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Journal Club for Condensed Matter Physics

Superconductivity in the repulsive Hubbard model: an asymptotically exact weak-coupling solution

Published May 27, 2010• 1 citation
Publisher

Abstract

The existence and nature of superconductivity (or superfluidity) of Fermi systems interacting (primarily) via repulsive interactions has been a long-standing subject, ever since BCS-theory was proposed.The pioneering works of Kohn and Luttinger (1965), Berk and Schrieffer (1966), Anderson and Brinkman(1975) explained correctly how an attractive ef-

DOI
10.36471/jccm_may_2010_03

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The existence and nature of superconductivity (or superfluidity) of Fermi systems interacting (primarily) via repulsive interactions has been a long-standing subject, ever since BCS-theory was proposed.The pioneering works of Kohn and Luttinger (1965), Berk and Schrieffer (1966), Anderson and Brinkman(1975) explained correctly how an attractive ef-
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We revisit the renormalization group (RG) analysis of the Kohn-Luttinger (KL) mechanism for superconductivity. The KL mechanism leads to superconductivity in a system with a repulsive bare interaction. The key ingredient is the screening effect that renders the induced interaction attractive in channels with nonzero angular momentum $\ensuremath{\ell}\ensuremath{\ne}0$, thereby triggering the Bardeen-Cooper-Schrieffer (BCS) instability. According to the original argument, the resulting gap is exponentially small, with its exponent scaling as $\ensuremath{-}{\ensuremath{\ell}}^{4}$. However, the KL mechanism was originally formulated within perturbation theory, where the series is known to converge poorly in certain cases---most notably, for the ppp-wave paring gap induced by a repulsive sss-wave contact interaction. This poor convergence may be attributed to a divergent integrand in a specific class of diagrams containing both the BCS logarithm and the Kohn anomaly, suggesting that one must resum the Kohn anomaly contributions separately from the BCS logarithm. In this work, we incorporate the Kohn anomaly contribution into the $\ensuremath{\beta}$ function of the RG equation governing the BCS instability near the Fermi surface. Our solution shows that the KL gap exponent is then proportional to $\ensuremath{-}\ensuremath{\ell}$, indicating a significant enhancement of the KL mechanism beyond the previously known result. To illustrate this, we study the spin-triplet ppp-wave pairing gap arising from a repulsive sss-wave contact interaction and compare our RG-based results with those obtained from the Bethe-Salpeter equation in perturbation theory.
In this short review, we first discuss the results, which are mainly devoted to the generalizations of the famous Kohn–Luttinger mechanism of superconductivity in purely repulsive fermion systems at low electron densities. In the context of repulsive-U Hubbard model and Shubin–Vonsovsky model we consider briefly the superconducting phase diagrams and the symmetries of the order parameter in novel strongly correlated electron systems including idealized monolayer and bilayer graphene. We stress that purely repulsive fermion systems are mainly the subject of unconventional low-temperature superconductivity. To get the high temperature superconductivity in cuprates (with T C of the order of 100 K) we should proceed to the t–J model with the van der Waals interaction potential and the competition between short-range repulsion and long-range attraction. Finally we note that to describe superconductivity in metallic hydrogen alloys under pressure (with T C of the order of 200 K) it is reasonable to reexamine more conventional mechanisms connected with electron–phonon interaction. These mechanisms arise in the attractive-U Hubbard model with static onsite or intersite attractive potential or in more realistic theories (which include retardation effects) such as Migdal–Eliashberg strong coupling theory or even Fermi–Bose mixture theory of Ranninger et al. and its generalizations.