Algorithmic Matsubara Integration for Hubbard-like models

TL;DR

This paper introduces an exact algorithm for evaluating Matsubara sums in Hubbard-like models, enabling analytic continuation and efficient exploration of phase space without numerical instability.

Contribution

The authors develop a symbolic algorithm for exact Matsubara sum evaluation, facilitating analytic continuation and comprehensive phase space analysis in Hubbard models.

Findings

Exact analytic results for Matsubara sums are obtained.

Method enables analytic continuation to real frequencies.

Compatible with diagrammatic Monte Carlo for efficient phase space exploration.

Abstract

We present an algorithm to evaluate Matsubara sums for Feynman diagrams comprised of bare Green's functions with single-band dispersions with local U Hubbard interaction vertices. The algorithm provides an exact construction of the analytic result for the frequency integrals of a diagram that can then be evaluated for all parameters $U$, temperature $T$, chemical potential $\mu$, external frequencies and internal/external momenta. This method allows for symbolic analytic continuation of results to the real frequency axis, avoiding any ill-posed numerical procedure. When combined with diagrammatic Monte-Carlo, this method can be used to simultaneously evaluate diagrams throughout the entire $T-U-\mu$ phase space of Hubbard-like models at minimal computational expense.