Configurational Thermometer for Lattice Gauge Theories

TL;DR

This paper introduces a gauge-invariant configurational temperature estimator for lattice gauge theories, derived from the gradient and Hessian of the Euclidean action and expressible as $\beta_M = \left\langle \nabla_{\mathbf{q}} \cdot \left( \frac{\mathbf{g}}{\mathbf{g}\cdot\mathbf{g}} \right) \right\rangle = \left\langle \frac{\mathrm{Tr}(\mathbb{H})}{|\mathbf{g}|^2} - 2 \frac{\mathbf{g}^T \mathbb{H} \mathbf{g}}{|\mathbf{g}|^4} \right\rangle$. It provides a direct, momentum-free check of thermodynamic consistency in Monte Carlo simulations and is shown to reproduce the input temperature across 1D, 2D, and 4D compact U(1) lattice gauge theories, while also diagnosing sampling inefficiencies and algorithmic artifacts. The estimator is computationally feasible, with linear scaling in volume, and does not rely on energy fluctuations. The work also discusses extensions to non-Abelian theories, highlighting challenges related to gauge-group geometry and Haar measure preservation, with future plans for SU($N$) implementations.

Abstract

We propose a diagnostic tool, a temperature estimator, for lattice gauge theory simulations. The estimator is obtained from the gradient and the Hessian of the Euclidean lattice action. It is gauge invariant, configuration-based, and independent of momentum-space information. These features enable direct checks of thermodynamic consistency in Monte Carlo simulations. We apply this tool to compact U(1) lattice gauge theories in one, two, and four dimensions. The results confirm the proposed estimator's ability to reproduce the input temperatures across different lattice ensembles. The estimator is sensitive to sampling inefficiencies and algorithmic artifacts, making it a useful diagnostic for large-scale simulations.

Figures (3)

• Figure 1: Thermodynamic observables for the one-dimensional U(1) lattice theory on a 48-site lattice. The energy $E(\beta)$ (left), specific heat $C(\beta)$ (middle), and the measured inverse temperature $\beta_M$ (right) are plotted against the input $\beta$.
• Figure 2: Thermodynamic observables for the two-dimensional U(1) lattice gauge theory on a $32 \times 32$ lattice. The expectation value of the plaquette (left), Wilson loops of sizes $A = R \times T$ (middle), and the estimated inverse temperature $\beta_M$ (right) are shown as functions of the input $\beta$.
• Figure 3: Thermodynamic observables for the four-dimensional U(1) lattice theory on an $8^4$ lattice. The plaquette $\langle P \rangle$ (left), the plaquette susceptibility $\chi_P$ (middle), and the estimated inverse temperature $\beta_M$ (right) are plotted against the input $\beta$.