Maximum Entropy Method Approach to $\theta$ Term

TL;DR

This paper explores the application of the maximum entropy method to address the sign problem in lattice field theory with a $ heta$ term, aiming to recover the free energy and phase structure despite flattening issues.

Contribution

It demonstrates that the maximum entropy method can effectively mitigate flattening in the free energy calculation, providing smoother results than Fourier transform methods in certain cases.

Findings

MEM produces smoother $f(\theta)$ than Fourier transform in flattened cases.

MEM results agree with Fourier transform when no flattening occurs.

Default models influence the error and flattening detection in MEM analysis.

Abstract

In Monte Carlo simulations of lattice field theory with a $\theta$ term, one confronts the complex weight problem, or the sign problem. This is circumvented by performing the Fourier transform of the topological charge distribution $P(Q)$. This procedure, however, causes flattening phenomenon of the free energy $f(\theta)$, which makes study of the phase structure unfeasible. In order to treat this problem, we apply the maximum entropy method (MEM) to a Gaussian form of $P(Q)$, which serves as a good example to test whether the MEM can be applied effectively to the $\theta$ term. We study the case with flattening as well as that without flattening. In the latter case, the results of the MEM agree with those obtained from the direct application of the Fourier transform. For the former, the MEM gives a smoother $f(\theta)$ than that of the Fourier transform. Among various default models investigated, the images which yield the least error do not show flattening, although some others cannot be excluded given the uncertainty related to statistical error.