# Sparsity pattern of the self-energy for classical and quantum impurity problems

## TL;DR

This paper rigorously proves that the self-energy matrix in classical and quantum impurity models exhibits a sparsity pattern dictated by impurity sites, which is fundamental for numerical methods in many-body physics.

## Contribution

It provides the first rigorous proof of the sparsity pattern of the self-energy matrix in impurity problems, extending known quantum results to classical and broader quantum systems.

## Key findings

- Self-energy matrix is sparse with pattern determined by impurity sites.
- Classical Gibbs measure precision matrix differs from Gaussian by a sparse matrix.
- Proof is non-perturbative and adaptable to various physical systems.

## Abstract

We prove that for various impurity models, in both classical and quantum settings, the self-energy matrix is a sparse matrix with a sparsity pattern determined by the impurity sites. In the quantum setting, such a sparsity pattern has been known since Feynman. Indeed, it underlies several numerical methods for solving impurity problems, as well as many approaches to more general quantum many-body problems, such as the dynamical mean field theory. The sparsity pattern is easily motivated by a formal perturbative expansion using Feynman diagrams. However, to the extent of our knowledge, a rigorous proof has not appeared in the literature. In the classical setting, analogous considerations lead to a perhaps less-known result, i.e., that the precision matrix of a Gibbs measure of a certain kind differs only by a sparse matrix from the precision matrix of a corresponding Gaussian measure. Our argument for this result mainly involves elementary algebraic manipulations and is in particular non-perturbative. Nonetheless, the proof can be robustly adapted to various settings of interest in physics, including quantum systems (both fermionic and bosonic) at zero and finite temperature, non-equilibrium systems, and superconducting systems.