Equivariant Flow-Based Sampling for Lattice Gauge Theory

We define a class of machine-learned flow-based sampling algorithms for lattice gauge theories that are gauge invariant by construction. We demonstrate the application of this framework to U(1) gauge theory in two spacetime dimensions, and find that, at small bare coupling, the approach is orders of magnitude more efficient at sampling topological quantities than more traditional sampling procedures such as hybrid Monte Carlo and heat bath.

Figure 1

Standard approaches (HMC and HB) to MCMC sampling for U(1) gauge theory explore the distribution of topological charge $Q$ very slowly compared with the flow-based approach introduced here. Results are shown for coupling $\beta =7$ on a $16\times 16$ lattice, see Eq. (6). The first (second) half of the Markov chain history is displayed for HMC (HB).

Figure 2

An example of a variable splitting based on a tiled $4\times 1$ pattern with an actively updated link $U^i\equiv U_{\mu }(x)$ and $1\times 1$ loop $U^i{S}^i\equiv P_{\mu \nu }(x)$ located at $x$, a passively updated $1\times 1$ loop at $\tilde{x}=x-\widehat{\nu }$, and two frozen traced $1\times 1$ loops at $x+\widehat{\nu }$ and $x+2\widehat{\nu }$ included in the set $I^i$.

Figure 3

Left: estimates of average Wilson loops $⟨W_{\ell \times \ell }⟩$ measured on the most frozen ensemble studied here ($\beta =7$). Right: estimates of topological susceptibility measured on the three most frozen ensembles studied here ($\beta =5$, 6, 7). All values are plotted as ratios to the exact results. The flow-based estimates are consistent with the exact values, while the HMC and heat bath estimates have larger and underestimated uncertainties due to long autocorrelation times on the order of the Markov chain length or longer.

Figure 4

Integrated autocorrelation time for the topological charge, ${\tau }_Q^{\mathrm{int}}$, measured on ensembles of $16\times 16$ lattices generated using HMC, heat bath, and the flow-based algorithm. Ten replicas of each ensemble were used to estimate errors, which are smaller than the plot markers for most points.