New Developments in the Green's-Function Method and Construction of the Strong-Coupling Theory for Superconductivity with Vertex Corrections

在一个强相关的系统中，我提出了两个古老层次理论的顺序，第一个阶段是建立一个初始框架，以提供没有注意到可操作性的实际结果，第二个是引入一个合理的方法在框架中的关键数量，并使用所有可用资源的全部帐户。自从我已经为第一阶段建立了自能量修订理论的操作员F，本研究的主要目标是第二阶段，命名为要找到一个很好的近似到F.最简单的近似是一个gauge-invariant self-consistent （GISC）方法中的一个I neglect the Vector vertex function能量，通过沃德的身份。I have improved on this GISC systematically by introducing the ratio function R， defined as the ratio between Γ and the longitudinal part of ≡イイD2·イエD2·イエD2·イエD2。我利用它的不对称性质在各种限制中构建了一个良好的近似形式，R.为了它的插图，我将此算法应用于电子气体，并找到后续： （i）靠近费米表面， RPA发现可以提供更好的结果，因为在Γ和Σ之间的更强的取消; （ii）这不是远离费米表面的工艺的情况;和（三）一个强的低能量转换可以在动态结构因子中观察到吸引式电子孔的多散射。我将应用相同的方法来实现真正的晶体，作为超行为，我将在GISC水平上开发理论，并在不同方面研究传统Eliashberg理论的有效性。I shall extend the theory so as to provide more accurate results with full attention to the role of Nambu-Goldstone modes in Γ。

In order to illuminate fundamental physics in dynamical processes in a strongly-correlated system, I propose a theory of twofold hierarchy The first stage is to set up a firm framework to provide exact results without attention to feasibility of computation and the second one is to introduce a relevant approximation to the key quantity in the framework with taking full account of available resources of computation. Since I have already established the theory of the self-energy revision operator F for the first stage, the main objective of this study is the second stage, namely, to obtain a good approximation to F.The simplest approximation is the gauge-invariant self-consistent (GISC) method in which I neglect the vector vertex function ΓィイD2νィエD2 in relating the scalar vertex function Γ with the self-energy Σ through the Ward identity. I have improved on this GISC systematically by introducing the ratio function R, defined as the ratio between Γ and the longitudinal part of ΓィイD2νィエD2. I exploit its asymptotic properties in various limits to construct a good approximate form for R.For its illustration, I apply this algorithm to the electron gas and find the following : (i) Near the Fermi surface, RPA is found to provide rather accurate results because of the strong cancellation between Γ and Σ due to the Panli principle; (ii) it is not the case for the processes far from the Fermi surface; and (iii) a strong lower-energy shift is observed in the dynamic structure factor owing to the attractive electron-hole excitonic multiple scatterings. I shall apply the same method to real crystals.As for superconductivity, I develop a theory on the GISC level and investigated the validity of the conventional Eliashberg theory in various aspects. I shall extend the theory so as to provide more accurate results with full attention to the role of Nambu-Goldstone modes in Γ.