# Spectral statistics and many-body quantum chaos with conserved charge

## TL;DR

This paper analytically and numerically studies spectral statistics in chaotic many-body quantum systems with a conserved charge, revealing diffusion-driven scaling and identifying the Thouless time using Bethe Ansatz techniques.

## Contribution

It introduces an analytical approach to spectral form factors in systems with conserved charge using a minimal Floquet model and Bethe Ansatz, connecting spectral statistics to spin chain physics.

## Key findings

- Spectral form factor exhibits diffusion-driven scaling for times less than the Thouless time.
- Bethe Ansatz allows extraction of the Thouless time in the model.
- Numerical results support the analytical predictions.

## Abstract

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 auxiliary 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' $t^{\vphantom{*}}_{\rm Th}$ demarcating the extent of random matrix behavior, and find scaling behavior governed by diffusion for $K(t)$ at $t\lesssim t^{\vphantom{*}}_{\rm Th}$. We also report numerical results for $K(t)$ in a generic Floquet spin model, which are consistent with these analytic predictions.