$\mathrm{SU(N)}$ lattice gauge theories with Physics-Informed Neural Networks

TL;DR

The paper develops adiabatic Physics-Informed Neural Networks (PINNs) to solve the spectral problem of $SU(N_c)$ lattice gauge theories in the Hamiltonian, gauge-invariant framework. Starting from analytically known strong-coupling eigenstates, the method flows to weaker couplings by enforcing the Schrödinger equation, normalization, and Gauss' law constraints, enabling unsupervised learning of eigenfunctions and eigenvalues across $g$. Validation on the single-plaquette, pure-gauge cases of $U(1)$ and $SU(2)$ reproduces the expected energy ladders and gauge-invariant wavefunctions, including reductions to the Mathieu equation in both theories. This approach offers a nonperturbative, scalable pathway to obtain spectral information without explicit Hilbert-space truncation, with potential extensions to fermions, larger lattices, and higher color numbers.

Abstract

We present an application of Physics-Informed Neural Networks (PINNs) to the study of $\mathrm{SU}(N_c)$ lattice gauge theories. Our method enables the learning of eigenfunctions and eigenvalues at arbitrary gauge couplings, smoothly moving from the analytically known strong-coupling regime towards weaker couplings. By encoding the Schrödinger equation and the symmetries of the eigenstates directly into the loss function, the network performs an unsupervised exploration of the spectrum. We validate the approach on the single-plaquette $\mathrm{U}(1)$ and $\mathrm{SU}(2)$ pure-gauge theories, showing that the PINNs successfully reproduce the hierarchy of energy levels and their corresponding wavefunctions.

Figures (12)

• Figure 1: Pictorial representation of the ideal adiabatic learning described in the text. The normalization of the state and the eigenvalue equation are imposed throughout, while we evolve the coupling smoothly from $g_1$ to $g_2$ (from left to right). The eigenfunction $\psi(g_1)$ remains in its instantaneous eigenstate, with final form given by $\psi(g_2)$.
• Figure 2: Illustrative representation of the Fully-Connected-Neural-Network used in this work in train the eigenfunctions. The inputs are the gauge links of the lattice (in the form of the angles needed to parametrize them), which are passed to the hidden layers $h^{(1)}$, $h^{(2)}$ to output the wavefunction $\psi$. Each layer uses a tanh activation function.
• Figure 3: First energy levels for the even and odd eigenfunctions of a $\mathrm{U}(1)$ single-plaquette system. The discrete points represent our prediction obtained with a PINN, while the dashed lines correspond to the exact analytic values.
• Figure 4: Learned eigenfunction $\psi$ for the ground state (left panel) and second excited state (right panel) of a single-plaquette $\mathrm{U}(1)$ system. On the horizontal axis, $\omega$ is such that $\cos{\omega}$ is the real part of the trace of the plaquette. Different curves correspond to different couplings, according to the colormap on the side. Each curve is obtained by fine-tuning the model from the previous coupling. At ${1/g \to 0}$ the network is trained on the analytic expression.
• Figure 5: Comparison of learned eigenfunction (open points) and exact result (dashed line) at the intermediate coupling $2/g^2 = 5.0$. The left and right panel correspond respectively to the ground state and 2nd excited state eigenfunctions, for a $\mathrm{U}(1)$ single-plaquette system.