Gradient and Hessian-Based Temperature Estimator in Lattice Gauge Theories: A Diagnostic Tool for Stability and Consistency in Numerical Simulations

TL;DR

This paper introduces a gauge-invariant temperature estimator based on the gradient and Hessian of the Euclidean action, providing a robust diagnostic tool for stability and consistency in lattice gauge theory simulations.

Contribution

The authors develop a novel temperature estimator from geometric formulations, validated across multiple dimensions, enhancing diagnostic capabilities in lattice gauge theory simulations.

Findings

Estimator accurately reproduces input temperature

Remains robust across lattice volumes and couplings

Applicable to non-Abelian and anisotropic lattices

Abstract

We present a field configuration-based temperature estimator in lattice gauge theories, constructed from the gradient and Hessian of the Euclidean action. Adapted from geometric formulations of entropy in classical statistical mechanics, this estimator provides a gauge-invariant, non-kinetic diagnostic of thermodynamic consistency in Monte Carlo simulations. We validate the method in compact U(1) lattice gauge theories across one, two, and four dimensions, comparing the estimated configurational temperature with the conventional temperature set by the temporal extent of the lattice. Our results show that the estimator accurately reproduces the input temperature and remains robust across a range of lattice volumes and coupling strengths. The temperature estimator offers a general-purpose diagnostic for lattice field theory simulations, with potential applications to non-Abelian theories, anisotropic lattices, and real-time monitoring in hybrid Monte Carlo algorithms.