Non-Perturbative Trivializing Flows for Lattice Gauge Theories

TL;DR

The paper addresses the slow convergence of traditional lattice gauge theory samplers by introducing a flexible, gauge-equivariant continuous normalizing flow defined on matrix Lie groups. It unifies Lüscher's trivializing maps with modern neural ODEs, employing adjoint sensitivity and a Lie-group–aware Crouch-Grossmann integrator to maintain gauge invariance and numerical stability. The authors demonstrate state-of-the-art effective sample sizes for two-dimensional SU(2) and SU(3) pure gauge theories and provide detailed architectural and training considerations. This work offers a scalable framework for flow-based lattice simulations that leverages symmetries to improve sampling efficiency and lays groundwork for extensions to larger lattices and fermionic theories.

Abstract

Continuous normalizing flows are known to be highly expressive and flexible, which allows for easier incorporation of large symmetries and makes them a powerful computational tool for lattice field theories. Building on previous work, we present a general continuous normalizing flow architecture for matrix Lie groups that is equivariant under group transformations. We apply this to lattice gauge theories in two dimensions as a proof of principle and demonstrate competitive performance, showing its potential as a tool for future lattice computations.

Figures (5)

• Figure 1: Loops used in the ODE architecture. The first two loops, highlighted in red, are used for the non-linear superposition function. The anchor in the dual lattice is displayed as a dot.
• Figure 2: Comparison of the ${\mathrm{SU(3)}}$ densities at $\beta=9$ in terms of angular coordinates for the three conjugation equivariant targets listed in table \ref{['tab:coefficients']} and the respectively trained continuous normalizing flows.
• Figure 3: Effective sample size along the flow time of trained normalizing flows for each of the three families of target distributions listed in table \ref{['tab:coefficients']} in comparison to the ESS achieved by the individual discrete flows from boyda2021SamplingUsing.
• Figure 4: Ratio of Monte Carlo estimates of observables to analytic values for flows on ${\mathrm{SU(2)}}$ on a $16 \times 16$ lattice. The trained flows of table \ref{['tab:lat16']} are used as proposal distribution for an independent Metropolis-Hastings Markov chain of length $20480$, and errors shown as vertical bars are estimated using a bootstrap method.
• Figure 5: Ratio of Monte Carlo estimates of observables to analytic values for flows on ${\mathrm{SU(3)}}$ on a $16 \times 16$ lattice. The trained flows of table \ref{['tab:lat16']} are used as proposal distribution for an independent Metropolis-Hastings Markov chain of length $20480$, and errors shown as vertical bars are estimated using a bootstrap method.