Analytic continuation by averaging Pad\'e approximants

TL;DR

This paper introduces an improved Padé approximants method for analytic continuation of Green's functions, averaging multiple fits to enhance stability and accuracy, and compares it favorably to other techniques across various test cases.

Contribution

The authors propose an averaging approach over multiple Padé approximants to address the ill-posed nature of analytic continuation, demonstrating improved robustness and resolution.

Findings

The averaging method outperforms traditional Padé approximants in noisy conditions.

It compares favorably with maximum entropy and Tikhonov methods in test cases.

The approach effectively resolves fine spectral features.

Abstract

The ill-posed analytic continuation problem for Green's functions and self-energies is investigated by revisiting the Pad\'{e} approximants technique. We propose to remedy the well-known problems of the Pad\'{e} approximants by performing an average of several continuations, obtained by varying the number of fitted input points and Pad\'{e} coefficients independently. The suggested approach is then applied to several test cases, including Sm and Pr atomic self-energies, the Green's functions of the Hubbard model for a Bethe lattice and of the Haldane model for a nano-ribbon, as well as two special test functions. The sensitivity to numerical noise and the dependence on the precision of the numerical libraries are analysed in detail. The present approach is compared to a number of other techniques, i.e. the non-negative least-square method, the non-negative Tikhonov method and the maximum entropy method, and is shown to perform well for the chosen test cases. This conclusion holds even when the noise on the input data is increased to reach values typical for quantum Monte Carlo simulations. The ability of the algorithm to resolve fine structures is finally illustrated for two relevant test functions.