Minimal Pole Representation and Controlled Analytic Continuation of Matsubara Response Functions

TL;DR

This paper introduces a systematic method for approximating Matsubara functions with minimal poles, enabling controlled and reliable analytic continuation to real frequencies in finite-temperature field theories.

Contribution

The authors develop a minimal pole representation approach that converges systematically and is robust to noise, improving the reliability of analytic continuation in many-body physics.

Findings

Converges systematically to the exact spectral function

Robust against noise in Matsubara data

Facilitates direct physical interpretation

Abstract

Analytical continuation is a central step in the simulation of finite-temperature field theories in which numerically obtained Matsubara data is continued to the real frequency axis for physical interpretation. Numerical analytic continuation is considered to be an ill-posed problem where uncertainties on the Matsubara axis are aplified exponentially. Here, we present a systematic and controlled procedure that approximates any Matsubara function by a minimal pole representation to within a predefined precision. We then show systematic convergence to the exact spectral function on the real axis as a function of our control parameter for a range of physically relevant setups. Our methodology is robust to noise and paves the way towards reliable analytic continuation in many-body theory and, by providing access to the analytic structure of the functions, direct theoretical interpretation of physical properties.