The Hubbard-Holstein model is a simple model including both electron-phonon interaction and
electron-electron correlations. We review a body of theoretical work investigating, the effects of
strong correlations on the electron-phonon interaction. We focus on the regime, relevant to high-
A wealth of materials, including the most challenging systems (cuprates, manganites, fullerenes, etc), present clear signatures of both electron-electron (e-e) and electron-phonon (e-ph) interactions, leading to a competition-or- interplay which can give rise to different physics according to the value of relevant control parameters and of the chemical and electronic properties of the materials. The results presented in this paper are mainly motivated by high-temperature superconductors, with the copper-oxide compounds (cuprates) in a prominent role, and an attention to the alkali-doped fullerides.
In the case of the cuprates, which are arguably the most accurately studied materials in the last twenty-five years, the signatures of electron-phonon interactions are nowadays clear, even though the overall scenario is far from ordinary [
On the other hand, phonons, which typically affect resistivity in standard metals, hardly appear in transport experiments on cuprates. For instance, the resistivity around optimal doping is ubiquitously linear in temperature (even in systems with relatively low critical temperature) [
Although there is a wide range of suggestions for the superconducting mechanism, it is almost universally recognized that a key player in the cuprate game is the e-e correlation. Electron-electron correlation makes the parent compounds Mott insulators, and is expected to be important at least in the pseudogap region. Therefore, it is not surprising that the signatures of e-ph interaction in the cuprates can hardly be understood in terms of the standard theory of e-ph interactions in weakly correlated metals, and a new theoretical framework including e-e correlations is needed. We will argue here that this change of perspective can indeed reconcile the different relevance of phonons in the various observables in correlated systems.
On the other hand, the superconducting members of the fulleride family, of composition A
In an extremely broad sense, these materials (cuprates and fullerides), as well as many others that we did not talk about, raise the same conceptual problem, namely the investigation of systems in which both e-e interaction and e-ph coupling are nonnegligible and the physics can be explained only taking both into account. On the other hand the same phenomenology suggests that this competition may result in completely different physics according to specific aspects of the materials. In general, we can expect different behaviors because of: (i) Different parameters within the same model (e.g., which is the largest scale between electron-phonon interaction and electron-electron repulsion); (ii) Different form for the interaction term, or more generally, different models.
Here we focus on point (i), and we choose an extremely simplified model, the Hubbard-Holstein model, in which one band of correlated electrons with local Coulomb repulsion is coupled with a dispersionless phonon mode and the coupling only involves the local electronic charge [
Our choice is to focus on the “strongly correlated” metallic phases, that is, on system in which the Coulomb repulsion is the largest energy scale, and the system is either at half-filling (number of electrons equal to the number of sites) or close. The polar star of this work is the understanding of the fate of electron-phonon interaction in systems that are dominated by electron-electron interactions such as the cuprates. Nonetheless, our discussion will also follow some detours, which will help us to build a more comprehensive picture of the competition between the two interactions. One of these detours will touch point (ii) addressing the role of the phonon symmetry in its interplay with correlations. This point is crucial for the understanding of the synergy between e-ph interaction and e-e correlation in the fullerenes.
The paper is organized as follows: In Section
The simplest model of a strongly correlated electron system coupled to the lattice is given by the single-band Hubbard-Holstein (HH) model. In this work, the HH model is not used as a microscopic model for the cuprates, but rather as an idealized description of the competition between e-ph interaction and e-e interaction. In physical terms, the most crucial limitations is the local nature of the interactions. We refer to previous literature for analyses of nonlocal e-ph interactions [
The single-band HH model reads
Even if our focus will be the strongly correlated HH model, we will discuss its results in comparison with some related models, like the Hubbard-tJ model, the three-band Hubbard model for the cuprates and a three-orbital Hubbard model with Jahn-Teller interactions for the doped fullerides. Our investigation will be mainly dedicated to the effects of e-ph interaction on the self-energy and the quasiparticle renormalization factor
In this section, we begin our analysis of the properties of e-ph interaction in a correlated metal within a Landau Fermi-liquid (FL) picture [
In order to characterize the fate of the e-ph interaction in a similar correlated metal, we need to consider also the vertex corrections introduced by e-e interactions, for which no Migdal theorem can be invoked.
We can gain a first insight on the way in which the e-ph interaction behaves in the presence of strong correlations by considering the effective dimensionless e-e interaction mediated by the exchange of a single-phonon
Within a Landau Fermi-liquid picture we can use the Ward identities that connect the vertex corrections
Plugging these results into (
The strong
Results for these models show that the product
The general Fermi liquid discussion and the specific analysis of models with strong correlations generically demonstrates the relevant role of dynamics in the screening effects that e-e correlations induce on the e-ph coupling. This strong dependence of the e-ph vertex on momentum and frequency (and on their ratio) makes the effects of the e-ph coupling rather subtle, since different physical quantities, involving different dynamical regimes, may display more or less suppressed e-ph effects. In particular the e-ph coupling (and the e-e interaction mediated by phonons) will be depressed by strong e-e interactions whenever small energy and large momentum transfer are involved (e.g., in transport). This suppression may be substantial, for instance, in the low-doping region of the superconducting cuprates, where e-e correlations are strong due to the relative proximity to a correlation-induced insulating phase. On the other hand different physical processes involving dynamical processes could experience a more pronounced e-ph coupling. Specific calculations carried out in a single-band Hubbard-Holstein model within a large-
Even if our focus is on the Hubbard-Holstein model, it can be useful to recall that in the case of phonons coupled to the electron current, Ward identities similar to those of (
A natural tool to address the screening effects beyond the small-
In this discussion we will consider the infinite-repulsion limit, which simplifies the formalism. In this limit we have a sharp constraint of no double occupancy on each lattice site
In the SB large-
Leading-order in
Static effective e-
On the other hand, as it is natural, the long-range potential screens out the long-range charge fluctuations thereby driving to zero the e-ph coupling at low momenta.
These findings can be obtained and confirmed with several different approaches. In particular, they reproduce exactly the results of large-
All these results show that e-ph scattering at large momenta is typically weaker than scattering at low momenta in the presence of short range forces only (which is the case of metallic phases far from phase separation charge instabilities, where, on the contrary, long-range interactions start to play a crucial role). This can be of obvious relevance when the relative importance of e-ph couplings between the quasiparticles and specific phononic modes is considered [
The previous subsection was focused on the screening of the static e-ph coupling by electronic processes. However the Fermi-liquid analysis carried out in Section
Figure
Effective e-ph vertex in units of the bare e-ph coupling as a function of the Matsubara frequency in the HH model at leading order in
Again this result is not specific of this single-band model (or of its treatment) and it has been confirmed by the analysis of a three-band Hubbard model for the cuprate CuO
In this section we extend our analysis beyond the mean-field level and discuss the fate of the e-ph interaction in a correlated metal under the sole assumption that the physics is governed by the Hubbard repulsion close to a Mott-Hubbard transition, without assuming a weak e-ph coupling, and/or any approximation as far as the adiabatic ratio is concerned. This means that we need a theoretical approach able to treat several energy scales simultaneously, without assuming that any of them is negligible or perturbative. A natural candidate for this purpose is the Dynamical Mean-Field Theory (DMFT) [
The central approximation behind DMFT is the locality of the self-energy (both the electronic and the phononic contributions), a condition which becomes exact when the coordination number becomes large. The original lattice enters in the calculation only through the density of states, which we always choose to be a semicircular one of half-bandwidth
DMFT allowed to obtain a complete characterization of the Mott-Hubbard transition in the pure Hubbard model, and the emerging physical picture is able to explain several properties of correlated oxides. While we refer to original papers [
We first consider the half-filled system and, in order to focus on pure correlation effects, we consider a paramagnetic phase. In this regime, for large repulsion, the ground state of the HH model can become a “Mott” insulator, in which the electrons are localized because the electron motion is energetically unfavorable. Starting from the uncorrelated systems and increasing the correlation strength
In the following part, we discuss the effect of a nonperturbative electron-phonon coupling in the strongly correlated metallic solution, that is, a system in which a quasiparticle peak at the Fermi level is separated from the Hubbard bands realizing a separation of energy scale. Starting from this situation, in which the quasiparticle bandwidth is The e-ph interaction can introduce a further quasiparticle renormalization, associated to the increase of the quasiparticle effective mass, which may eventually lead to polaronic effects for very strong coupling. This effect leads to a decrease of The e-ph interaction mediates an attractive density-density interaction (the density-density form is specific for a Holstein coupling), which directly contrasts the Hubbard repulsion. If we integrate out the phonon degrees of freedom, the fermions interact through a dynamical (retarded) interaction [
The balance between this two effects is not generic and it depends on the adiabatic ratio and on the precise value of the interactions. Yet, important conclusions can be drawn in the correlated regime, in which, also in the presence of e-ph interaction, the separation of energy scales determined by correlations survives. In this regard, it is important to recall that, within DMFT, the quasiparticle weight is associated to a Kondo resonance of an Anderson-Holstein impurity model. Assuming that the Hubbard
These results can be tested through a DMFT solution of the Hubbard-Holstein model. We start our discussion from half-filling, where the separation of energy scales characteristic of correlated systems is clearer and more solid. We first computed the quasiparticle weight
In Figure
Effect of the electron-phonon interaction on the quasiparticle weight
This behavior confirms that, when the e-e correlation dominates, the leading effect of the e-ph interaction is a reduction of the effective
Assuming a form
Effective static electron-electron interaction for the low-energy properties of the Hubbard-Holstein model. The picture shows the coefficient
On the basis of our knowledge about the DMFT results for the pure Hubbard model, the quasiparticle weight completely characterizes the low-energy quasiparticle peak. Therefore, our analysis should imply that the low-energy part of the spectral function of our Hubbard-Holstein model can be described by means of the effective Hubbard model that we introduced. This is strikingly confirmed by a direct comparison, as shown in Figure
Comparison between the DMFT momentum-integrated spectral function
The picture that emerges for the half-filled model can be summarized as follows: Quasiparticle motion arises from virtual processes in which doubly occupied sites are created. Obviously, these processes are not so frequent, since the energy scale involved is large, but they are extremely rapid (the associated time scale is
According to what we have just described, we can conclude that strong correlations reduce the effect of the e-ph interaction on the low-energy properties, associated to quasiparticle propagation, while the high-energy properties present more standard phonon signatures, such as the satellites at the phononfrequency scale.
It has to be underlined that DMFT is not able to introduce momentum-dependent corrections to the electronic properties. The above analysis therefore, shows indeed that the “standard” electron-phonon interaction is heavily screened (and it actually loses its dynamical nature) when the low-energy quasiparticle properties are considered. Only “nonstandard” effects, such as the prevalence of forward scattering that we discussed in Section
The above scenario has been carried out at half-filling, where the presence of e-e correlations has its most striking effects, both in terms of the phase diagram, as it can give rise to a Mott transition, and in terms of the separation of energy scales, which is clearly sharper than for doped systems. Therefore, as soon as we dope the Hubbard-Holstein model, even for
Nonetheless, if we consider reasonably large values of
Effective mass
The role of the adiabatic ratio is illustrated by Figure
Renormalization of the quasiparticle electron-phonon coupling as a function of
The scenario which emerges from DMFT calculations at finite
Renormalization of the effective mass due to electron-phonon coupling for infinite
The above analysis suggests that, as expected, strong e-e correlation essentially opposes to e-ph coupling, even though “anomalous” signatures of e-ph coupling still survive even in regions of parameters for which correlations prevail. Yet, these results are limited to the metallic paramagnetic state, in which no broken symmetry is allowed. At half-filling and for some finite doping region, strong correlations lead to an antiferromagnetically ordered state, and it is expected that finite-range antiferromagnetic correlations survive in a wider doping region. The relation between antiferromagnetism and e-ph interaction is hinted by the experimental framework. Indications of phonon signatures in high-
From a theoretical point of view, several investigations indeed suggest that the e-ph interaction is particularly effective for a hole in an antiferromagnetic background [
As we discussed in the previous sections, in the paramagnetic phase the effect of increasing correlations is a strong reduction of the quasiparticle weight
Direct DMFT calculations in the antiferromagnetic phase at half-filling confirm these expectations [
Effect of the Coulomb repulsion on the linear contribution in
The comparison with the uncorrelated system is shown in Figure
Quasiparticle renormalization as a function of
The above discussion has been limited to half-filling. It is worth to briefly discuss the effect of doping in the antiferromagnetic state. In Figure
Quasiparticle renormalization as a function of
In the previous sections we focused on the metallic phases, without discussing the possible instabilities, either directly driven by the interaction terms, or favored by the weakness of the correlated metallic state. One can indeed expect, on very general grounds, that the reduced kinetic energy characteristic of the strongly correlated metal can be easily overcome by different localizing effects thereby destabilizing the metal in favor of ordered phases.
As we did in Section
It is important to stress that the results of Section
Moreover, near the instability condition
This is what indeed happens in the HH model treated within the SB-large-
In the presence of long-range Coulomb interactions the electrostatic cost of the charge-rich regions would become infinite, and the thermodynamic phase separation cannot take place. However, inhomogeneous charge-density wave ordering can establish as a compromise between the charge segregation tendency and the homogenizing effect of long-range interactions. This mechanism for charge ordering is the so-called “frustrated phase separation” [
Phonon dispersion curves (a) in the instability (0.28,0.86) direction and (b) in the (1,1) direction, for
Of course it is quite tempting to relate these anomalies, to the anomalies observed by inelastic neutron scattering [
All the above analysis has been carried out for the HH model. It has to be emphasized that some of the effects we discussed may be less general than what the simple form of the Hamiltonian may suggest. As we discussed in details, the HH model is indeed characterized by two interaction terms which are both related to the charge degrees of freedom, and they indeed directly compete, as clearly shown by (
The first situation can obviously have relevance for the cuprates, in which different phonon modes with specific symmetries may play a role, or for system dominated by the so-called Su-Schrieffer-Heeger coupling in which the phonons modulate the nearest-neighbor hopping. The second situation occurs instead in the fullerenes, where the relevant conduction band is a three-fold degenerate manifold of
A three-band model which includes a strong Coulomb repulsion, a Hund's rule splitting and a moderate Jahn-Teller e-ph coupling has been studied in several papers [
Here we do not discuss the physics of this model in details, since we are mainly interested in contrasting its behavior with the HH model. The key point is that the Jahn-Teller interaction does not touch the total charge on each molecule, as it couples with a combination of local spin and orbital degrees of freedom [
As a matter of fact, the effective interaction between quasiparticles obtained in a nonperturbative DMFT study of the model corresponds to a severely screened Hubbard repulsion plus an essentially unscreened phonondriven attraction that can be parameterized as
This enhancement is explicitly found by solving the three-orbital model within DMFT in the s-wave superconducting phase. If we follow the evolution of the superconducting order parameter as a function of
A full DMFT solution of the model has allowed both to predict the experimental observation only later provided in [
In the context of this paper, our solution is a clear example of the crucial role of the phonon symmetry. In our multiband model it is possible to consider phonons which are by symmetry unharmed by correlations, as opposed to the Holstein model. The result is confirmed by investigations of simplified two-orbital models which share the same properties [
The focus of the paper is on the effects of strong electron-electron correlations on the electron-phonon coupling. We mostly considered the Hubbard-Holstein model, where both e-e and e-ph interactions locally couple to electron density fluctuations. In this case the competition between these interactions is quite effective, particularly in the proximity of the Mott-Hubbard transition.
In particular, we study strongly correlated metallic phases, where the system is either at half-filling or at small doping, and the correlation strength is large enough to put the system close to a Mott-Hubbard transition. In these conditions, since the e-e Hubbard repulsion makes the density fluctuations stiffer, the Holstein e-ph coupling
Along this paper we have discussed how this suppression depends on exchanged momentum and frequency, and on other physical parameters. A first observation is that the suppression is strong whenever the quasiparticle residuum
The nearly static case
All the above findings are only partly peculiar of the specific HH model, where the electron density locally involves both the e-e and e-ph coupling: The analysis of other models like the Su-Schrieffer-Heeger [
The work that we review has been carried out in collaboration with G. Sangiovanni, P. Barone, S, Ciuchi, C. Di Castro, J. Lorenzana, G. Seibold, A. Di Ciolo, F. Becca, M. Fabrizio, E. Koch, O. Gunnarsson, P. Paci, and R. Raimondi. We acknowledge financial support by MIUR PRIN 2007 Prot. 2007FW3MJX003. M.Capone's activity is funded by the European Research Council under FP7/ERC Starting Independent Research Grant “SUPERBAD” (Grant Agreement no. 240524). M. Grilli acknowledges financial support from the Vigoni Program of the Ateneo Italo-Tedesco.