Normalizing flows for SU($N$) gauge theories employing singular value decomposition

TL;DR

This work addresses efficient, symmetry-preserving sampling of SU($N$) lattice gauge theories using normalizing flows (NFs). It introduces an SVD-based, gauge-equivariant NF construction that operates on gauge-invariant blocks built from the sum of adjacent staples, transforming links through $\tilde{U}_\mu(x)= V^\dagger_\mu(x) U_\mu(x) W_\mu(x) e^{-i\phi_\mu(x)}$ and parametrizing SU(3) eigenvalues via $(\theta,\varphi)$ with spline-based mappings, all while targeting $p \propto e^{-S_\text{W}}$ and minimizing $D_\text{KL}(q\|p)$. The authors compare Haar priors with a trivializing-map–inspired prior and evaluate on a $4^4$ lattice at $\beta=1$, finding that the SVD-based NF yields higher effective sample size and faster training than a plaquette-based spectral flow baseline. By updating links in a two-step even/odd schedule and cascading multiple blocks, the approach substantially improves efficiency in 4D, while preserving gauge symmetry and avoiding inadvertent gauge fixing. This framework offers a scalable, symmetry-consistent route to accelerate lattice gauge theory simulations with neural networks.

Abstract

We present a progress report on the use of normalizing flows for generating gauge field configurations in pure SU(N) gauge theories. We discuss how the singular value decomposition can be used to construct gauge-invariant quantities, which serve as the building blocks for designing gauge-equivariant transformations of SU(N) gauge links. Using this novel approach, we build representative models for the SU(3) Wilson action on a \( 4^4 \) lattice with \( β= 1 \). We train these models and provide an analysis of their performance, highlighting the effectiveness of the new technique for gauge-invariant transformations. We also provide a comparison between the efficiency of the proposed algorithm and the spectral flow of Wilson loops.

Figures (4)

• Figure 1: Evolution of $\mathrm{SU}(3)$ gauge links due to the (scaled) Wilson flow \ref{['eq:triv-map']}.
• Figure 2: Block diagram for the method of normalizing flows. $V_\mu(x)$ and $U_\mu(x)$ are the prior and transformed fields, and $r(V)$ and $q(U)$ are the corresponding probability densities. See Ref. Komijani:2023fzy for more information.
• Figure 3: Left panel shows the principal cell of the parametrization introduced in Eq. \ref{['eq:SU3:eig:parametrization']}. Middle and right panels show more cells and conjugacy volume as a color coding. The dotted white lines separate different cells.
• Figure 4: The ESS, as defined in Eq. \ref{['eq:ess:definition']}, as a function of the epoch number is shown for various models applied to a lattice of size $4^4$ using the Wilson action for $\mathrm{SU}(3)$ gauge links with $\beta = 1$. The ESS values are measured every 10 epochs, with batch sizes set to 16364, 8192, 1024, and 8192 for the models labeled as "Triv Map", "NF:1", "NF:1+1", and "Triv Map + NF:1", respectively. The larger fluctuations in the "NF:1+1" data points (green circles) are due to the smaller batch size used in that case. This plot can be directly compared to Fig. 6 in Abbott:2023thq.