A rigorous treatment of the perturbation theory for many-electron systems

TL;DR

This paper rigorously analyzes the perturbation theory for many-electron systems at finite temperature, providing bounds on convergence and implementing second-order calculations for 2D lattices.

Contribution

It introduces a rigorous framework for perturbation theory in many-electron systems, including convergence bounds and numerical implementation up to second order.

Findings

Established lower bounds on the radius of convergence.

Derived upper bounds on the perturbation series.

Numerically implemented second-order perturbation calculations for 2D systems.

Abstract

Four point correlation functions for many electrons at finite temperature in periodic lattice are analyzed by the perturbation theory with respect to the coupling constant. The correlation functions are characterized as a limit of finite dimensional Grassmann integrals. A lower bound on the radius of convergence and an upper bound on the perturbation series are obtained. The perturbation series up to second order is numerically implemented along with the volume-independent upper bounds on the sum of the higher order terms in 2 dimensional case.