Efficient Learning of Lattice Gauge Theories with Fermions

TL;DR

This work introduces a versatile SD-based learning framework to recover lattice action parameters from observed expectation values, by constructing convex losses from Schwinger-Dyson constraints within an exponential-family setting $p(x) \propto e^{-S_0(x) - \sum_i \theta_i Q_i(x)}$. It unifies and extends score matching as a special case and generalizes the approach to Grassmann-valued fermions, including a concrete lattice Thirring-model demonstration, as well as to gauge theories with gauge-invariant SD relations applicable to lattice QCD. The method provides a principled path to infer couplings like $\beta$ and fermion masses from expectation data, potentially offering lower-variance estimators and enabling efficient exploration of renormalization flows. It also clarifies limitations, such as sensitivity to integration contours and the need to address statistical costs and nonperturbative physics beyond SD constraints. Overall, the framework broadens the toolkit for inverse learning in lattice field theories, with practical implications for studying nonperturbative phenomena and renormalization in gauge-fermion systems.

Abstract

We introduce a learning method for recovering action parameters in lattice field theories. Our method is based on the minimization of a convex loss function constructed using the Schwinger-Dyson relations. We show that score matching, a popular learning method, is a special case of our construction of an infinite family of valid loss functions. Importantly, our general Schwinger-Dyson-based construction applies to gauge theories and models with Grassmann-valued fields used to represent dynamical fermions. In particular, we extend our method to realistic lattice field theories including quantum chromodynamics.

Figures (2)

• Figure 1: Workflow for the demonstration of learning of Grassmannian actions in the case of the lattice Thirring model. We begin with an action written purely in terms of Grassmann fields. Bosonization (via Hubbard-Stratonovich) yields a theory in which the fermions can be integrated out, leaving only c-number fields. Expectation values in this theory are obtained by Monte Carlo, and from those expectation values we use Schwinger-Dyson-based learning to recover directly the original (purely Grassmann) theory.
• Figure 2: Estimates of the bare coupling of the Thirring model, as the bare coupling used for simulation is changed. All calculations are performed at a bare mass of $m = 0.1$ on a $12 \times 12$ lattice.