Sign problem and MEM in lattice field theory with the $\theta$ term

TL;DR

This paper explores the application of the maximum entropy method (MEM) to address the sign problem in lattice field theory with a $ heta$ term, comparing it to the Fourier transform method (FTM) and analyzing flattening effects.

Contribution

It demonstrates that MEM can produce accurate partition function images for small $ heta$, and systematically investigates flattening phenomena related to the sign problem.

Findings

MEM yields results close to FTM for small $ heta$

Flattening behavior depends on the default model in MEM

MEM can identify the sign problem effects in lattice field theory

Abstract

Lattice field theory with the $\theta$ term suffers from the sign problem. The sign problem appears as flattening of the free energy. As an alternative to the conventional method, the Fourier transform method (FTM), we apply the maximum entropy method (MEM) to Monte Carlo data obtained using the CP$^3$ model with the $\theta$ term. For data without flattening, we obtain the most probable images of the partition function ${\hat{\cal Z}}(\theta)$ with rather small errors. The results are quantitatively close to the result obtained with the FTM. Motivated by this fact, we systematically investigate flattening in terms of the MEM. Obtained images ${\hat{\cal Z}}(\theta)$ are consistent with the FTM for small values of $\theta$, while the behavior of ${\hat{\cal Z}}(\theta)$ depends strongly on the default model for large values of $\theta$. This behavior of ${\hat{\cal Z}}(\theta)$ reflects the flattening phenomenon.