Connecting Tikhonov regularization to the maximum entropy method for the analytic continuation of quantum Monte Carlo data

TL;DR

This paper introduces a Tikhonov regularization approach for analytic continuation of quantum Monte Carlo data, showing its close relation to the maximum entropy method and providing practical noise estimation techniques.

Contribution

It establishes a connection between Tikhonov regularization and MaxEnt, offering a simple implementation and noise estimation method for improved analytic continuation.

Findings

Tikhonov regularization yields results similar to MaxEnt.

The method is easy to implement in standard linear algebra packages.

A practical noise estimation technique for QMC data is proposed.

Abstract

Analytic continuation is an essential step in extracting information about the dynamical properties of physical systems from quantum Monte Carlo (QMC) simulations. Different methods for analytic continuation have been proposed and are still being developed. This paper explores a regularization method based on the repeated application of Tikhonov regularization under the discrepancy principle. The method can be readily implemented in any linear algebra package and gives results surprisingly close to the maximum entropy method (MaxEnt). We analyze the method in detail and demonstrate its connection to MaxEnt. In addition, we provide a straightforward method for estimating the noise level of QMC data, which is helpful for practical applications of the discrepancy principle when the noise level is not known reliably.