Spectral properties of four-time fermionic Green's functions

Shvaika Shvaika

doi:10.5488/cmp.19.33004

Abstract

The spectral relations for the four-time fermionic Green's functions are derived in the most general case. The terms which correspond to the zero-frequency anomalies, known before only for the bosonic Green's functions, are separated and their connection with the second cumulants of the Boltzmann distribution function is elucidated. The high-frequency expansions of the four-time fermionic Green's functions are provided for different directions in the frequency space.

Highlights

One of the main tasks of the quantum many-body theory is, on the one hand, to calculate the observable quantities that could be measured directly by experiment, and, on the other hand, to provide connections between the measured quantities and microscopic properties of a system
Kubo [1] that linear transport coefficients are expressed in terms of the Fourier transforms of appropriate correlation functions that relate by spectral relations to the two-time Green’s functions
Green’s function method has been admitted and extensively developed [2,3,4]. In his seminal article Kubo [1] pointed out the difference between the isothermal and adiabatic response of the many-body system and its connection with the ergodic properties of a system. Later on it was noticed by Stevens and Toombs [6] that spectral relations should be completed by a special treatment of an additional contribution at zero frequency connected with the presence of conserved quantities [7, 8]

Summary

Introduction

One of the main tasks of the quantum many-body theory is, on the one hand, to calculate the observable quantities that could be measured directly by experiment, and, on the other hand, to provide connections between the measured quantities and microscopic properties of a system Spectral relations for multi-time, i.e., three-time Green’s functions of Kubo type, were originally introduced by Bonch-Bruevich [4, 30] before the zero-frequency anomaly problem was noticed. Spectral relations for three-time bosonic Matsubara Green’s functions taking into account zero-frequency anomalies were considered by Shvaika [31] and there were obtained solutions of the reverse problem, i.e., finding of spectral densities from the known Green’s functions. In this article we consider spectral relations for the four-time fermionic Matsubara Green’s functions with special emphasis on zero-frequency anomalies. We provide the high-frequency asymptotics in section 4 and in the last section we conclude

Four-time correlation functions and spectral densities

Four-time Matsubara Green’s function

Zero-frequency anomaly and cumulants

Analytic continuation and reverse engineering problem

High frequency asymptotics

Summary