Jahn-Teller Polaron and its Superconductivity

在讨论电子-声子耦合系统中的极化子效应和超导性时，通常采用Froelich模型或Holstein模型作为理论基础。然而，由于简并态的存在和伴随的内部结构，Jahn-Teller晶体中的电子-声子复合系统表现出类似于夸克-胶子系统中的约束问题的新特征。一般来说，内部结构可以用伪角动量的概念来描述，它的守恒定律对电子-声子相互作用的工作方式施加了限制。考虑到这样一个明确的限制，我写下了系统的哈密顿量，在此基础上，我用Migdal-Eliashberg理论证明了在这个系统中只允许某些类型的顶点修正。这种对可能顶点修正的限制使我发现，与荷尔斯泰因模型相比，简-泰勒极化子的有效质量在c…中急剧下降。特别地，当电子-声子耦合常数α较大时，我解析地推导出有效质量增强因子不是通常的荷尔斯坦因子e^<2α>，而是（2/πα）^<1/2>e^α。我将上述关于单电子问题的研究扩展到多电子问题，并从Eliashberg强耦合超导理论的角度研究了顶点修正对超导转变温度T_c的影响。结果，我发现在双极化子超导的情况下，由于有效质量非常大，即使在Jahn-Teller系统中，当α变得如此之高以至于顶点修正不能忽略时，T_c也不会变得很高。对于α小于该值的情况，顶点修正根本不起作用，因此传统的麦克米伦公式（或Allen-Dynes公式）足以估计T_c。顺便提一下，一阶顶点校正在Jahn-Teller系统中严格消失，这使得Eliashberg理论的适用范围非常广泛。基于这个理由，我可以否认经常提到的论点，即扬-泰勒系统中的超导性与其他电子-声子系统中的超导性有很大不同。在总结这些结果时，我得出的结论是，我们应该从处理hubard - holstein （HH）模型开始，以追求解决超导性的深刻问题，即在顶点修正不能忽略的强相关电子-声子系统中构建超导性的强耦合理论。从这个角度出发，我建立了一种新的处理顶点修正的通用方法，并成功地应用于充分纳入相关效应的均质电子液体。我还成功地利用变位移Lang-Firsov变换将HH模型的两点解扩展到一维的无限点系统，以及Lieb-Wu精确解到Hubbard模型。少

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