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Scaling flow-based approaches for topology sampling in $\mathrm{SU}(3)$ gauge theory

Claudio Bonanno, Andrea Bulgarelli, Elia Cellini, Alessandro Nada, Dario Panfalone, Davide Vadacchino, Lorenzo Verzichelli

TL;DR

The paper tackles topological freezing in lattice SU(3) Yang-Mills theory near the continuum limit by pairing Open Boundary Conditions with out-of-equilibrium Monte Carlo simulations that gradually switch on Periodic Boundary Conditions, thereby removing unphysical boundary effects while preserving ergodicity. It provides a detailed scaling analysis of NE-MCMC costs as a function of defect size and evolution steps, and shows how to keep sampling efficiency in the continuum limit by scaling $n_{\mathrm{between}}$ and $n_{\mathrm{step}}$ with $a^{-2}$ and $a^{-3}$ respectively. The authors then introduce Stochastic Normalizing Flows with defect-focused coupling layers to accelerate the non-equilibrium transformations, demonstrating a ~3x efficiency gain over purely stochastic NE-MCMC and achieving superior progression of $\hat{ESS}$ and the dissipated work $\langle W_d\rangle_f$. They validate the approach on lattices down to $a\simeq 0.045$ fm and show consistent results for the topological susceptibility $a^4\chi_L$ compared with established benchmarks, indicating robust control over the continuum extrapolation. The work lays a foundation for efficient topology sampling in SU(3) gauge theories and paves the way for extending the methods to full QCD with dynamical fermions and more advanced flow architectures.

Abstract

We develop a methodology based on out-of-equilibrium simulations to mitigate topological freezing when approaching the continuum limit of lattice gauge theories. We reduce the autocorrelation of the topological charge employing open boundary conditions, while removing exactly their unphysical effects using a non-equilibrium Monte Carlo approach in which periodic boundary conditions are gradually switched on. We perform a detailed analysis of the computational costs of this strategy in the case of the four-dimensional $\mathrm{SU}(3)$ Yang-Mills theory. After achieving full control of the scaling, we outline a clear strategy to sample topology efficiently in the continuum limit, which we check at lattice spacings as small as $0.045$ fm. We also generalize this approach by designing a customized Stochastic Normalizing Flow for evolutions in the boundary conditions, obtaining superior performances with respect to the purely stochastic non-equilibrium approach, and paving the way for more efficient future flow-based solutions.

Scaling flow-based approaches for topology sampling in $\mathrm{SU}(3)$ gauge theory

TL;DR

The paper tackles topological freezing in lattice SU(3) Yang-Mills theory near the continuum limit by pairing Open Boundary Conditions with out-of-equilibrium Monte Carlo simulations that gradually switch on Periodic Boundary Conditions, thereby removing unphysical boundary effects while preserving ergodicity. It provides a detailed scaling analysis of NE-MCMC costs as a function of defect size and evolution steps, and shows how to keep sampling efficiency in the continuum limit by scaling and with and respectively. The authors then introduce Stochastic Normalizing Flows with defect-focused coupling layers to accelerate the non-equilibrium transformations, demonstrating a ~3x efficiency gain over purely stochastic NE-MCMC and achieving superior progression of and the dissipated work . They validate the approach on lattices down to fm and show consistent results for the topological susceptibility compared with established benchmarks, indicating robust control over the continuum extrapolation. The work lays a foundation for efficient topology sampling in SU(3) gauge theories and paves the way for extending the methods to full QCD with dynamical fermions and more advanced flow architectures.

Abstract

We develop a methodology based on out-of-equilibrium simulations to mitigate topological freezing when approaching the continuum limit of lattice gauge theories. We reduce the autocorrelation of the topological charge employing open boundary conditions, while removing exactly their unphysical effects using a non-equilibrium Monte Carlo approach in which periodic boundary conditions are gradually switched on. We perform a detailed analysis of the computational costs of this strategy in the case of the four-dimensional Yang-Mills theory. After achieving full control of the scaling, we outline a clear strategy to sample topology efficiently in the continuum limit, which we check at lattice spacings as small as fm. We also generalize this approach by designing a customized Stochastic Normalizing Flow for evolutions in the boundary conditions, obtaining superior performances with respect to the purely stochastic non-equilibrium approach, and paving the way for more efficient future flow-based solutions.

Paper Structure

This paper contains 11 sections, 52 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Non-equilibrium Monte Carlo simulations in lattice field theory
  3. Some insights on NE-MCMC and its metrics
  4. Lattice setup and topological observables
  5. Scaling of NE-MCMC in the boundary conditions
  6. Understanding the scaling with the degrees of freedom
  7. Accelerating NE-MCMC with Stochastic Normalizing Flows
  8. Coupling layers for a defect
  9. Sampling topology towards the continuum limit
  10. Conclusions
  11. Interpolation strategy for defect coupling layer parameters

Figures (10)

  • Figure 1: Scheme of a typical non-equilibrium simulation. A thermalized configuration (black circle) is sampled from the prior distribution every $n_{\mathrm{between}}$ MCMC steps (black squares); and an out-of-equilibrium evolution starts from it, following a given protocol $\lambda$ for $n_{\mathrm{step}}$ MCMC steps (red diamonds) until the desired target distribution is reached. The work $W$ of Eq. \ref{['eq:work']} is computed along each evolution, while the value of the desired observable(s) is calculated in the last configuration (red circle). The estimators of Eqs. \ref{['eq:estimator']} and \ref{['eq:jar']} are obtained by taking the average $\langle \dots \rangle_\mathrm{f}$ across different evolutions.
  • Figure 2: Results for the Kullback-Leibler divergence of Eq. \ref{['eq:kl']} for NE-MCMC in the boundary conditions as a function of the number of steps in the flow $n_{\mathrm{step}}$ (left panel) and as a function of $n_{\mathrm{step}}$ divided by the size of the defect (right panel). All results obtained on a $16^4$ lattice at $\beta=6.0$.
  • Figure 3: Results for the effective sample size of Eq. \ref{['eq:ess']} for NE-MCMC in the boundary conditions as a function of the number of steps in the flow $n_{\mathrm{step}}$ (left panel) and as a function of $n_{\mathrm{step}}$ divided by the size of the defect (right panel). All results obtained on a $16^4$ lattice at $\beta=6.0$.
  • Figure 4: Effective Sample Size $\hat{\mathrm{ESS}}$ as a function of the number of links on the defect, for three fixed NE-MCMC architectures.
  • Figure 5: Results for the Kullback-Leibler divergence of Eq. \ref{['eq:kl']} for different flows in the boundary conditions as a function of the number of steps in the flow divided by the volume of the defect. Both NE-MCMC (circles) and SNFs with defect coupling layers (squares) are shown. All results obtained on a $16^4$ lattice at $\beta=6.0$.
  • ...and 5 more figures