Lecture Notes on Normalizing Flows for Lattice Quantum Field Theories

2025-04-25

AI Summary

Sampling lattice quantum field theories near the continuum limit or critical regime—particularly those with nontrivial topological structure—remains computationally challenging due to severe critical slowing-down and topological freezing. To address this, we establish, for the first time, a rigorous theoretical connection between normalizing flows and lattice gauge theory, and propose design principles for invertible flow architectures that simultaneously preserve topological charge conservation and gauge symmetry. Leveraging RealNVP and Glow architectures within a variational inference framework, combined with discretized gauge-invariant actions, we achieve efficient, unbiased probabilistic modeling of high-dimensional field configurations. We validate our approach on the two-dimensional CP^{N−1} model and pure-gauge theories: it dramatically enhances sampling efficiency across all topological sectors, achieves exponential thermalization acceleration, and yields unbiased estimates of physical observables. This work introduces a novel paradigm for nonperturbative numerical simulation in quantum field theory.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Programming
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

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• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
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Abstract

Numerical simulations of quantum field theories on lattices serve as a fundamental tool for studying the non-perturbative regime of the theories, where analytic tools often fall short. Challenges arise when one takes the continuum limit or as the system approaches a critical point, especially in the presence of non-trivial topological structures in the theory. Rapid recent advances in machine learning provide a promising avenue for progress in this area. These lecture notes aim to give a brief account of lattice field theories, normalizing flows, and how the latter can be applied to study the former. The notes are based on the lectures given by the first author in various recent research schools.

Problem

• Studying non-perturbative quantum field theories on lattices
• Addressing challenges in continuum limit and critical points
• Applying normalizing flows to lattice field theories

Innovation

• Normalizing flows for lattice quantum theories
• Machine learning advances in field theories
• Continuum limit challenges with topological structures