Spectral representation of Matsubara n-point functions: Exact kernel functions and applications

TL;DR

This paper derives universal kernel functions for spectral representations of Matsubara n-point functions, enabling simplified calculations in quantum many-body physics across various systems and operator types.

Contribution

It provides the complete form of kernel functions for arbitrary n, operator types, and anomalous terms, extending the spectral representation framework.

Findings

Derived explicit kernel functions for arbitrary n-point functions.

Applied the framework to bosonic 3- and 4-point functions in specific models.

Facilitated simplified calculations of correlation functions in quantum systems.

Abstract

In the field of quantum many-body physics, the spectral (or Lehmann) representation simplifies the calculation of Matsubara n-point correlation functions if the eigensystem of a Hamiltonian is known. It is expressed via a universal kernel function and a system- and correlator-specific product of matrix elements. Here we provide the kernel functions in full generality, for arbitrary n, arbitrary combinations of bosonic or fermionic operators and an arbitrary number of anomalous terms. As an application, we consider bosonic 3- and 4-point correlation functions for the fermionic Hubbard atom and a free spin of length S, respectively.