Onset of Random Matrix Behavior in Scrambling Systems

TL;DR

This paper investigates the time scale at which random matrix behavior emerges in strongly chaotic quantum systems, revealing how it depends on system size, locality, and conservation laws through numerical and analytical methods.

Contribution

It provides a detailed analysis of the onset time for random matrix statistics in various scrambling systems, highlighting the roles of locality and conservation laws.

Findings

For local systems with conservation laws, the ramp time scales with diffusion time, about N^2 for 1D chains.

In k-local systems with conservation laws, the ramp time scales as log N, but differs from the scrambling time.

Without conservation laws, the ramp time is approximately log N, regardless of connectivity.

Abstract

The fine grained energy spectrum of quantum chaotic systems is widely believed to be described by random matrix statistics. A basic scale in such a system is the energy range over which this behavior persists. We define the corresponding time scale by the time at which the linearly growing ramp region in the spectral form factor begins. We call this time $t_{\rm ramp}$. The purpose of this paper is to study this scale in many-body quantum systems that display strong chaos, sometimes called scrambling systems. We focus on randomly coupled qubit systems, both local and $k$-local (all-to-all interactions) and the Sachdev--Ye--Kitaev (SYK) model. Using numerical results for Hamiltonian systems and analytic estimates for random quantum circuits we find the following results. For geometrically local systems with a conservation law we find $t_{\rm ramp}$ is determined by the diffusion time across the system, order $N^2$ for a 1D chain of $N$ qubits. This is analogous to the behavior found for local one-body chaotic systems. For a $k$-local system with conservation law the time is order $\log N$ but with a different prefactor and a different mechanism than the scrambling time. In the absence of any conservation laws, as in a generic random quantum circuit, we find $t_{\rm ramp} \sim \log N$, independent of connectivity.