A scalable flow-based approach to mitigate topological freezing

TL;DR

Topological freezing in lattice SU(3) Yang–Mills simulations leads to large autocorrelations near the continuum limit, where the Boltzmann weight is $p(U)\propto e^{-S[U]}$. The authors introduce a defect-based Stochastic Normalizing Flow (SNF) that interleaves non-equilibrium MCMC steps with localized, gauge-equivariant defect layers to transport configurations from an Open Boundary Condition (OBC) prior to a fully periodic target, with Jarzynski-type reweighting to preserve exactness. Training minimizes the average dissipated work $\langle W_d \rangle_f$, increasing trajectory reversibility and the effective sample size $\hat{\mathrm{ESS}}$. Across lattice tests at fixed $n_{\mathrm{step}}/n_{\mathrm{dof}}$, defect SNFs show better reversibility and about a threefold speedup over purely stochastic NE-MCMC, and they reproduce the topological susceptibility in agreement with reference results. The method provides a scalable path toward accurate topology sampling near the continuum, with potential extensions to dynamical-fermion simulations and more sophisticated flow architectures.

Abstract

As lattice gauge theories with non-trivial topological features are driven towards the continuum limit, standard Markov Chain Monte Carlo simulations suffer for topological freezing, i.e., a dramatic growth of autocorrelations in topological observables. A widely used strategy is the adoption of Open Boundary Conditions (OBC), which restores ergodic sampling of topology but at the price of breaking translation invariance and introducing unphysical boundary artifacts. In this contribution we summarize a scalable, exact flow-based strategy to remove them by transporting configurations from a prior with a OBC defect to a fully periodic ensemble, and apply it to 4d SU(3) Yang--Mills theory. The method is based on a Stochastic Normalizing Flow (SNF) that alternates non-equilibrium Monte Carlo updates with localized, gauge-equivariant defect coupling layers implemented via masked parametric stout smearing. Training is performed by minimizing the average dissipated work, equivalent to a Kullback--Leibler divergence between forward and reverse non-equilibrium path measures, to achieve more reversible trajectories and improved efficiency. We discuss the scaling with the number of degrees of freedom affected by the defect and show that defect SNFs achieve better performances than purely stochastic non-equilibrium methods at comparable cost. Finally, we validate the approach by reproducing reference results for the topological susceptibility.

Figures (3)

• Figure 1: Schematic representation of the defect coupling layer used in the SNF. The deterministic transformation updates only the links on and in the immediate neighborhood of the defect (green strip); the red dashed links indicate the subset of boundary links whose couplings are modified during the boundary-condition evolution.
• Figure 2: Comparison of the dissipated work (left) and $\hat{\mathrm{ESS}}$ (right) for NE-MCMC (triangles) and defect SNFs (circles) as a function of $n_{\mathrm{step}}/n_{\mathrm{dof}}$ at $\beta=6$ and $L/a=16$. Figure taken from Bonanno:2025pdp.
• Figure 3: Topological susceptibility in lattice units $a^{4}\chi_{_{{\rm L}}}$ versus proxy effective sample size $\hat{\mathrm{ESS}}$ for boundary-condition flows at $\beta=6.4$ ($30^{4}$ lattice) and $\beta=6.5$ ($34^{4}$ lattice). Horizontal bands show reference results from Bonanno:2023pleBonanno:2025eeb. Figure taken from Bonanno:2025pdp.