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

TL;DR

This work develops a configurational temperature estimator for lattice gauge theories based on the gradient and Hessian of the Euclidean action, rooted in Rugh's geometric thermodynamic framework. The method yields a momentum-free, gauge-invariant thermometer expressed as $\frac{1}{k_B T} = \left\langle \nabla_{\mathbf{q}} \cdot \left( \frac{\nabla_{\mathbf{q}} H}{\nabla_{\mathbf{q}} H \cdot \nabla_{\mathbf{q}} H} \right) \right\rangle$ or equivalently $\frac{{\rm Tr}(\mathbb{H})}{|\mathbf{g}|^2} - 2 \frac{\mathbf{g}^T \mathbb{H} \mathbf{g}}{|\mathbf{g}|^4}$ with $\mathbf{g}=\nabla_{\mathbf{q}} \Phi$ and $\mathbb{H}=\nabla_{\mathbf{q}} \nabla_{\mathbf{q}}^T \Phi$. The estimator is validated in compact U(1) lattice gauge theories in 1D, 2D, and 4D, showing recovery of the input temperature across a range of volumes and couplings; in 4D it remains informative near a phase transition, and off-diagonal Hessian contributions become negligible at large volumes, enabling simpler implementations. The results establish the estimator as a practical diagnostic for thermalization, sampling consistency, and potential real-time monitoring in HMC and beyond, with clear pathways to non-Abelian extensions and anisotropic finite-temperature applications.

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.

Figures (7)

• 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 $\beta$. $E(\beta)$ and $C(\beta)$ are benchmarked against the analytical results given in Eqs. \ref{['eq:energy_ana_1d']} and \ref{['eq:spec_ana_1d']}, respectively. The plot of $\beta_M$ against $\beta$ shows the expected behavior.
• Figure 2: Thermodynamic observables of the one-dimensional U(1) lattice gauge theory on a lattice with 48 sites. The figure presents histograms of the energy (left), the measured inverse temperature $\beta_M$ (right), and scatter plots in the form of a heat-map illustrating the relationship between energy and $\beta_M$ (middle). The top three plots correspond to $\beta = 10$, while the bottom three are for $\beta = 1$. These plots highlight the fluctuations around the average values, with the heat-map scatter plots demonstrating how energy shifts as the temperature deviates from the true value.
• Figure 3: Time series of $\exp(-\Delta H)$ and the estimated inverse temperature $\beta_M$ for the one-dimensional U(1) lattice gauge theory on a 48-site lattice. The input inverse temperature is $\beta = 3$.
• Figure 4: Two-dimensional U(1) lattice gauge theory. The expectation value of plaquette, Wilson loops of sizes $A = R \times T$ and estimated inverse temperature $\beta_M$ as functions of input $\beta$ are shown on a $32 \times 32$ lattice.
• Figure 5: (Left) Comparison of the full estimator $\beta_M$ and the directional estimator $\beta_x$ as a function of lattice size $L_x$ for $L_x \times L_x$ lattices at $\beta = 1$. (Right) The Monte Carlo time history of both estimators on a $16 \times 16$ lattice illustrates fluctuations and near agreement between the two estimators over simulation time. The dotted lines correspond to the average values of the respective observables over the Monte Carlo history.