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

TL;DR

A fixed point action for four-dimensional SU(3) gauge theory using convolutional neural networks with exact gauge invariance is obtained using machine learning methods to revisit the question of how to parametrize fixed point actions.

Abstract

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.

Figures (15)

• Figure 1: A sketch of the renormalization group flow and the renormalized trajectory (RT) in the infinite-dimensional coupling space, with the gauge coupling as the only relevant direction. The fixed point is on the critical surface $\beta \rightarrow \infty$ where $\xi/a = \infty$ for any physical scale $\xi$, with the values of the critical couplings $c^{\rm FP}_n$ determined by the specific form of the RG blocking. The FP action uses the same coupling values at finite $\beta$, tracking the RT closely at sufficiently weak coupling.
• Figure 2: The leading couplings $\rho_{\mu \nu}(r)$ of the perturbative FP action, from Blatter:1996np. The blocking kernel $T[U,V]$ is designed to maximize the exponential decay of the couplings, with $\exp(-3.4 r/a)$ shown as a visual guide.
• Figure 3: Fixed point action density as a function of $\beta_\mathrm{wil}$ on a $4^4$ lattice. We show the ensemble-averaged FP action $\mathcal{A}^\mathrm{FP}[V]$, the blocking kernel $T[U,V]$, and the parametrized FP action $\mathcal{A}^\mathrm{FP}[U]$ used on the right-hand side of Eq. \ref{['eq:fp']}, normalized to the coarse lattice volume. The mean values are obtained by averaging over the ensemble at a given $\beta_\mathrm{wil}$. The shaded regions indicate the standard deviation.
• Figure 4: Examples of minimization on $8^4$ lattice configurations with $\beta_{\rm wil} = 6.0$ (top) and 5.4 (bottom). The insets show the decrease of $\mathcal{A}[U]+T[U,V]$ in each iteration.
• Figure 5: Minimization of instanton configurations. In the upper panel, an instanton of size $\rho/a = 3.0$ on a $16^4$ lattice persists after being blocked to the $8^4$ lattice. The lower panel shows an instanton with $\rho/a = 1.5$ which is too small to survive the RG blocking, i.e., the instanton falls through the lattice.