Jahn-Teller Polaron and its Superconductivity

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In discussing the polaron effect and superconductivity in an electron-phonon coupled system, either the Froelich or the Holstein model is usually adopted as a theoretical basis. However, the electron-phonon complex system in a Jahn-Teller crystal exhibits a new feature similar to the confinement problem in the quark-gluon system due to the existence of degenerate states and the concomitant internal structure.Generally the internal structure can be described in terms of a concept of pseudo-angular momentum and its conservation law imposes a restriction on the way how the electron-phonon interaction works. Taking explicitly such a restriction into account, I have written down the Hamiltonian of the system, based on which I have shown using the Migdal-Eliashberg theory that only some class of vertex corrections is allowed in this system. This restriction on the possible vertex corrections leads me to the finding that the effective mass of the Jahn-Teller polaron decreases drastically in c … More omparison with that of the Holstein model. In particular, when the electron-phonon coupling constant α is large, I have derived analytically that the effective-mass enhancement factor is not the usual Holstein factor e^<2α> but (2/πα)^<1/2>e^α.I have extended the above study on the one-electron problem to the many-electron one and investigated the effect of the vertex corrections on the superconducting transition temperature T_c from a viewpoint of the Eliashberg's theory on strong-coupling superconductivity. As a result, I have found that T_c does not become high in the scenario of bipolaron superconductivity due to the very heavy effective mass even in the Jahn-Teller system, when α becomes so high that the vertex corrections cannot be neglected. For the case of α smaller than that, the vertex corrections do not contribute at all and thus the conventional McMillan formula (or the Allen-Dynes one) suffices in estimating T_c. Incidentally, the first-order vertex correction vanishes rigorously in the Jahn-Teller system, which makes the applicable range of the Eliashberg theory very wide. On this ground, I can deny the frequently-mentioned argument that superconductivity in the Jahn-Teller system differs very much from that in other electron-phonon systems.In summarizing those results, I come to the conclusion that we should start with treating the Hubbard-Holstein (HH) model in the pursuit of solving the profound problem in superconductivity, namely, constructing a strong-coupling theory of superconductivity in a strongly-correlated electron-phonon system where the vertex corrections cannot be neglected. From this perspective, I have established a new general method of treating the vertex corrections and successfully applied it to the homogeneous electron liquid with fully incorporating correlation effects. I have also succeeded in extending the two-site solution to the HH model to the infinite-site system in one dimension by using a variable-displacement Lang-Firsov transformation as well as the Lieb-Wu exact solution to the Hubbard model. Less