Numerical analytic continuation: Answers to well-posed questions

TL;DR

This paper presents a new approach to numerical analytic continuation that reliably assesses the accuracy and ambiguity of spectral functions derived from input data, using stochastic optimization and maximum entropy methods.

Contribution

It introduces a formulation that enables meaningful conclusions about spectral functions from data, addressing accuracy and ambiguity with novel methods.

Findings

Effective characterization of spectral function accuracy

Application to Fermi polaron problem

Critical analysis of spectral property accessibility

Abstract

We formulate the problem of numerical analytic continuation in a way that lets us draw meaningful conclusions about properties of the spectral function based solely on the input data. Apart from ensuring consistency with the input data (within their error bars) and the {\it a priori} and {\it a posteriori} (conditional) constraints, it is crucial to reliably characterize the accuracy---or even ambiguity---of the output. We explain how these challenges can be met with two approaches: stochastic optimization with consistent constraints and the modified maximum entropy method. We perform illustrative tests for spectra with a double-peak structure, where we critically examine which spectral properties are accessible and which ones are lost. For an important practical example, we apply our protocol to the Fermi polaron problem.