Spectral Statistics and Many-Body Quantum Chaos with Conserved Charge

We investigate spectral statistics in spatially extended, chaotic many-body quantum systems with a conserved charge. We compute the spectral form factor $K(t)$ analytically for a minimal Floquet circuit model that has a $U(1)$ symmetry encoded via spin-$1/2$ degrees of freedom. Averaging over an ensemble of realizations, we relate $K(t)$ to a partition function for the spins, given by a Trotterization of the spin-$1/2$ Heisenberg ferromagnet. Using Bethe ansatz techniques, we extract the “Thouless time” $t_{\text{Th}}$ demarcating the extent of random matrix behavior, and find scaling behavior governed by diffusion for $K(t)$ at $t\lesssim t_{\text{Th}}$. We also report numerical results for $K(t)$ in a generic Floquet spin model, which are consistent with these analytic predictions.

Figure 1

Behavior of $K(t)$. Upper figure (main panel): $\ln K(t)/t$ vs $t/L^2$. The scaling collapse of data for $L=14$, 16, and 18 indicates that the Thouless time $t_{\text{Th}}$ is controlled by diffusion; small deviations for $L=12$ are presumably a finite-size effect. The full line is a fit to the scaling function (11) with $D=0.05$. Inset: same data vs $t$ for comparison. Lower panel: $K(t)$ for $L=12$; the small system size narrows the relative extent of the ramp regime $K(t)=t$, highlighting the short-time, pre-RMT behavior.