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

Okuto Morikawa, Hiroshi Suzuki

TL;DR

This work develops a direct, reweighting-free approach to measure the 't Hooft partition function $Z_{tH}$ by embedding dynamical background fluxes $B$ into a halfway-updating HMC framework. By sampling both gauge links and discrete flux sectors, the method directly estimates $Z[B]$ and the discrete Fourier transform $Z_{tH}$ across all flux sectors, enabling probing of confinement, Higgs, Coulomb, and oblique-confining phases on $T^4$ for SU(2) YM and revealing the Witten effect at $\theta=2\pi$. The authors demonstrate the approach with SU(2) on $T^4$ at $\beta=2.6$ and present a preliminary finite-temperature analysis, while discussing thermalization/separability challenges and the need for larger statistics and optimized flux updates. Overall, the work builds a concrete link between center symmetry, fractional topological charge, and phase structure, offering a direct lattice route to compare with symmetry-TFT/anomaly inflow descriptions of YM dynamics.

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 $θ=2π$. We also present a preliminary finite-temperature study and discuss outstanding issues on thermalization and separability between different flux sectors.

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

TL;DR

This work develops a direct, reweighting-free approach to measure the 't Hooft partition function by embedding dynamical background fluxes into a halfway-updating HMC framework. By sampling both gauge links and discrete flux sectors, the method directly estimates and the discrete Fourier transform across all flux sectors, enabling probing of confinement, Higgs, Coulomb, and oblique-confining phases on for SU(2) YM and revealing the Witten effect at . The authors demonstrate the approach with SU(2) on at and present a preliminary finite-temperature analysis, while discussing thermalization/separability challenges and the need for larger statistics and optimized flux updates. Overall, the work builds a concrete link between center symmetry, fractional topological charge, and phase structure, offering a direct lattice route to compare with symmetry-TFT/anomaly inflow descriptions of YM dynamics.

Abstract

The 't Hooft partition function is a discrete Fourier transform of Yang--Mills partition functions in background 2-form gauge fields and encodes information on confinement, Higgs, Coulomb and oblique-confining phases. We report a direct Monte Carlo strategy to measure 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 lattice Yang--Mills on and observe the characteristic ``light/heavy'' behavior expected in the confining phase, together with the shift implied by the Witten effect at . We also present a preliminary finite-temperature study and discuss outstanding issues on thermalization and separability between different flux sectors.
Paper Structure (8 sections, 11 equations, 4 figures)

This paper contains 8 sections, 11 equations, 4 figures.

Table of Contents

  1. Motivation and overview
  2. Center symmetry, background 2-form fields, and fractional topological charge
  3. The 't Hooft partition function
  4. Halfway-updating HMC with dynamical $B$ fields
  5. Novel HMC algorithm for $SU(N)/\mathbb{Z}_N$ theory
  6. Estimating $Z[B]$ and $Z_{\mathrm{tH}}$ by counting
  7. Numerical results: confining phase and Witten effect
  8. Finite temperature: preliminary study and outlook

Figures (4)

  • Figure 1: Confining-phase data for $Z_{\mathrm{tH}}[E;B]$ in $SU(2)$ YM on $T^4$.
  • Figure 2: Averaged $Z_{\mathrm{tH}}[E;B]$ by Euclidean rotations.
  • Figure 3: Shift relation \ref{['eq:witten']} and oblique confinement (Witten effect).
  • Figure 4: Distributions of plaquette and action. $L^3\times T$ with $L=8$, $\beta=2.4$, $B_{34}\in\{0,1\}$. The blue histogram denotes those of $T=L$, $B_{34}=1$, the red one is $T=L$, $B_{34}=0$ (zero-temperature); the green one is $T=L/2$, $B_{34}=1$, and the gray one is $T=L/2$, $B_{34}=0$ (finite-temperature).