[Paper Review] Flow-based sampling in the lattice Schwinger model at criticality

Michael S. Albergo, Denis Boyda+8 more|arXiv (Cornell University)|Feb 23, 2022

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper introduces a flow-based sampling method for the lattice Schwinger model at criticality, using normalizing flows to model field configurations and bypass topological freezing that plagues conventional HMC. The approach achieves robust sampling with reliable uncertainties and over 1,000x higher effective sampling efficiency than HMC, demonstrating a critical advance for fermionic gauge theories in non-perturbative regimes.

ABSTRACT

Recent results suggest that flow-based algorithms may provide efficient sampling of field distributions for lattice field theory applications, such as studies of quantum chromodynamics and the Schwinger model. In this work, we provide a numerical demonstration of robust flow-based sampling in the Schwinger model at the critical value of the fermion mass. In contrast, at the same parameters, conventional methods fail to sample all parts of configuration space, leading to severely underestimated uncertainties.

Motivation & Objective

• To address the failure of conventional Markov Chain Monte Carlo (MCMC) methods like HMC in sampling fermionic gauge theories at criticality due to topological freezing.
• To demonstrate that flow-based sampling can effectively explore configuration space in the Schwinger model at the critical fermion mass.
• To provide a robust, efficient alternative to HMC for non-perturbative lattice field theory calculations with fermions.
• To validate the method using the chiral condensate as a key observable, showing convergence to a baseline value with correct uncertainty scaling.

Proposed method

• The method employs normalizing flows to learn the target distribution of the lattice Schwinger model’s field configurations via a bijective, differentiable transformation from a simple base distribution.
• The flow model is trained using maximum likelihood estimation on a set of reference configurations generated via HMC at non-critical parameters.
• The trained flow model enables direct, independent sampling from the target distribution, avoiding the random walk behavior and autocorrelations inherent in MCMC.
• Sampling efficiency is quantified by measuring the average number of steps between topological sector transitions, with flow-based sampling showing a transition every ~6 steps versus ~20,000 for HMC.
• The approach uses exact evaluation of the fermion determinant, ensuring accuracy, and is compatible with hybrid MCMC schemes for potential future improvements.
• Numerical experiments are conducted on a 16×16 lattice at β=2.0 and κ=0.276, corresponding to the critical fermion mass.

Experimental results

Research questions

• RQ1 Can flow-based sampling overcome topological freezing in a fermionic lattice gauge theory at criticality?
• RQ2 How does the sampling efficiency of flow-based methods compare to HMC in terms of topological sector transitions?
• RQ3 Do flow-based samplers produce unbiased estimates of observables with statistically consistent uncertainties in the presence of critical slowing down?
• RQ4 Can normalizing flows effectively model long-range correlations in the Schwinger model at the critical point?
• RQ5 Is the flow-based approach scalable to larger volumes and more complex theories like QCD?

Key findings

• Flow-based sampling successfully samples the full configuration space in the Schwinger model at criticality, while HMC fails to explore all topological sectors, leading to biased results.
• HMC estimates of the chiral condensate appear to converge but suffer from severely underestimated uncertainties, as evidenced by sudden jumps upon rare tunneling events.
• Flow-based sampling converges to the correct baseline value with statistical uncertainties scaling as 1/√N, confirming reliable error estimation.
• The effective sampling efficiency of flow-based sampling exceeds that of HMC by more than three orders of magnitude, with topological sector transitions occurring every ~6 steps versus ~20,000 steps for HMC.
• The method demonstrates robustness and accuracy in modeling long-range correlations in the critical Schwinger model, even when HMC fails.
• The results suggest that flow-based sampling can serve as a foundation for hybrid or hierarchical MCMC schemes, enabling broader applications in lattice field theory.