Eigenstate Thermalization in 2+1 dimensional SU(2) Lattice Gauge Theory

TL;DR

This paper provides preliminary numerical evidence that the 2+1 dimensional SU(2) lattice gauge theory obeys the Eigenstate Thermalization Hypothesis, using simplified models and matrix element analysis to support thermalization behavior.

Contribution

The study offers the first numerical checks of ETH in 2+1D SU(2) lattice gauge theories across different approximations and boundary conditions.

Findings

Results are consistent with ETH predictions across studied models.

Off-diagonal matrix elements exhibit RMT behavior in certain frequency windows.

Preliminary evidence supports thermalization in simplified SU(2) lattice gauge models.

Abstract

We present preliminary numerical evidence for the hypothesis that the Hamiltonian SU(2) gauge theory discretized on a lattice obeys the Eigenstate Thermalization Hypothesis (ETH). To do so we study three approximations: (a) a linear plaquette chain in a reduced Hilbert space limiting the electric field basis to $j=0,\frac{1}{2}$ , (b) a two-dimensional honeycomb lattice with periodic or closed boundary condition and the same Hilbert space constraint, and (c) a chain of only three plaquettes but such a sufficiently large electric field Hilbert space ($j \leq \frac{7}{2})$ that convergence of all energy eigenvalues in the analyzed energy window is observed. While an unconstrained Hilbert space is required to reach the continuum limit of SU(2) gauge theory, numerical resource constraints do not permit us to realize this requirement for all values of the coupling constant and large lattices. In each of the three studied cases we check first for random matrix theory (RMT) behavior in the eigenenergy spectrum and then analyze the diagonal as well as the off-diagonal matrix elements between energy eigenstates for a few operators. Within current uncertainties all results for (a), (b) and (c) agree with ETH predictions. Furthermore, we find the off-diagonal matrix elements of the electric energy operator exhibit RMT behavior in frequency windows that are small enough in (b) and (c). To unambiguously establish ETH behavior and determine for which class of operators it applies, an extension of our investigations is necessary.