Entanglement Properties of SU(2) Gauge Theory

TL;DR

This work analyzes real-time thermalization in (2+1)D SU(2) lattice gauge theory through exact diagonalization of truncated Hamiltonians on plaquette chains and honeycomb lattices. It demonstrates that highly excited states obey the Eigenstate Thermalization Hypothesis and exhibit GOE-level statistics, with the spectral form factor displaying slope–ramp–plateau behavior and diffusive hydrodynamic corrections. Entanglement studies reveal Page-curve behavior, area-to-volume transitions, two-step entanglement growth, and a BW-entanglement Hamiltonian that closely approximates the true modular Hamiltonian, along with a non-Gaussian magic signature that peaks during thermalization. The results collectively support robust thermalization in nonabelian gauge theories and highlight the potential and challenges of quantum computation for simulating gauge-theoretic dynamics; nevertheless, the work is constrained by small lattice sizes and the absence of fermions, pointing to clear directions for future extension to larger systems, other gauge groups, and quantum-enabled simulations.

Abstract

We review recent and present new results on thermalization of nonabelian gauge theory obtained by exact numerical simulation of the real-time dynamics of $(2+1)$-dimensional SU(2) lattice gauge theory. We discuss: (1) tests confirming the Eigenstate Thermalization Hypothesis; (2) the entanglement entropy of sublattices, including the Page curve, the transition from area to volume scaling with increasing energy of the eigenstate and its time evolution that shows thermalization of localized regions to be a two-step process; (3) the absence of quantum many-body scars when higher gauge field representations are taken into account; (4) the spectral form factor, which exhibits the expected slope-ramp-plateau structure for late times; (5) the entanglement Hamiltonian for SU(2), which has properties in accordance with the Bisognano-Wichmann theorem; and (6) a measure for non-stabilizerness or ``magic'' that is found to reach its maximum during thermalization. We conclude that the thermalization of nonabelian gauge theories is a promising process to establish quantum advantage.

Figures (10)

• Figure 1: (a) Plaquette chain segment in the vicinity of the cut separating the system into left (black) and right (red) parts, i.e., $A$ and $A^c$. The state on the left is labeled by the collection of $j$ values $\{j_L\}$ on the uncut links and the $j$ values $j_{1L}$, $j_{2L}$ for the two dangling links. Similarly for the state on the right. $j_{1L}=j_{1R}$ and $j_{2L}=j_{2R}$ since they form the same link before the cut. (b) Honeycomb lattice parallelogram with $N_x=N_y=3$ plaquettes.
• Figure 2: (a) Histogram of the restricted gap ratio (blue) compared to the GOE (black) and the Poisson (dashed red) prediction for the converged region of the spectrum of the plaquette chain with $N=3$, $ag^2=0.6$ and $j_{\mathrm{max}}=4$Ebner_2024. (b) Mean fluctuations $|\Delta_i|$ of the $i$-plaquette Wilson loop operator around the microcanonical value for different system sizes $N=9,12,16,20$ of the honeycomb lattice with $j_{\mathrm{max}}=\frac{1}{2}$, $ag^2=1$ and all momentum sectors up to $k_x=\lfloor N_x/2 \rfloor$, $k_y=\lfloor N_y/2 \rfloor$Ebner_2024.
• Figure 3: (a) Double logarithmic plot of the small $\omega$ region of the spectral function $f_{\mathrm{el}}(E,\omega)$ calculated via the second moments (red) of the matrix elements and the Gaussian fit method (blue) for a system with $N=3$ plaquettes, $j_{\mathrm{max}}=4$ and $ag^2=0.6$. (b) $\Lambda^T$ measure as a function of $T$ for the total electric energy operator, the plaquette operator and their sign randomized versions, averaged over 100 samples, in the energy window around $E=23.5$ with width $\Delta E=0.1$ for a system with $N=3$ plaquettes, $j_{\mathrm{max}}=4$ and $ag^2=0.6$.
• Figure 4: (a) Averaged SFF (black) over 500 Gaussian windows with each $\sigma=0.01$ and the SFF of single sample (red) at $\mu=14$ with the same $\sigma$. The linear ramp (blue) is given in Eq. (\ref{['eq:SFF_ramp']}). (b) Averaged SFF (black) over 100 Gaussian windows with each $\sigma=1$ and fitted to it the "hydro-corrected" ramp (red) of Eq. (\ref{['eq:SFF_hydro']}) and linear ramp (blue). The systems in (a) and (b) are five-plaquette chains with $j_{\mathrm{max}}=\frac{3}{2}$, $ag^2=1.05$ and open boundary conditions $j=1,1,1,0$. The sampling of the Gaussian means is performed between $E=[10,22]$ (roughly the FWHM), which for this region of the spectrum results in a Heisenberg time $t_H/a = 3.818 \times 10^4$ (gray).
• Figure 5: (a) Entanglement entropy as a function of the subsystem size $N_A$ on a periodic nine-plaquette chain in the zero-momentum and reflection symmetric sector with $j_{\mathrm{max}}=1$. The states are labeled by their energies relative to the ground state energy $\Delta E=E-E_0$ in lattice units. (b) Comparison of entanglement entropy and associated thermal entropy of all eigenstates of the periodic nine-plaquette chain with $j_{\mathrm{max}}=1$ and $ag^2=1.2$ in a subsystem of length $N_A=4$. The vertical line indicates $\beta=0$ and everything on its left ($\beta>0$) is part of the physical spectrum.