Equivariant flow-based sampling for lattice gauge theory (2003.06413v1)

Abstract: 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 near critical points in parameter space 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.

• The paper introduces novel gauge-equivariant flow-based sampling methods for lattice gauge theory, designed to efficiently sample from complex distributions while respecting inherent gauge symmetry.
• Numerical results demonstrate that these flow-based models significantly outperform traditional methods like Hybrid Monte Carlo (HMC) near critical points, achieving up to 1500x greater efficiency in calculating topological quantities by mitigating critical slowing down.
• This work offers a promising approach to enable more precise calculations in theories like lattice QCD and potentially extend to non-Abelian gauge theories, advancing the application of machine learning in theoretical physics.

Equivariant Flow-based Sampling for Lattice Gauge Theory

The paper authored by Gurtej Kanwar et al. presents a significant advancement in the application of machine learning to lattice gauge theory through the development of equivariant flow-based sampling methods. These methods, which adhere strictly to gauge invariance, are shown to significantly improve on existing algorithms like Hybrid Monte Carlo (HMC) and Heat Bath when sampling from lattice gauge theories, particularly near critical points of parameter space.

Contributions and Methodology

The primary contribution of this paper is the introduction of flow-based generative models capable of efficiently sampling from the complex distributions associated with lattice gauge theories. These models leverage the concept of gauge equivariance, ensuring that they respect the gauge symmetry inherent in these theories. Flow-based models operate by transforming samples from a simple prior distribution through an invertible function to match the desired distribution. The key advantage is the ability to efficiently evaluate the Jacobian of this transformation, thus enabling accurate and fast sampling.

To ensure gauge invariance, the authors construct a series of coupling layers. These layers split the input space, transforming only a subset while ensuring gauge equivariance through conditions on the kernel functions. For the application to U(1) gauge theory in two-dimensional space-time, the authors achieve efficient sampling from a distribution that is known to suffer from critical slowing down.

Numerical Results and Implications

Numerical experiments demonstrate that these flow-based models can outperform established sampling methods significantly when approaching the continuum limit. The autocorrelation time for the topological charge remains manageable with flow-based models even as the theory approaches critical points, unlike HMC and Heat Bath, where it grows prohibitively large. Specifically, the flow-based approach is shown to be approximately 1500 times more efficient than HMC and 200 times more efficient than Heat Bath for estimating topological quantities on a $16 \times 16$ lattice at the highest values of $\beta$ considered.

These results have important implications for lattice QCD and other non-Abelian gauge theories. By mitigating the effects of critical slowing down, flow-based models can enable more precise calculations in these theories. This could potentially lead to breakthroughs in our understanding of the non-perturbative aspects of quantum field theories, as well as practical applications in condensed matter physics where similar mathematical structures arise.

Future Directions

The paper suggests several avenues for future research. Extending these techniques to non-Abelian gauge theories, such as quantum chromodynamics (QCD), remains a critical step. This would involve constructing suitably expressive and invertible kernels for the coupling layers, potentially using recent advancements in flows on spherical manifolds and Lie groups. Additionally, integrating flow-based approaches with traditional ones could yield further efficiencies across various observables.

Overall, this work stands as a useful contribution to the ongoing effort to apply machine learning methods to complex physical systems, opening pathways for the exploration of new computational techniques in theoretical physics.