Effective electron–electron and electron–phonon interactions in the Hubbard–Holstein model

doi:10.1016/j.nuclphysb.2006.03.021

Nuclear Physics B

Volume 744, Issue 3, 12 June 2006, Pages 277-294

https://doi.org/10.1016/j.nuclphysb.2006.03.021 Get rights and content

Abstract

We investigate the interplay between the electron–electron and the electron–phonon interaction in the Hubbard–Holstein model. We implement the flow-equation method to investigate within this model the effect of correlation on the electron–phonon effective coupling and, conversely, the effect of phonons in the effective electron–electron interaction. Using this technique we obtain analytical momentum-dependent expressions for the effective couplings and we study their behavior for different physical regimes. In agreement with other works on this subject, we find that the electron–electron attraction mediated by phonons in the presence of Hubbard repulsion is peaked at low transferred momenta. The role of the characteristic energies involved is also analyzed.

Introduction

The electron–phonon (e–ph) interaction can play an important role in many physical systems, where it can be responsible for instabilities of the metallic state leading, e.g., to charge density waves and superconductivity. When the e–ph is particularly strong polaronic effects can also be responsible for a huge increase in the effective electronic masses. On the other hand the issue of strong electron–electron (e–e) correlations is of relevance for many solid state systems like, e.g., (doped) Mott insulators, heavy fermions, and high-temperature superconductors. Therefore the interplay between the e–e correlation and the e–ph coupling is far from being academic and its understanding is of obvious pertinence in several physically interesting cases. In particular, in the superconducting cuprates the e–e correlations are strong and are customarily related to the marked anomalies of the metallic state, including the linear temperature dependence of the resistivity [1], [2], where phonons are not apparent. This led to believe that the e–ph interaction plays a secondary role in these materials, and that the main physics is ruled by electron correlation. However, the presence of polaronic spectroscopic features [3] as well as various isotopic effects [4] in underdoped superconducting cuprates shows that the e–ph coupling is sizable in these systems. Recent results obtained with angle resolved photo-emission spectroscopy (ARPES) [5] have also been interpreted in terms of electrons substantially coupled to phononic modes. These contradictory evidences naturally raise the issue of why correlations can reduce the e–ph coupling in some cases and not in other. In this regard it has been often advocated [7], [8], [9], [11] that strong electron correlation suppresses the e–ph large angles scattering, which mainly enters in transport properties, much more than the forward scattering, which is instead relevant in determining charge instabilities [8]. Therefore the momentum dependence of the e–ph interaction can give rise to important effects and select relevant screening processes due to correlations.

In this paper we study the effect of (a weak) electronic correlation on a two-dimensional Holstein e–ph system using the flow-equation technique developed by Wegner [12], [13]. The flow-equations describe the evolution of the parameters of the Hamiltonian under an infinite series of infinitesimal canonical transformations labeled by a parameter l that rules the flow. The generator of the canonical transformation is chosen in such a way to eliminate some interactions in favor of other interactions. Shall one solve the flow equations exactly the initial Hamiltonian at $l = 0$ and the final Hamiltonian at $l = \infty$ are guaranteed to have the same eigenvalues. Of course in general the flow-equations are not solvable, but suitable approximations can be introduced to obtain qualitative trends. Under certain approximations one can map the interacting problem into a noninteracting one. In this case the method can be roughly seen as a way to derive a Fermi liquid theory where bare fermions evolve into quasiparticles under the flow. Since one can choose what parts of the Hamiltonian remain, the method can also be exploited to get effective interactions. Of course, in getting a fully diagonal Hamiltonian one is naturally led to discover the “low-energy” phsysics of the problem. This is the approach (and the spirit of it) previously adopted by Wegner and coworkers to analyze the instabilities of various forms of the Hubbard model [14], [15], [16]. Another application of the method is the derivation of effective interactions. For example this scheme was used to obtain an effective e–e interaction by eliminating phononic degrees of freedom in the absence of the initial e–e interaction [17]. Rather than obtaining the low-energy physics one obtains an interacting Hamiltonian which still has to be solved. Analyzing the structure of the resulting effective interaction we will discuss the physics of the e–ph interaction in the presence of the e–e interaction and vice versa. We will see that this procedure provides the correct energy scales for the phonons dressing the effective e–e interaction and, conversely, of the e–e interaction dressing the e–ph interaction.

The paper is organized as follow: first, in Section 2, we introduce the Hubbard–Holstein Hamiltonian and, in Section 3, we present the formalism of the flow equations technique. Then in Section 4.1 we validate the method by applying it to a two atom system. In Section 4.2 we compute the effective e–ph interaction in a two-dimensional system and in Section 5 we eliminate the e–ph interaction to get an effective e–e coupling, thus providing a model Hamiltonian suitable to discuss different physical regimes. Finally the summary and the conclusions are provided in Section 6.

Section snippets

Model

We consider a system of interacting electrons and phonons described by the Hubbard–Holstein model, where the electron density is locally coupled with a dispersionless phonon. For the sake of simplicity and since this case is relevant in physical systems of broad interest like the cuprates and the manganites, we consider the model in a two-dimensional lattice $H = \sum_{i, j, σ} t_{i, j} c_{i, σ}^{†} c_{j, σ} - μ \sum_{i σ} n_{i σ} + U \sum_{i} n_{i ↑} n_{i ↓} + g_{0} \sum_{i} n_{i} (a_{i} + a_{i}^{†}) + ω_{0} \sum_{i} a_{i}^{†} a_{i},$ where $n_{i} \equiv n_{i ↑} + n_{i ↓}$ , with $n_{i σ} \equiv c_{i, σ}^{†} c_{i, σ}$ . a $(c)$ and $a^{†}$ $(c^{†})$ are the phonon

Method

The flow-equation method [12], [13] is based on an algebraic technique, the Jacobi method, to (block) diagonalize Hamiltonians. A series of infinitesimal rotations is performed such that at each step the non-diagonal elements get smaller while the others asymptotically approach a finite limit. If l is the parameter ruling the flow, this technique performs a unitary transformation $U (l)$ such that $H (l) = U (l) H (l = 0) U^{†} (l)$ . One can absorb all the evolution in the parameters of the Hamiltonian

Two-site lattice

As mentioned above, our first aim is to eliminate the e–e interaction term dressing the e–ph one. In order to achieve this result we use the generator $η (l) = [T + G, V] .$ In this case we are not properly trying to diagonalize the Hamiltonian but we want to eliminate some off-diagonal elements corresponding to the e–e interaction to see how other off-diagonal elements, the ones corresponding to the e–ph interaction, renormalize. Thus it is necessary to check if the generator given above yields a

Effective electron–electron interaction

An effective e–e coupling in the static limit can be obtained from the effective e–ph coupling of Eq. (33) considering the following two contributions diagrammatically depicted in Fig. 5 $V_{k, k^{'}, q}^{eff} = - \frac{g_{k, q} (l = \infty) g_{k^{'} - q, q} (l = \infty)}{ω_{0}} - \frac{g_{k + q, - q} (l = \infty) g_{k^{'}, - q} (l = \infty)}{ω_{0}} .$ If one restricts the external momenta k, $k^{'}$ to the domain I such that the condition of energy conservation $ϵ_{k^{'}} - ϵ_{k^{'} - q} = ϵ_{k + q} - ϵ_{k}$ is satisfied, one obtains a simple expression for $V_{k, k^{'}, q}^{eff}$ $V_{k, k^{'}, q}^{eff} = - \frac{2 g_{0}^{2}}{ω_{0}} - \frac{4 g_{0}^{2} U}{ω_{0} N} \sum_{p} (n_{p - q} - n_{p}) [2 (ϵ_{k + q} + ϵ_{p - q} - ϵ_{p} - ϵ_{k}) + (ϵ_{p} - ϵ_{p - q}$

Conclusions

This paper considered two main issues related to the interplay between e–e correlations and e–ph coupling. We first investigated the effect of the e–e Hubbard repulsion on the Holstein e–ph coupling, showing that this latter is generically suppressed. This finding is consistent with previous results based on general Fermi-liquid arguments [8], on large-N expansions based on the slave-boson [7], [8] or Hubbard-operator [9] techniques and on numerical quantum Monte Carlo approaches [11].

Acknowledgements

One of us (C.D.C.) gratefully acknowledges stimulating discussions with F. Wegner. We all acknowledge interesting discussions with M. Capone, C. Castellani, and R. Raimondi. This work was financially supported by the Ministero Italiano dell'Università e della Ricerca with the Projects COFIN 2003 (prot. $2003020230_{0} 06$ ) and COFIN 2005 (prot. $205022492_{0} 01$ ). One of us (C.D.C.) also thanks the von Humboldt Fundation.

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Effective electron–electron and electron–phonon interactions in the Hubbard–Holstein model

Abstract

Introduction

Section snippets

Model

Method

Two-site lattice

Effective electron–electron interaction

Conclusions

Acknowledgements

Physica C

Nucl. Phys. B

Phys. Rev. Lett.

Phys. Rev.

Phys. Rev. B

Phys. Rev. Lett.

Europhys. Lett.

Phys. Rev. Lett.

Structure and Bonding

Nature (London)

Proc. R. Soc. London A

Phys. Rev. Lett.

Phys. Rev. B

Phys. Rev. B

Phys. Rev. B

Phys. Rev. B