Onset of Random Matrix Behavior in Scrambling Systems

TL;DR

The paper investigates the onset of random-matrix behavior in scrambling, strongly chaotic many-body systems by focusing on the ramp time t_ramp in the spectral form factor. Using both time-independent Hamiltonians (SYK, RCQ variants, XXZ) and random/Brownian circuit models, it demonstrates that t_ramp is controlled by diffusion in geometrically local systems with conservation laws (t_ramp ∼ N^2 in 1D) and by logarithmic scalings (t_ramp ∼ log N) in k-local systems with conservation as well as in generic non-conserving random circuits. The work connects t_ramp to two-point function decay and diffusion processes, clarifying mechanisms distinct from scrambling, and provides analytic estimates (via Markov chains) supported by extensive numerics. These results illuminate how spectral rigidity emerges in many-body quantum chaos and inform expectations for related holographic/black-hole dynamics, while highlighting subtleties due to density fluctuations and edge effects in finite-size spectra.

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.

Figures (30)

• Figure 1: Pair correlation function $R(E_1, E_2)$ plotted when $E_2=0$ and $\Delta E =E_1$ is varied. For simplicity, we have used $L=100$ and have suppressed the delta function in equation \ref{['sinekernel']}. The oscillations here are at the scale of the nearest neighbor eigenvalue spacing.
• Figure 2: [Left] Equal width slices of the energy spectrum marked with different colors. [Right] Ramp and plateau coming from two slices marked with the corresponding colors.
• Figure 3: $h(\alpha,t)\equiv |Y(\alpha,t)|^2/|Y(\alpha,t=0)|^2$ at $N=32$, with various values of $\alpha$. Note that $g(t) = h(\alpha =0, t)$.
• Figure 4: $g(t)=|Z(t)|^2/|Z(0)|^2$ and $h(\alpha,t)=|Y(\alpha,t)|^2/|Y(\alpha,t=0)|^2$ with $\alpha=2.9$ for $N=32$, 1500 samples.
• Figure 5: The density of shifted and renormalized eigenvalues for the 2-local RCQ model for $N=9,10,\ldots,16$, and the averaged variance of eigenvalues with a straight line fit. $(2^{16-N}\times 100)$ samples have been used for each $N$.