| Research Abstract |
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.
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