The Average Spectrum Method for Analytic Continuation: Efficient Blocked Modes Sampling and Dependence on Discretization Grid

TL;DR

This paper improves the average spectrum method for analytic continuation by introducing an efficient implementation using singular value decomposition, analyzing the impact of discretization, and providing practical guidance for default model selection.

Contribution

It presents a computationally efficient implementation of the average spectrum method and analyzes how discretization grid choices affect the results, enhancing its reliability.

Findings

Discretization biases spectral results.

Grid point distribution acts as a default model.

Regularization depends on the number of grid points.

Abstract

The average spectrum method is a promising approach for the analytic continuation of imaginary time or frequency data to the real axis. It determines the analytic continuation of noisy data from a functional average over all admissible spectral functions, weighted by how well they fit the data. Its main advantage is the apparent lack of adjustable parameters and smoothness constraints, using instead the information on the statistical noise in the data. Its main disadvantage is the enormous computational cost of performing the functional integral. Here we introduce an efficient implementation, based on the singular value decomposition of the integral kernel, eliminating this problem. It allows us to analyze the behavior of the average spectrum method in detail. We find that the discretization of the real-frequency grid, on which the spectral function is represented, biases the results. The distribution of the grid points plays the role of a default model while the number of grid points acts as a regularization parameter. We give a quantitative explanation for this behavior, point out the crucial role of the default model and provide a practical method for choosing it, making the average spectrum method a reliable and efficient technique for analytic continuation.