Machine learning a fixed point action for SU(3) gauge theory with a gauge equivariant convolutional neural network

Aug 6, 2026

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Fixed point lattice actions are engineered to preserve key continuum classical properties while reducing discretization artifacts in quantum simulations. Such actions can enable continuum physics extraction from coarser lattices, mitigating critical slowing down and topological freezing near the continuum limit. Because fixed point actions are only implicitly defined, an accurate and compact parametrization is crucial for practical use. This work applies machine learning to re-express these parametrizations, constructing a gauge-invariant SU(3) fixed point action in four dimensions using convolutional neural networks with exact gauge equivariance.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages22

Reading Time55 min

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Introduction

Critical slowing down and topological freezing

Open boundary conditions and master fields

Normalizing flows and related machine-learning approaches

Fixed point (quantum perfect) lattice actions and renormalization group ideas

arXiv :2401 .0648 1v2 [hep-lat] 2 Oct 2024 Machine learning a fixed point action for SU(3) gauge theory with a gauge equivariant convolutional neural network Kieran Holland∗ University of the Pacific, 3601 Pacific Ave., Stockton, CA 95211, USA Andreas Ipp† and David I. M¨uller‡ Institute for Theoretical Physics, TU Wien, Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria Urs Wenger§ Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstraße 5, 3012 Bern, Switzerland (Dated: October 4, 2024) Fixed point lattice actions are designed to have continuum classical properties unaffected by discretization effects and reduced lattice artifacts at the quantum level. They provide a possible way to extract continuum physics with coarser lattices, thereby allowing one to circumvent problems with critical slowing down and topological freezing toward the continuum limit. A crucial ingredient for practical applications is to find an accurate and compact parametrization of a fixed point action, since many of its properties are only implicitly defined. Here we use machine learning methods to revisit the question of how to parametrize fixed point actions. In particular, we obtain a fixed point action for four-dimensional SU(3) gauge theory using convolutional neural networks with exact gauge invariance. The large operator space allows us to find superior parametrizations compared to previous studies, a necessary first step for future Monte Carlo simulations and scaling studies. I. INTRODUCTION Lattice regularization is the tool of choice to study nonperturbative properties of quantum field theories starting from first principles [1] . Modern lattice QCD simulations have attained a high level of precision and for some important Standard Model quantities, e.g., the QCD coupling at the electroweak scale αS (µ = mZ ), they provide the current most accurate determination [2] . Increased precision has amplified systematic issues relevant to any lattice calculation, such as the extrapolation to the continuum limit. Numerical simulations become rapidly more costly as the lattice spacing is reduced, not only due to the increased resolution at fixed physical volume, but also due to the increased autocorrelation times (critical slowing down) in generating statistically independent samples in Monte Carlo Markov chains and the related problem of suppressed tunneling between sectors of different topological charge (topological freezing) [3] . Fora robust continuum prediction, a range of lattice spacings is necessary, requiring a delicate balance between the control of discretization artifacts on coarse latticeson the one hand and the increased cost of simulating on finer lattices on the other. Several different approaches are currently being followed to deal with the problems of critical slowing down and topological freezing. Simulations employing open ∗ [kholland@pacific.edu](kholland@pacific.edu)[ ](kholland@pacific.edu)† [ipp@hep.itp.tuwien.ac.at](ipp@hep.itp.tuwien.ac.at)[ ](ipp@hep.itp.tuwien.ac.at)‡ [dmueller@hep.itp.tuwien.ac.at](dmueller@hep.itp.tuwien.ac.at) § [wenger@itp.unibe.ch](wenger@itp.unibe.ch) boundary conditions in time [4] or huge master fields [5, 6] both circumvent topological freezing, but they do not address critical slowing down. Approaches using trivializing or normalizing flows [7] attempt to solve both problems by finding invertible maps from a simple probability distribution for the lattice configurations, which allows efficient sampling, to the target one. Recently, the use of machine-learning tools for parametrizing normalizing flows has roused anew attention in this approach [8– 12], however, these attempts are so far restricted to simple field theories, low dimensions or, in four-dimensional SU(3) gauge theories, to very small and coarse systems [13] . Here we propose to follow a complementary approach in order to solve both critical slo

FAQs

为什么固定点晶格作用（fixed point lattice actions）对逼近连续极限很有价值？

它们在离散化时尽量保持连续极限的经典性质，并降低量子层面的晶格伪影，从而可能在更粗的格点上提取连续物理。

本文要解决的关键数值难题是什么？

主要是临界减速（critical slowing down）以及连续极限附近的拓扑冻结（topological freezing），它们会使蒙特卡洛采样变得困难。

本文如何为四维 SU(3) 构造固定点作用并进行参数化？

使用具有精确规范不变性的卷积神经网络，并通过规范等变（gauge equivariant）的结构学习参数化，在更大的算符空间中获得比以往更优的参数化。

Aug 6, 2026

Upload by Finn