Quantum Lyapunov Spectrum

TL;DR

This work defines a quantum analogue of the classical Lyapunov spectrum and demonstrates its utility by applying it to the SYK model and XXZ spin chains. By analyzing the full spectrum, it uncovers random-matrix universality in chaotic regimes and connects the sum of positive exponents to entanglement-driven entropy production, suggesting black holes maximize this rate. The results show that the largest exponent alone misses important structure, and that the spectrum provides deeper insights into scrambling, chaos strength, and potential holographic signatures. The findings pave the way for a spectrum-based chaos diagnostic across quantum many-body systems and motivate further exploration of holographic connections and experimental probes.

Abstract

We introduce a simple quantum generalization of the spectrum of classical Lyapunov exponents. We apply it to the SYK and XXZ models, and study the Lyapunov growth and entropy production. Our numerical results suggest that a black hole is not just the fastest scrambler, but also the fastest entropy generator. We also study the statistical features of the quantum Lyapunov spectrum and find universal random matrix behavior, which resembles the recently-found universality in classical chaos. The random matrix behavior is lost when the system is deformed away from chaos, towards integrability or a many-body localized phase. We propose that quantum systems holographically dual to gravity satisfy this universality in a strong form. We further argue that the quantum Lyapunov spectrum contains important additional information beyond the largest Lyapunov exponent and hence provides us with a better characterization of chaos in quantum systems.

Figures (22)

• Figure 1: Suppose we know that the initial condition is contained in the blue disk in the upper left corner. As this region evolves with time, although the volume is conserved due to the Liouville theorem, the shape changes nontrivially, in particular, it is stretched exponentially in several directions. The rate is governed by the positive Lyapunov exponents.
• Figure 2: Matrix configurations for one big black hole (left), two black holes (middle) and the gas of D0-branes (right).
• Figure 3: The generalized Majorana SYK model. $d_1$ and $d_2$ vs $j/L$. Averages over 50 samples ($N=12,16$) and 5 samples ($N=20$). [Left] $K=0.01$, [Right] $K=10$.
• Figure 4: The XXZ model. $d\equiv \frac{1}{N_{\rm site}} \sum_{k=1}^{N_{\rm site}} \sum_{i=0}^j |\langle E_{i,1/2}| \sigma_k^+ |E_{0,0}\rangle|^2$. Averages over 100 samples ($N_{\rm site}=6, 8$) and 5 samples ($N_{\rm site}=10$). The horizontal axis is $j/L_{S_z=1/2}$, where $L_{S_z=1/2}$ is the dimension of $S_z= 1/2$ Hilbert space.
• Figure 5: The generalized SYK model with $K=0.01$. [Left] The Lyapunov growth from OTOC, the almost linear dependence on time $t$ of $\lambda^{(\mathrm{OTOC})}t=\frac{1}{2}\log\left(\frac{1}{N}\sum_{i=1}^N e^{2\lambda_i t}\right)$. 10 samples for $N=24, 22$ and 1000 samples for $N=20, 18, \ldots, 6$ are used. Vertical lines correspond to 20% and 80% for $N=20$. [Middle] The Lyapunov exponent estimated from OTOC, $\lambda^{\mathrm{(OTOC)}}\equiv\frac{1}{2t}\log\left(\frac{1}{N}\sum_{i=1}^N e^{2\lambda_i t}\right)$. [Right] At each $N$, the exponent $\lambda^{\mathrm{(OTOC)}}$ at $t=2$ is shown as the function of the energy $E_i$. The horizontal axis is $(i+1/2)/L$, so that the left and right corresponds to low and high energies.