Barycentric rational function approximation made simple: A fast analytic continuation method for Matsubara Green's functions

Abstract

Analytic continuation is a critical step in quantum many-body computations, connecting imaginary-time or Matsubara Green's functions with real-frequency spectral functions, which can be directly compared to experimental results. However, due to the ill-posed nature of the analytic continuation problems, they have not been completely solved so far. In this paper, we suggest a simple, yet highly efficient method for analytic continuations of Matsubara Green's functions. This method takes advantage of barycentric rational functions to directly interpolate Matsubara Green's functions. At first, the nodes and weights of the barycentric rational functions are determined by the adaptive Antoulas-Anderson algorithm, avoiding reliance on the non-convex optimization. Next, the retarded Green's functions and the relatively spectral functions are evaluated by the resulting interpolants. We systematically explore the performance of this method through a series of toy models and realistic examples, comparing its accuracy and efficiency with other popular methods, such as the maximum entropy method. The benchmark results demonstrate that the new method can accurately reproduce not only continuous but also discrete spectral functions, irrespective of their positive definiteness. It works well even in the presence of intermediate noise, and outperforms traditional analytic continuation methods in computational speed. We believe that this method should stand out for its robustness against noise, broad applicability, high precision, and ultra efficiency, offering a promising alternative to the maximum entropy method.