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arXiv:1809.01671 · Journal of High Energy Physics

Quantum Lyapunov spectrum

Hrant Gharibyan, Masanori Hanada, Brian G. Swingle, Masaki Tezuka

Published Apr 1, 2019• 46 citations
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Abstract

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.

DOI
10.1007/jhep04(2019)082

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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.
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.
We study two classes of quantum phenomena associated with classical chaos in a variety of quantum models: (i) dynamical localization and its extension and generalization to interacting few- and many-body systems and (ii) quantum exponential divergences in high-order correlators and other diagnostics of quantum chaos. Dynamical localization (DL) is a subtle phenomenon related to Anderson localization. It hinges on quantum interference and is typically destroyed in presence of interactions. DL often manifests as a failure of a driven system to heat up, violating the foundations of statistical physics. Kicked rotor (KR) is a prototypical chaotic classical model that exhibits linear energy growth with time. The quantum kicked rotor (QKR) features DL instead: its energy saturates. Multiple attempts of many-body generalizations faced difficulties in preserving DL. Recently, DL was shown in a special integrable many-body model. We study non-integrable models of few- and many-body QKR-like systems and provide direct evidence that DL can persist there. In addition, we show how a novel related concept of localization landscape can be applied to study transport in rippled channels. Out-of-time-ordered correlator (OTOC) was proposed as an indicator of quantum chaos, since in the semiclassical limit, this correlator's possible exponential growth rate (CGR) resembles the classical Lyapunov exponent (LE). We show that the CGR in QKR is related, but distinct from the LE in KR. We also show a singularity in the OTOC at the Ehrenfest time tᴱ due to a delay in the onset of quantum interference. Next, we study scaling of OTOC beyond tᴱ. We then explore how the OTOC-based approach to quantum chaos relates to the random-matrix-theoretical description by introducing an operator we dub the Lyapunovian. Its level statistics is calculated for quantum stadium billiard, a seminal model of quantum chaos, and aligns perfectly with the Wigner-Dyson surmise. In the semiclassical limit, the Lyapunovian reduces to the matrix of uncorrelated finite-time Lyapunov exponents, connecting the CGR at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference. Finally, we consider quantum polygonal billiards: their classical counterparts are non-chaotic. We show exponential growth of the OTOCs in these systems, sharply contrasted with the classical behavior even before quantum interference develops.
A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-N behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed to random matrix behavior and quantum chaos. In particular, the spectral form factor exhibits a dip-ramp-plateau structure after a running time average, in qualitative agreement with SYK. But we also observe that even though the spectrum has a unique ground state, it has a huge (quasi-?)degeneracy of intermediate energy states, not seen in SYK. If one ignores the delta function due to the degeneracies however, there is level repulsion in the unfolded spacing distribution hinting chaos. Furthermore, there are gaps in the spectrum. The system also has a spectral mirror symmetry which we trace back to the presence of a unitary operator with which the Hamiltonian anticommutes. We use it to argue that to the extent that the model exhibits random matrix behavior, it is controlled not by the Dyson ensembles, but by the BDI (chiral orthogonal) class in the Altland-Zirnbauer classification.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
We investigate the bulk-boundary correspondence in the SYK model from a quantum information perspective. The SYK model describes a system of Majorana fermions with random all-to-all interactions, whose disorder average-typically taken over a Gaussian ensemble-admits a dual description in terms of JT gravity in the large-NNN, low-energy limit. This framework provides a minimal setting for exploring holography and emergent spacetime in nearly AdS2_22​. We probe the holographic principle through diagnostics of quantum chaos and entanglement. In the early-time regime, the SYK model saturates the universal bound on the Lyapunov exponent, signaling maximal chaos consistent with semiclassical black hole dynamics. In the late-time regime, its spectral statistics are governed by random matrix theory, reflecting universal features of strongly chaotic quantum systems. These dynamical properties establish a concrete link between boundary quantum chaos and bulk semiclassical gravity. In parallel, we analyze quantum entanglement and the structure of operator algebras to investigate transitions in the associated von Neumann algebras and their implications for emergent geometry. To explore the robustness of these phenomena, we consider deformations of the SYK model through modified matter couplings and alternative random distributions. Our results clarify how quantum information-theoretic structures encode bulk gravitational dynamics and provide insight into the mechanism of spacetime emergence.
We propose a characterization of quantum many-body chaos: given a collection of simple operators, the set of all possible pair correlations between these operators can be organized into a matrix with a random-matrix-like spectrum. This approach is particularly useful for locally interacting systems, which do not generically show exponential Lyapunov growth of out-of-time-ordered correlators. We demonstrate the validity of this characterization by numerically studying the Sachdev-Ye-Kitaev model and a one-dimensional spin chain with random magnetic field (XXZ model).