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

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 h 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 functionΓD2νD2 in relating the scalar tex verfunctionΓwith the self-energy∑through the Widentity。I have improved on this GISC systematically by introducing the ratio function R，defined as the ratio betweenΓand the longitudinal part ofΓD 2.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 betweenΓandΣdue to the Panli principle；(Ii)it is not the case for the processes far from the Fermi surface；and(Iii)ong struct of the fermi surface；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 inΓ。

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.