Enhanced Sampling Techniques for Lattice Gauge Theory

Abstract

In theories with topological sectors, such as lattice QCD and four-dimensional SU(N) gauge theories with periodic boundary conditions, conventional update algorithms suffer from topological freezing due to large action barriers separating distinct sectors. With appropriately constructed bias potentials, Metadynamics and related enhanced sampling techniques can mitigate this problem and significantly reduce the integrated autocorrelation times of the topological charge and associated observables. We test strategies to accelerate the buildup of bias potentials and the possibility of extrapolating potentials from small to large volumes. We also investigate the effectiveness of orthogonal algorithmic improvements, such as longer HMC trajectories and HMC variants, which may benefit conventional simulations as well.

Figures (4)

• Figure 1: Evolution of the bias parameters (upper panel) and the CV (lower panel) in a VES run for four-dimensional SU(3) gauge theory with the DBW2 action at $L/a = 16$ and $\beta = 1.25$. One SGD iteration corresponds to $50$ HMC trajectories of length $4$. The reference values for $\alpha_1$ and $\alpha_2$ were obtained by fitting \ref{['eq:fit_ansatz']} to a bias potential generated in an independent Metadynamics simulation.
• Figure 2: Convolution-based volume extrapolation of bias potentials at $\beta = 6.1912$ in four-dimensional SU(3) gauge theory. The $L/a = 24$ reference potential was reconstructed from an unbiased simulation from Durr:2025qtq. The extrapolation from $L/a = 12$ to $L/a = 24$ is inaccurate due to sizeable finite volume effects. Since the volume ratio $24^4/20^4 = 2.0736$ is close to $2$, a good estimate for the bias potential at the larger volume can be obtained by a single convolution.
• Figure 3: Effect of the HMC trajectory length on the sampling efficiency for the energy density $E$ and the squared topological charge $Q^2$. The additional computational overhead of longer trajectories has already been taken into account.
• Figure 4: Effect of the HMC trajectory length on the integrated autocorrelation times of the energy density $E$ and the squared topological charge $Q^2$. The additional computational overhead of longer trajectories has already been taken into account. The solid lines are fits of the form $c / \sqrt{T} + 0.5$ to all data points, while the dashed lines correspond to the same functional form with $c = \tau_{\mathrm{int}, T = 1}$.