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arXiv:cond-mat/9609065 · Annalen der Physik

Similarity renormalization of the electron‐phonon coupling

Andreas Mielke

Published Jan 1, 1997• 40 citations
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Abstract

Abstract We study the problem of the phonon‐induced electron‐electron interaction in a solid. Starting with a Hamiltonian that contains an electron‐phonon interaction, we perform a similarity renormalization transformation to calculate an effective Hamiltonian. Using this transformation singularities due to degeneracies are avoided explicitly. The effective interactions are calculated to second order in the electronphonon coupling. It is shown that the effective interaction between two electrons forming a Cooper pair is attractive in the whole parameter space. For a simple Einstein model we calculate the renormalization of the electronic energies and the critical temperature of superconductivity.

DOI
10.1002/andp.19975090305

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Annalen der Physik•Nov 14, 2003•0 citations
Abstract We study the problem of the phonon‐induced electron‐electron interaction in a solid. Starting with a Hamiltonian that contains an electron‐phonon interaction, we perform a similarity renormalization transformation to calculate an effective Hamiltonian. Using this transformation singularities due to degeneracies are avoided explicitly. The effective interactions are calculated to second order in the electronphonon coupling. It is shown that the effective interaction between two electrons forming a Cooper pair is attractive in the whole parameter space. For a simple Einstein model we calculate the renormalization of the electronic energies and the critical temperature of superconductivity.
We have investigated the ground-state properties of a model system with a strong electron-phonon interaction. It is found that as long as the renormalized intersite electron correlation is attractive, a new type of Cooper pairing will occur whether the renormalized on-site electron correlation is repulsive or attractive. Furthermore, we have shown that a new phonon coherent state, named a two-phonon coherent state, must be introduced to serve as the ground state of the phonon subsystem, so that the ground-state energy of the interacting system can be made a stable minimum and an appropriate condensation energy of the superconducting ground state may be obtained. We have performed numerical calculations for the n=1 case.
Higher-order terms of the transformed electron-phonon Hamiltonian ${H}_{e\text{\ensuremath{-}}\mathit{ph}}$, obtained by performing the Fr\"ohlich transformation, are investigated. The influence of terms discarded by Fr\"ohlich (in particular those proportional to the third power of electron-phonon coupling) on the effective Hamiltonian is examined. To this end a second Fr\"ohlich-type transformation is performed, which yields, among others, an effective four-electron interaction. This interaction is reduced to a form admitting solution of thermodynamics. The form of the coupling of the four-electron interaction is found. By applying standard approximations, it is shown that this interaction is attractive with the interaction coupling given by $\ensuremath{-}{D}_{{\mathbf{k}}_{F}}^{6}∕{\ensuremath{\omega}}_{{\mathbf{k}}_{F}}^{5}$, where Dk{D}_{\mathbf{k}}Dk​ is electron-phonon coupling, ${\ensuremath{\omega}}_{\mathbf{k}}$ is the phonon energy, and kF{\mathbf{k}}_{F}kF​ is the Fermi momentum. The form of higher-order terms of the original Fr\"ohlich-transformed ${H}_{e\text{\ensuremath{-}}\mathit{ph}}$ is also found, up to terms proportional to the sixth power of the coupling---that is, up to those which yield the effective four-electron interactions.
We have addressed the problem of the influence of Coulomb correlations on the electron-phonon interaction in the two-dimensional Hubbard model. Both the energy and momentum dependence of the renormalization factor have been taken into account. We find that the superconducting properties of the system depend strongly on the phononic momentum and energy transfer values. This dependence is related to the symmetry of the superconducting order parameter. On one hand, the renormalization of the electron-phonon coupling function acts to the detriment of superconductivity in the s-wave channel. On the other hand, an enhancement of pairing correlations for anisotropic extended s-wave and d-wave superconductivity is observed.
This article presents a systematic theoretical enquiry concerning the conceptual foundations and the nature of phonon-mediated electron-electron interactions. Starting from the fundamental many-body Hamiltonian, we propose a simple scheme to decouple the electrons and nuclei of a crystalline solid via effective interactions. These effective interactions, which we express in terms of linear response functions, are completely symmetric under the exchange of electrons and nuclei. Correspondingly, we derive concrete formulae for both the effective electron interaction mediated by phonons and the effective core interaction mediated by electrons. In particular, we rederive from our fundamental ansatz the well-known general expressions of the effective electron-electron interaction in terms of the elastic Green function and the phonon dispersion relation. We further show that the effective core interaction coincides in the instantaneous limit with the dynamical matrix as calculated in electronic structure theory. If combined with the Kubo formalism, our general formulae lend themselves to the calculation of effective interactions from first principles. By showing the compatibility of our approach with the functional integral formalism, this work also paves the way for the derivation of ab initio initial interactions for functional renormalization group applications.
We investigate the interaction of correlated electrons with acoustical phonons using the extended Hubbard-Holstein model. The Lang-Firsov canonical transformation allows us to obtain mobile polarons for which a diagram technique and generalized Wick's theorem is used. In the strong-coupling limit of the electron-phonon interaction, and in the normal as well as in the superconducting phase, chronological thermodynamical averages of products of acoustical phonon-cloud operators can be expressed by one-cloud propagators. The pairing of electrons in the presence of acoustical phonon clouds is discussed and the critical temperature of the superconducting phase transition is determined for the special case U<0U<0U<0, where UUU is the renormalized value of the on-site Coulomb repulsion. The renormalization is caused by the electron-phonon interaction.
Abstract We present a general method to study weak‐coupling instabilities of a large class of interacting electron models in a controlled and unbiased way. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high‐energy spin fluctuations create an effective attraction between the quasi‐particles. As an example, we show the occurrence of d ‐wave pairing in the repulsive Hubbard model in two dimensions. In one dimension or if the Fermi surface is nested, there are several competing instabilities. The required renormalization group formalism for this case is presented to lowest (one‐loop) order on a most elementary level, connecting the idea of the “parquet summation” to the more modern concept of Wilson's effective action. The validity and restrictions of the one‐loop approximation are discussed in detail. As a result, three different renormalization group approaches known in the literature are shown to be equivalent within the regime of applicability. We also briefly discuss the open problem of a two‐dimensional Fermi system at Van Hove filling without nesting.