroelectron-electron interaction, DMFT shows a continuoustransition35. Thus the first-order nature of the metal-BPI transition at zero and small U is due to the limita-tions of our variational approach. Indeed more restrictivevariational choices, like the VLF and the VSQ predict afirst-order transition for γ . γ1 (for the VSQ the limitingγ1 is only slightly smaller than for VLF). The full solu-tion of our GPW equations lowers the value of γ1, but amore sophisticated wavefunction, able to capture nonlo-cal character of the ground state of e-ph coupled system,is required in adiabatic regimes, when retardation effectsbetween the motion of electrons and lattice distortion ofthe lattice become particularly relevant.

On the other hand a change in the order of the tran-sition, from second to first one with increasing U , hasbeen reported within exact DMFT at U/Uc ≃ 0.5 forγ = 0.112, and it was associated in Ref. 15 to an abruptchange in the electronic configuration from the Mott state

6

where all the sites are singly occupied to the BPI whereall the sites are either empty or doubly occupied. There-fore our GPW is able to correctly reproduce this highlynon trivial evolution of the order of the transition as afunction of U . We underline that more limited variationalchoices such as VLF and VSQ always find a first-ordertransition regardless the value of U . This result witnessesthat the improvement brought by the GPW is particu-larly important in the presence of correlation effects.

Our analysis shows a remarkable agreement betweenour phase diagram and previous DMFT results for γ =0.2. Such an accuracy can only increase for larger γ,where we observe, e.g., a metallic phase which intrudesbetween the two insulators for large U and λ. Unfor-tunately to our knowledge there are no detailed DMFTstudies of the phase diagram for larger values of γ tocompare with our variational predictions. Recently, thephase diagram of the one-dimensional Hubbard-Holsteinmodel at different values of the adiabaticity parameterhas been obtained, and an intermediate metallic phasehas been observed getting larger between the two insu-lating phases as γ increases37, in qualitative agreementwith our results.

IV. QUASIPARTICLE WEIGHT, PHONON

DISPLACEMENT AND BIPOLARON

FORMATION

Beside the metal-insulator transition associated tobipolaronic binding, a strong electron-phonon couplingcan drive the formation of polaronic and bipolaronicstates. A polaron is an electron strongly entangledwith phonons, thus moving with a significantly enhancedeffective mass. The residual interaction between po-larons is attractive, leading to the formation of pairs,called bipolarons. Strong Coulomb repulsion or largelattice fluctuations can avoid the metal-insulator tran-sition due to bipolaron localization, and they can leadto a poorly metallic state of bipolarons. It has been re-peatedly shown that the formation of polarons and bipo-larons in the absence of electron-electron interaction oc-curs through a continuous crossover. In the followingwe introduce and discuss two quantities that allow us tocharacterize the (bi)polaronic character of the correlatedmetallic phase from the point of view of both electronicand phononic observables.

The metallic or insulating character of the variationalsolutions can be addressed by computing the quasiparti-cle renormalization factor, which corresponds to the in-verse effective mass m∗ in the Gutzwiller approach:

Z =m

m∗ = 8d(1 − 2d)|〈φ1|φ2〉|2 = (1 − u2)|〈φ1|φ2〉|2,(27)

that is zero when the system is a MI or a BPI and anonvanishing quantity ≤ 1 for metallic solutions. Theeffect of the e-ph coupling on the mass renormalizationis not only comprised in the overlap 〈φ1|φ2〉, as in the

infinite-U limit31, but it is also hidden in the effectivecorrelation parameter u.

In the absence of U it is clear from Eq. (27) that themain correction to Z for weak e-ph coupling effects comesfrom the overlap 〈φ1|φ2〉, which actually results in an en-hancement of the effective mass that is linear in λ, asshown in the left panels of Fig. 2; however the rapid sup-pression of Z as the BPI is approached by increasing λmainly comes from the phonon-induced attractive inter-action, at least for γ > 0.5 (top and middle panels of theleft column in Fig. 2). Approaching the adiabatic regimethe relevance of the effective phonon-mediated attractionis reduced, with Z almost completely determined by thebehavior of the overlap 〈φ1|φ2〉 as a function of λ (bottompanel of the left column in Fig. 2). Yet, the sudden de-crease of the overlap drives a sharp first-order transitionto the BPI for γ . 0.5.

Moving to the correlated metallic phase with U =0.8Uc, we find that, for all the considered values of γ,Z has a non monotonic behavior. For small coupling Zincreases with λ, it reaches a maximum before decreas-ing to eventually reach the BPI. Also in this case Eq.(27)allows to explain in a very intuitive way the observedincrease of Z when the e-ph coupling is turned on in acorrelated regime12,14. In fact, as long as the effectiveinteraction stays repulsive, the main effect of λ is to par-tially screen the bare Hubbard repulsion, the effectivenessof such screening being strongly dependent on the adia-baticity regime14 (cfr. the right panels of Fig. 2); theelectron motion, even if slowed down by e-ph coupling,can take advantage of the smaller energy cost associatedto double occupation, and this results in a decrease ofm∗

with respect to the λ = 0 value. As the phonon-inducedattraction overcomes the Hubbard repulsion, the two ef-fect of phonons (localizing tendency to self-trap and toform local pairs) are cooperative, leading to a sharp first-order transition to the BPI for strong enough U and smallγ12.

The above discussion clearly shows that Z is not theright quantity to identify the bipolaron crossover in thepresence of competing interactions. While in the ab-sence of U bipolaron formation is usually associated toan abrupt enhancement of the effective mass, as shownin the left panels of Fig. 2, a more direct measure of po-laronic features in the groundstate has to be used whenthe effective mass results from the competition betweenattractive and repulsive interactions. More precisely, weneed a quantity that measures the entanglement betweenthe electron motion and the lattice distortion. A goodcandidate is the groundstate phonon distribution func-tion

P (x) ≡ 〈ψ0|x〉〈x|ψ0〉 =∑

ν

Pν |φν(x)|2, (28)

|x〉 being an eigenstate of the displacement operators,and |ψ0〉 the groundstate vector. P (x) measures the ef-fective (ground-state) displacement of the lattice. In fact,at least in the U = 0 limit the development of polaronic

7

features is associated to the evolution from a monomodalshape of P (x) centered around an equilibrium displace-ment (x = 0 at half-filling) to a bimodal distribution withmaxima displaces of ±x0, signaling the development offinite lattice distortions35. Focusing on the half-fillingcase, x = 0 changes from a maximum in the monomodaldistribution to a minimum in the bimodal one. Thus wecan associate the appearance of bipolarons in our modelto a change of sign in the second derivative of P (x) forx = 0, i.e.,

4d

[

φ2(x = 0)d2φ2

dx2

∣

∣

∣

∣

x=0

+

(

dφ2

dx

)2

x=0

]

+

2(1 − 2d)φ1(x = 0)d2φ1

dx2

∣

∣

∣

∣

x=0

≥ 0. (29)

We notice that the evaluation of the condition (29) isparticularly sensitive to the choice of the variational pro-jected wavefunctions. The evolution of P (x) for ourGutzwiller function is shown in Fig. 3 in comparisonwith the VSQ ansatz in a moderately adiabatic situa-tion γ = 0.6. We notice that some care has to be takenin using the condition (29) to pinpoint the bipolaroncrossover in the correlated regime (right panels), sincethe effect of U on the P (x) can result in shoulders oreven in a three-peak structure rather than the simple bi-modal shape. However our criterion captures the correctorder of magnitude of the critical λ for the onset of abipolaronic groundstate.

In the antiadiabatic limit γ → ∞ no bipolaroncrossover occurs before the system turns insulating byforming incoherent local pairs: Replacing φν in Eq. (29)with the proper displaced harmonic oscillators, one finds,in agreement with the DMFT of Ref. 35, that bipo-laron formation occurs at the transition line only whenα2 > 1/4, which implies a very large λ when γ is large.However, at finite values of the adiabaticity parameter,the condition (29) can be realized, since the lattice dis-tortions induced by the e-ph coupling, which are roughlyproportional to 1/

√γ, can push the system to polaron

formation before the BPI instability takes place. Actu-ally, for all the considered values of γ we found bipo-laron formation before the metal-BPI transition line, ina region whose size depends again on both correlationand adiabaticity parameter. On the weakly correlatedside, we notice (see Fig. 1) that, starting from small γ,the distance between the bipolaronic crossover and themetal-insulator transition first increases before decreas-ing in the antiadiabatic regime. This is in qualitativeagreement with DMFT35, even if for γ = 4 the GPW stillpredicts the bipolaronic crossover before the BPI transi-tion, in contrast with DMFT, where for the same valueof γ the opposite occurs. As shown in Fig. 1, at γ = 4,the presence of U slightly enlarge the region where theground state displays polaronic features.

V. DISCUSSION OF THE RESULTS.

In this section we discuss the behavior of the phononwavefunctions, that represent the main novelty intro-duced by our Gutzwiller method with respect to previousschemes where the functional form was assumed a priori.

A comparison with VLF and VSQ schemes helps us tobenchmark our method. Due to the enlarged variationalspace, the ground-state energy of the GPW is alwayssmaller than that given in Eqs. (17) and (20). The gaus-sian ansatz for the φν ’s (VLF) results in a significantlyhigher energy, especially in the intermediate to strongcoupling regime, whereas the VSQ energy stays very closeto the general GPW one, even though the GPW becomesvisibly more accurate than VSQ in the presence of size-able correlations, as shown in Fig. 4. However, evenwhen GPW provides only a slight improvement on thevariational energy with respect to VSQ, it gives a moreaccurate description of the physical properties of the lat-tice; in fact, the projected phonon wavefunctions of GPWcan provide a better estimates of the different energy con-tributions, i.e. the potential energy gain and the kineticenergy loss due to lattice distortion, whose balance re-sults in the overall ground-state energy.

To get a flavor of the improvement introduced by thewavefunction Eq. (2) we plot in Fig. 5 φ2 and φ1 asobtained by means of our general equations, compar-ing them with the simpler variational ansatzs, VLF andVSQ. In the absence of Coulomb repulsion, the squeezedstates stay relatively close to the self-consistent solution,whereas VLF underestimates the effective lattice distor-tion and therefore the potential energy gain. On theother hand, in the strongly-correlated regime the GPWfunctions are rather different from the other methods andunderline the non-trivial way in which anharmonic fluc-tuations of the lattice states are influenced by the corre-lated electron motion. Even if significantly more accuratethan VLF, the VSQ underestimates the anomalous lat-tice fluctuations that we find in GPW; since the overlapof the phonon wavefunctions related to different chargeconfigurations is essential in capturing the way e-ph cou-pling renormalizes the effective hopping, this means thatVSQ overestimates the kinetic energy loss.

A deeper understanding of the differences between thestandard variational ansatzs and the self-consistent so-lution for the phonon wavefunctions can be traced outby looking at the degree of effective displacement, givenby 〈x〉ν , and of the related quantum fluctuations, thatcan be estimated by ∆x2

ν = 〈[xν − 〈xν〉]2〉. These twoquantities are, roughly speaking, related respectively tothe potential energy gain and to the kinetic energy lossinduced by the e-ph coupling, and their inspection willhelp us to identify the way in which the GPW improveswith respect to the other approaches. It is readily foundthat 〈x〉ν = ±

√2αf for ν = 0, 2 and 〈x〉1 = 0 for

both VLF and VSQ ansatz, with f given respectivelyby Eqs. (18) and (21), whereas ∆x2

ν = 1/2 in the har-monic approximation and ∆x2

ν = 1/2τ2 for any ν in

8

the squeezed phonon state approximation. In Fig. 6we compare the effective displacement and ∆x2 obtainedwithin the different approaches as a function of λ in thepresence of a sizeable U , namely U = 0.8Uc. By look-ing at 〈x〉ν , we find that both harmonic and squeezedansatzs reproduce the weakly interacting regime quiteaccurately, while VLF deviates in the intermediate tostrongly-coupled regime, falling suddenly onto the atomicsolution (which in the present mean-field approach actu-ally translates in the BPI). The origin of such a discrep-ancy is the relevance of phonon quantum fluctuationsthat are crucial in the crossover region and are poorlydescribed within VLF. However within GPW such effectis strongly dependent on the charge state associated tothe different wavefunctions: the phonon wavefunctionsprojected onto empty and doubly-occupied sites displaya maximum in ∆x2

0(2) and then tend to the atomic so-

lutions as the BPI is approached, whereas ∆x21 increases

monotonically. Therefore, as anticipated, the simple two-phonon coherent state Eq. (19) reproduces only a kindof average squeezing effect; to stress this point we showan averaged ∆x2, namely ∆x2 =

∑

ν Pν∆x2ν , that indeed

stays very close to the VSQ solution.In conclusion, the improved accuracy of our variational

phonon wavefunctions proves to be relevant especially forintermediate-to-strong e-ph coupling and in the presenceof sizeable correlation, in particular close to the BPI in-stability, when quantum fluctuations of the phonons playan important role. Indeed, by looking at Fig. 6, oneimmediately realizes that the evaluation of the phonondistribution function strongly relies on the quality of theprojected wavefunctions, and the fulfillment of the con-dition (29) can lead, as it actually does, to completelydifferent bipolaron crossover lines for different variationalansatzs when considering adiabatic regimes. To be moreexplicit, we find that the simple variational Lang-Firsov(or harmonic) ansatz never predicts the onset of a bipo-laronic groundstate for any U at γ < 1, being P (x) al-ways monomodal (results not shown). On the other handVSQ wavefunctions reproduce a bimodal P (x), displayedin the bottom panels of Fig. 3, at least at U = 0, but atlarger λ with respect to GPW; however they completelymiss the complex structures that P (x) develops in thepresence of a sizeable U , and they do not reproduce anybipolaron crossover in the correlated regime.

VI. CONCLUSIONS

In this work we have extended the analysis of a newtype of Gutzwiller variational wavefunction introducedto describe correlated electron systems in the presenceof e-ph coupling. In a previous paper31 we benchmarkedthe wavefunction in the infinite-U limit by comparing itto exact DMFT calculations, showing that polaron for-mation occurs smoothly for any value of the adiabaticityparameter γ and electron density n, as opposed to theadiabatic first-order transition found in standard varia-

tional approaches27,29. In the infinite-U limit the effectof correlation on polaronic physics can be analyzed bytuning electron filling, and we found that polaron forma-tion is inhibited and pushed to higher, though finite, λwhen moving from low-density to half-filled system.

Relaxing the assumption of infinite repulsion betweenelectrons, in this paper we turned our attention to thehalf-filling regime and considered, together with bipo-laron formation, the competition between U and thephonon-mediated bipolaron instability, ruled out by aninfinite U . We found, in excellent agreement with pre-vious works14,28,29, that the Mott metal-insulator transi-tion is always robust with respect to polaron formationand that screening of the bare repulsion due to the cou-pling with the lattice is less and less effective as adiabaticregimes are attained. On the other hand we observedthat both bipolaron crossover and metal-BPI transitiondepend not only on the adiabaticity parameter but alsoon the strength of Hubbard repulsion. Two localizingmechanisms are involved at intermediate-to-strong e-phcoupling, one resulting in self-trapping of electrons cou-pled with Holstein phonons, and the other related to thephonon-mediated e-e attraction. If on one hand correla-tion pushes to larger values of λ both the onset of bipola-ronic features and the metal-BPI transition, it influencesthe two processes in a different way, which depend onthe adiabaticity regime, giving rise to different scenar-ios. For example, when γ > 1, U is more effective inscreening the attractive interaction than in preventingthe bipolaron crossover14. On the other hand at small γthe bipolaron crossover is inhibited by U faster than bipo-laronic transition, and polaronic features are observed ina narrow region that rapidly shrinks with increasing U ;at the same time a change in the order of the transitionoccurs depending on the strength of correlation.

A critical comparison with standard variational ap-proaches to Holstein phonons shows that the proposedwavefunction can be viewed as a proper generalizationapt to include correlation effects, underlying the impor-tance of a reliable variational description of anharmonicfluctuations in the presence of U . Our wavefunction im-proves on previous variational approaches already in theabsence of electron correlations (VLF and VSQ can in-deed be obtained as restricted solutions of our methodlimiting the variational freedom), but it still has somelimitations in the deep adiabatic limit due to the natureof our phonon wavefunctions, that are associated to thedifferent local electronic states. For the same reasons,our method becomes considerably more accurate in thepresence of strong correlations, that make the electronicphysics more local.

Acknowledgments

We are grateful with S. Ciuchi for important discus-sions and a careful reading of the manuscript, and withG. Sangiovanni for discussions. We acknowledge financial

9

support from MIUR PRIN, Prot. 2005022492.

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10

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

λ

U/Uc

M MI

BPI

γ = 4

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

λ

U/Uc

M MI

BPI

γ = 0.6

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1 1.2 1.4

λ

U/Uc

M

MI

BPI γ = 0.2

0 0.1 0.2 1

1.2

1.4

FIG. 1: Phase diagram of the half-filled Hubbard-Holsteinmodel in the U − λ plane as obtained in GPW for three dif-ferent values of γ. Solid lines represent the metal-insulatorstransitions, dashed lines are the bipolaron formation lines ob-tained by imposing condition Eq.(29) as discussed in SectionIV (the inset shows the narrow region where a bipolaronicmetal is found for γ = 0.2). Thin dotted lines are the antia-diabatic (γ → ∞) transition lines symmetric to the U = λDline, also displayed. The vertical arrows in the middle panelmark the boundary of the region in which the transition tothe BPI is of first order.

11

1

0.8

0.6

0.4

0.2

0 0 0.5 1 1.5 2 2.5

Z

λ

U = 0

γ = 4

1

0.8

0.6

0.4

0.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5λ

U/Uc = 0.8

γ = 4

1

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Z

λ

U = 0

γ = 0.6

1

0.8

0.6

0.4

0.2

0 0.5 1 1.5 2 2.5 3 3.5λ

U/Uc = 0.8

γ = 0.6

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Z

λ

U = 0

γ = 0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3λ

U/Uc = 0.8

γ = 0.2

FIG. 2: (Color online) Typical evolution of Z as a functionof λ with and without Hubbard repulsion. Parameters areγ = 4, 0.6, 0.2 from top to bottom and U = 0, U = 0.8Uc

in the left and right panel respectively. Open circles are thephonon contribution to the hopping renormalization comingfrom |〈φ1|φ2〉|2.

12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-3 -2 -1 0 1 2 3

P(x)

x

GPW

U=0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-4 -3 -2 -1 0 1 2 3 4

P(x)

x

GPW

U/Uc = 0.8

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-3 -2 -1 0 1 2 3

P(x)

x

VSQ

U=0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-4 -3 -2 -1 0 1 2 3 4

P(x)

x

VSQ

U/Uc = 0.8

FIG. 3: GPW results (top panels) for P (x) as a function ofe-ph coupling constant at γ = 0.6 in the presence of Hubbardrepulsion (right panel, U/Uc = 0.8 and λ ranging from 2.8 to3.12) and with U = 0 (left panel, λ ranging from 1 to 1.3).The bottom panels show the lattice distribution function asobtained with VSQ for the same parameters.

-0.52-0.51-0.5

-0.49-0.48-0.47-0.46-0.45-0.44-0.43-0.42

0 0.2 0.4 0.6 0.8 1

E0/

D

λ

U = 0

VLFVSQGPW

-0.06-0.055-0.05

-0.045-0.04

-0.035-0.03

-0.025-0.02

-0.015

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

E0/

D

λ

U/Uc=0.8

VLFVSQGPW

FIG. 4: (Color online) Comparison of the variational on-siteenergies obtained for the paramagnetic solution in the absenceof Hubbard repulsion (upper panel) and close to the criticalvalue Uc = 8|ε0| for the purely electronic Mott transition(lower panel) for γ = 0.2.

13

-6 -5 -4 -3 -2 -1 0 1 2 3

φ 2

x

U = 0GPWVLFVSQ

-4 -3 -2 -1 0 1 2 3 4

φ 1

x

U = 0GPWVLFVSQ

-6 -5 -4 -3 -2 -1 0 1 2 3

φ 2

x

U/Uc = 0.8 GPWVLFVSQ

FIG. 5: (Color online) Comparison of the projected phononwavefunctions for γ = 0.2 as obtained with GPW and withthe variational ansatzs discussed in the text (top panel is U =0, λ = 1; bottom panel is U = 0.8Uc, λ = 2.8). The thindotted line is the corresponding atomic solution.

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5

<x>

2

λ

GPWVLFVSQ

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.5 1 1.5 2 2.5 3 3.5λ

∆x22

∆x21

∆x2VSQ

∆x2

FIG. 6: (Color online) Comparison of the effective displace-ment and width of projected phonon wavefunctions in themetallic solution before the BPI is reached as obtained withGPW and with the variational ansatzs discussed in the text(parameters are γ = 0.6 and U = 0.8Uc). The thin dottedline is the atomic solution.