Chaos, Complexity, and Random Matrices

TL;DR

The paper investigates chaos and scrambling through Gaussian Unitary Ensemble (GUE) dynamics, linking out-of-time-ordered correlators (OTOCs) and spectral form factors to quantify scrambling, randomness, and complexity. It introduces k-invariance as a precise, basis-independent criterion for when dynamics are effectively described by random-matrix theory, and analyzes how GUE dynamics become approximate k-designs at intermediate times before deviating at late times. By deriving exact expressions for 2-point and 4-point spectral form factors and relating them to frame potentials, the authors quantify time scales for Haar-like behavior and complexity growth, highlighting both the successes and limitations of random matrices as models of chaotic quantum dynamics. The work emphasizes a dynamical transition from early-time chaos to late-time random-matrix universality, and proposes k-invariance as a useful tool to characterize this transition and its implications for information scrambling and holography.

Abstract

Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an $\mathcal{O}(1)$ scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce $k$-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate $k$-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.

Figures (14)

• Figure 1: The 2-point spectral form factor for SYK with $N=26$ Majoranas at inverse temperature $\beta=5$, computed for 1000 random samples. The slope, dip, ramp, and plateau are labeled.
• Figure 2: The $2$-point spectral form factor at infinite temperature, as given in Eq. \ref{['eq:R2func']}, plotted for various values of $L$ and normalized by the initial value $L^2$. We observe the linear ramp and scaling of the dip and plateau with $L$.
• Figure 3: The $2$-point spectral form factor at finite temperature as per Eq. \ref{['eq:R2beta']}, on the left plotted at different values of $L$, and on the right plotted at different temperatures, normalized by the initial value. We see that the dip and plateau both scale with $\beta$ and $L$ and that lowering the temperature smooths out the oscillations in ${\cal R}_2$.
• Figure 4: The GUE $4$-point spectral form factor at infinite temperature, plotted for different values of $L$ and normalized by their initial values. We observe the scaling of the dip and plateau, and the quadratic rise $\sim t^2$.
• Figure 5: The 2-point form factor and the 2-point functions $\langle{A_j A_j(t)}\rangle$ of Pauli operators for $H_{\rm RNL}$ for $n=5$ sites and averaged over $500$ samples. The thick blue line is ${\cal R}_2/L^2$ and surrounding bands of lines are all 1024 Pauli 2-point functions of different weight.