Ruelle-Pollicott Decay of Out-of-Time-Order Correlators in Many-Body Systems

TL;DR

This work investigates how information scrambling, as diagnosed by the out-of-time-order correlator (OTOC), relates to the spectral properties of a weakly open many-body quantum system. By studying the kicked Ising spin chain, the authors show that the long-time OTOC decay rate in the closed system equals approximately twice the intrinsic Liouvillian gap of its weakly dissipative extension, a relation that persists across integrable and chaotic regimes. They validate this connection with parity-resolved Liouvillian analyses and an efficient Arnoldi-Lindblad computation, demonstrating a unified framework where Liouvillian spectroscopy captures both relaxation and irreversibility in many-body dynamics. The results extend RP-resonance-inspired ideas to strongly interacting systems, suggesting practical diagnostics of intermediate-time dynamics via open-system spectra.

Abstract

The out-of-time-order correlator (OTOC) quantifies information scrambling in quantum systems and serves as a key diagnostic of quantum chaos. In one-body systems with a classical counterpart, the relaxation of the OTOC is governed by Ruelle-Pollicott resonances. For many-body systems lacking a semiclassical limit, recent studies have identified an analogous role played by the Liouvillian spectrum of weakly open extensions of the dynamics, where the slowest decay rate -- the Liouvillian gap -- encodes relaxation. Here we study the kicked Ising spin chain and show that the long-time exponential decay of the OTOC in the isolated system occurs at a rate equal to twice this intrinsic gap. This correspondence persists even in crossover regimes between integrability and chaos, demonstrating that the Liouvillian spectrum provides a unified framework for understanding relaxation and irreversibility in closed many-body quantum systems.

Figures (4)

• Figure 1: The absolute value of $O_1^{zz}(1,t)$ for a transverse field $h_x$ in the chaotic regime. The data corresponds to a spin chain of size $L = 12$ and $h_x=0.8168$. The blue line with circles represents $O_1^{zz}(1,t)$, while the dashed line is the exponential fit to the intermediate time behavior of $|O_1^{zz}(1,t)|$, with a decay exponent $\alpha =$$0.2888$.
• Figure 2: Liouvillian gap $g(\gamma)$ as a function of the dissipation strength $\gamma$ for system sizes $L = 6$ (blue circles), $L = 8$ (orange squares), and $L = 10$ (green triangles) and $h_x = 0.8168$. The dashed line indicate quadratic fit. For $L = 10$, the extrapolated value of $\bar{g}$ is $0.1429$
• Figure 3: Comparison of Liouvillian gaps for two values of the transverse field $h_x$. (a) $h_x = 0.7854$. The parity-even subspace is shown in blue empty circles, the parity-odd subspace in orange empty squares, and the global gap obtained via the Arnoldi-Lindblad method in green crosses. (b) $h_x = 0.2749$. The parity- even subspace is shown in blue empty circles, the parity- odd subspace in orange empty squares, and the global gap obtained via the Arnoldi-Lindblad method in green crosses.
• Figure 4: (a) Chaos indicators $\eta$ and $\bar{\xi}_E$ as functions of the transverse field $h_x$, computed from the Floquet spectrum in the parity-even subspace for system size $L=12$. Blue upward triangles denote $\eta$ and cyan diamonds denote $\bar{\xi}_E$. (b) Comparison between the OTOC decay exponent $\alpha$ (red crosses) and twice the Liouvillian gap $2\bar{g}$ (orange empty circles) as functions of $h_x$. The Liouvillian gap $\bar{g}$ is extrapolated for $L=10$, while $\alpha$ is obtained from the isolated chain with $L=12$. The two panels together illustrate how both dynamical and spectral quantities capture the transition between integrable and chaotic regimes.