Nevanlinna Analytic Continuation for Migdal-Eliashberg Theory

TL;DR

This paper introduces a novel Nevanlinna-based analytic continuation method for Migdal-Eliashberg theory, enabling accurate real-frequency spectral functions from imaginary frequency Green's functions, thus bridging theoretical calculations and experimental observations.

Contribution

The work develops a Nevanlinna analytic continuation approach tailored for ME Green's functions, overcoming previous limitations and improving the extraction of real-frequency properties.

Findings

Accurate real-frequency spectral functions for MgB₂ and LaBeH₈ superconductors.

Enhanced compatibility of analytic continuation with causal Green's functions.

Demonstrated effectiveness of the method in practical superconductivity models.

Abstract

In this work, we present a method to reconstruct real-frequency properties from analytically continued causal Green's functions within the framework of Migdal-Eliashberg (ME) theory for superconductivity. ME theory involves solving a set of coupled equations self-consistently in imaginary frequency space, but to obtain experimentally measurable properties like the spectral function and quasiparticle density of states, it is necessary to perform an analytic continuation to real frequency space. Traditionally, the ME Green's function is decomposed into three fundamental complex functions, which are analytically continued independently. However, these functions do not possess the causal properties of Green's functions, complicating or even preventing the application of standard methods such as Maximum Entropy. Our approach overcomes these challenges, enabling the use of various analytic continuation techniques that were previously impractical. We demonstrate the effectiveness of this method by combining it with Nevanlinna analytic continuation to achieve accurate real-frequency results for ME theory, which are directly comparable to experimental data, with applications highlighted for the superconductors MgB$_2$ and LaBeH$_8$.