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Physical review. B, Condensed matter

Correlation-renormalized electron-phonon interaction in the two-dimensional Hubbard model

Marcin Mierzejewski, J. Zieliński, P. Entel

Published Oct 1, 1999• 7 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.60.10442

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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.
We address the problem of superconductivity in a system where the only many-body interactions are repulsive. The Eliashberg equations have been generalized to account for possible pairing correlations in the two-dimensional Hubbard model. Details of the two-dimensional band structure have been explicitly taken into account when considering the symmetry of the superconducting state. The pairing kernels have been discussed at low doping and at temperatures close to the superconducting transition temperature. We have proved that local Coulomb repulsion leads to attractive pairing correlations in the ddd-wave channel. Extended sss-wave superconductivity is less likely to occur within this purely electronic model, at least in the physically interesting region of doping.
The problem of phonon-mediated and phonon-free contributions to d-wave superconductivity in the two-dimensional Hubbard model has been addressed. Strong local correlations (U\ensuremath{\rightarrow}\ensuremath{\infty} limit) have been incorporated in the slave boson formulation. In order to consider electron-boson and electron-phonon coupling on equal footing, fluctuations of the auxiliary boson fields over their mean-field value have been introduced. The magnitude of effective pairing interactions derived with the help of a canonical transformation indicates that within the weak-coupling theory, phonon-mediated pairing plays a marginal role and can be dominated by a phonon-free pairing mechanism. On the other hand, in the strong-coupling limit (Eliashberg formulation), electron-phonon and electron-phonon-boson interactions can significantly enhance the superconducting transition temperature. It is also shown that Coulomb correlations together with the electron-phonon interaction can effectively stabilize a d-wave superconducting state. \textcopyright{} 1996 The American Physical Society.
The renormalization of the electron-phonon interaction by strong electronic correlations is studied using a one-band Hubbard model with infinite repulsion and nearest (t model) and nearest and second-nearest (tt\ensuremath{'} model) neighbor hopping terms and an on-site electron-phonon coupling. Using Hubbard's X operators and an extension from 2 to N degrees of freedom for the electrons the leading contributions for the electron self-energy and the vertex function in 1/N and the electron-phonon coupling constant are given and numerically evaluated for a square lattice. We find that the momentum dependence of the vertex function depends strongly on doping: For large dopings it is rather weak, with decreasing doping it becomes more and more pronounced leading to a strong reduction of the vertex at larger momentum transfers until, for very small dopings, the vertex essentially consists of a forward scattering peak with a width proportional to the doping. This behavior occurs both in the t and the tt\ensuremath{'} models and also in one dimension where analytic expressions are derived. Correlation effects also change ${\mathrm{\ensuremath{\alpha}}}^{2}$F in general: The full-symmetric component ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ is in the t model somewhat, in the tt\ensuremath{'} model rather strongly suppressed, especially near half-filling; the other symmetry components ${\mathrm{\ensuremath{\alpha}}}^{2}$Fi{\mathit{F}}_{\mathit{i}}Fi​ with i=2,...,5 increase strongly with decreasing doping and are of similar magnitude as ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ near half-filling.Including also direct Coulomb repulsion nontrivial symmetries such as d wave become more stable than the s-wave order parameter below a critical value for the doping even for the considered phonon-mediated superconductivity. Most dramatic, however, is the quenching of the resistivity due to electron-phonon scattering both in the t and the tt\ensuremath{'} models at intermediate and small dopings. This result may explain the absence of phonon features in the experimental transport coefficients of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compounds. \textcopyright{} 1996 The American Physical Society.
The properties of the electron-phonon interaction in the presence of a sizable electronic repulsion at finite doping are studied by investigating the metallic phase of the Hubbard-Holstein model with dynamical mean-field theory. Analyzing the quasiparticle weight at finite doping, we find that a large Coulomb repulsion reduces the effect of electron-phonon coupling at low energy, while this reduction is not present at high energy. The renormalization of the electron-phonon coupling induced by the Hubbard repulsion depends in a surprisingly strong and nontrivial way on the phonon frequency. Our results suggest that phonon might affect differently high-energy and low-energy properties and this, together with the effect of phonon dynamics, should be carefully taken into account when the effects of the electron-phonon interaction in a strongly correlated system, like the superconducting cuprates, are discussed.
A microscopic theory of superconductivity in the extended Hubbard model which takes into account the intersite Coulomb repulsion and electron-phonon interaction is developed in the limit of strong correlations. The Dyson equation for normal and pair Green functions expressed in terms of the Hubbard operators is derived. The self-energy is obtained in the noncrossing approximation. In the normal state, antiferromagnetic short-range correlations result in the electronic spectrum with a narrow bandwidth. We calculate superconducting Tc by taking into account the pairing mediated by charge and spin fluctuations and phonons. We found the d-wave pairing with high-Tc mediated by spin fluctuations induced by the strong kinematic interaction for the Hubbard operators. Contributions to the d-wave pairing coming from the intersite Coulomb repulsion and phonons turned out to be small.