Spectral reconstruction techniques, their shortcomings and relevance to the electric conductivity coefficient

TL;DR

This paper evaluates machine learning and novel spectral reconstruction methods for ill-posed inverse problems, applying them to lattice QCD data to estimate electric conductivity, highlighting their effectiveness and limitations.

Contribution

It introduces a machine learning framework and a new multipoint method for spectral reconstruction, comparing them with existing techniques using mock and lattice data.

Findings

Machine learning methods show promise in spectral reconstruction.

The multipoint method provides insights near zero frequency.

Reconstructed spectral functions enable estimation of electric conductivity.

Abstract

Spectral reconstruction is a well studied numerically ill-posed problem which arises due to the relation of the Euclidean correlator to the spectral function via an inhomogeneous Fredholm equation of the first kind. Several different methods are on the market to resolve this issue, each taking different approaches and assumptions. In this proceedings we focus on implementing and testing a machine learning framework for spectral reconstruction, as well as implementing a novel method of estimating the behavior of the spectral function in the vicinity of vanishing frequency, which we denote as multipoint method, and compare these methods to well established spectral reconstruction techniques from the literature using mock data. As a physics application, we apply the reconstruction techniques to quenched lattice data for the correlation function in the vector channel at non-zero external magnetic field to extract the spectral function and the electric conductivity through its behaviour at vanishing frequency via a Kubo formula.