Direct numerical simulation of the 't Hooft partition function and (de)confining phases

TL;DR

This paper introduces a Monte Carlo method to directly measure the 't Hooft partition function in lattice Yang--Mills theory, revealing phase behaviors and flux sector characteristics in confinement and finite-temperature regimes.

Contribution

It develops a novel Monte Carlo approach to measure the 't Hooft partition function without reweighting, enabling detailed flux sector analysis in lattice gauge theories.

Findings

Observed characteristic 'light/heavy' behavior in the confining phase

Detected flux sector shifts due to the Witten effect at θ=2π

Performed preliminary finite-temperature analysis

Abstract

The 't Hooft partition function $Z_{\mathrm{tH}}[E_i;B_{ij}]$ is a discrete Fourier transform of Yang--Mills partition functions in background $\mathbb{Z}_N$ 2-form gauge fields and encodes information on confinement, Higgs, Coulomb and oblique-confining phases. We report a direct Monte Carlo strategy to measure $Z_{\mathrm{tH}}$ without reweighting, by extending hybrid Monte Carlo to include dynamical updates of the background flux variables. As a first application we measure all flux sectors of four-dimensional $SU(2)$ lattice Yang--Mills on $T^4$ and observe the characteristic ``light/heavy'' behavior expected in the confining phase, together with the shift implied by the Witten effect at $\theta=2\pi$. We also present a preliminary finite-temperature study and discuss outstanding issues on thermalization and separability between different flux sectors.