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arXiv:2207.11980 · Physical Review Letters

Onset of Quantum Chaos in Random Field Theories

Micha Berkooz A, Adar Sharon, Navot Silberstein, Erez Y. Urbach

Published Aug 9, 2022• 10 citations
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Abstract

We study the quantum Lyapunov exponent λ_{L} in theories with spacetime-independent disorder. We first derive self-consistency equations for the two- and four-point functions for products of N models coupled by disorder at large N, generalizing the equations appearing in SYK-like models. We then study families of theories in which the disorder coupling is an exactly marginal deformation, allowing us to follow λ_{L} from weak to strong coupling. We find interesting behaviors, including a discontinuous transition into chaos, mimicking classical KAM theory.

DOI
10.1103/physrevlett.129.071601

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We study the quantum Lyapunov exponent λ_{L} in theories with spacetime-independent disorder. We first derive self-consistency equations for the two- and four-point functions for products of N models coupled by disorder at large N, generalizing the equations appearing in SYK-like models. We then study families of theories in which the disorder coupling is an exactly marginal deformation, allowing us to follow λ_{L} from weak to strong coupling. We find interesting behaviors, including a discontinuous transition into chaos, mimicking classical KAM theory.
We study the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ in quantum field theories with spacetime-independent disorder interactions. Generically ${\ensuremath{\lambda}}_{L}$ can only be computed at isolated points in parameter space, and little is known about the way in which chaos grows as we deform the theory away from weak coupling. In this paper, we describe families of theories in which the disorder coupling is an exactly marginal deformation, allowing us to follow ${\ensuremath{\lambda}}_{L}$ from weak to strong coupling. We find surprising behaviors in some cases, including a discontinuous transition into chaos. We also derive self-consistency equations for the two- and four-point functions for products of NNN nontrivial conformal field theories deformed by disorder at leading order in 1/N1/N1/N.
A bstract We discuss aspects of the quantum Lyapunov exponent λ L in theories with an exactly marginal SYK-like random interaction, where λ L can be computed as a continuous function of the interaction strength J \mathcal{J} J J . In 1 d , we prove a conjecture from [1] which states that at small J \mathcal{J} J J , λ L can be found by considering a specific limit of the four-point function in the decoupled theory. We then provide additional evidence for the 2 d version of this conjecture by discussing new examples of Lyapunov exponents which can be computed at weak coupling.
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.
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.
Recent examples have confirmed the common belief that quantum chaos is always suppressed in the presence of an environment. Here we show that this is not always the case. We compute the Lyapunov exponent of a qqq-body Majorana Sachdev-Ye-Kitaev (SYK) model coupled to a Markovian bath. Provided that the jump operators describing the bath are isotropic random k>2k > 2k>2-body Majoranas fields, the Lyapunov exponent is positive for any strength of the coupling to the bath. Interestingly, this also applies to q=2q=2q=2 where the SYK is integrable which indicates that many-body quantum chaos can be induced by the environment. Moreover, for q>2q > 2q>2, where the unitary dynamics is quantum chaotic, and sufficiently large kkk, the Lyapunov exponent increases with the coupling to the bath. Explicit analytical results are obtained in the q=2q=2q=2 and large qqq limits. Our results put forward an alternate route to induce and control the generation of scrambling in quantum many-body systems which is of potential relevance in the design of quantum information devices.
We study scrambling in a model consisting of a number NNN of MMM-component quantum rotors coupled by random infinite-range interactions. This model is known to have both a paramagnetic phase and a spin glass phase separated by second order phase transition. We calculate in perturbation theory the squared commutator of rotor fields at different sites in the paramagnetic phase, to leading non-trivial order at large NNN and large MMM. This quantity diagnoses the onset of quantum chaos in this system, and we show that the squared commutator grows exponentially with time, with a Lyapunov exponent proportional to 1M\frac{1}{M}M1​. At high temperature, the Lyapunov exponent limits to a value set by the microscopic couplings, while at low temperature, the exponent exhibits a T4T^4T4 dependence on temperature TTT.