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arXiv:2301.01353 · Journal of High Energy Physics

More on chaos at weak coupling

Rohit R. Kalloor, Adar Sharon

Published Jul 11, 2025• 0 citations
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Abstract

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 $$ \mathcal{J} $$ J . In 1 d , we prove a conjecture from [1] which states that at small $$ \mathcal{J} $$ 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.

DOI
10.1007/jhep07(2025)139

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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 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.
The Sachdev-Ye-Kitaev (SYK) model shows chaotic behavior with a maximal Lyapunov exponent. In this paper, we investigate the four-point function of a SYK-type model numerically, which gives us access to its Lyapunov exponent. The model consists of two sets of Majorana fermions, called A and B, and the interactions are restricted to being exclusively pairwise between the two sets, not within the sets. We find that the Lyapunov exponent is still maximal at strong coupling. Furthermore, we show that even though the conformal dimensions of the A and B fermions change with the population ratio, the Lyapunov exponent remains constant, not just in the conformal limit where it is maximal, but also in the intermediate and weak coupling regimes.
Abstract A simple probe of chaos and operator growth in many-body quantum systems is the out of time ordered four point function. In a large class of local systems, the effects of chaos in this correlator build up exponentially fast inside the so called butterfly cone. It has been previously observed that the growth of these effects is organized along rays and can be characterized by a velocity dependent Lyapunov exponent, λ(v). We show that this exponent is bounded inside the butterfly cone as λ(v) ≤ 2πT (1 − | v |/v B ), where T is the temperature and v B is the butterfly speed. This result generalizes the chaos bound of Maldacena, Shenker and Stanford. We study λ(v) in some examples such as two dimensional SYK models and holographic gauge theories, and observe that in these systems the bound gets saturated at some critical velocity v * < v B . In this sense, boosting a system enhances chaos. We discuss the connection to conformal Regge theory, where λ(v) is related to the spin of the leading large N Regge trajectory, and controls the four point function in an interpolating regime between the Regge and the light cone limit. Finally, we comment on the generalization of the chaos bound to boosted and rotating ensembles and clarify some recent results on this in the literature.
Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.