A two-way approach to out-of-time-order correlators

TL;DR

This work develops a two-way mean-field framework to compute out-of-time-order correlators (OTOCs) in all-to-all interacting systems, notably the SYK model, by propagating forward and backward scrambling modes on a doubled thermofield double background. By decomposing scrambling into coherent and incoherent parts and introducing size operators, the authors derive a nonlinear OTOC equation that can be solved exactly in the large-$q$ limit, yielding a closed-form late-time expression featuring a confluent hypergeometric function. A kinetic-equation approach on the double Keldysh contour connects perturbations to a solvable Liouville-type equation, clarifying the scramblon dynamics and their black-hole analogies. The work also establishes a link between the Lyapunov exponent and high-frequency tails of the spectral function, providing a nontrivial bound that ties chaotic growth to spectral features. Overall, the paper offers a concrete, tractable route to exact OTOCs in a prototypical chaotic quantum many-body system and deepens the connection between quantum chaos and gravitational scattering pictures.

Abstract

Out-of-time-order correlators (OTOCs) are a standard measure of quantum chaos. Of the four operators involved, one pair may be regarded as a source and the other as a probe. A usual approach, applicable to large-$N$ systems such as the SYK model, is to replace the actual source with some mean-field perturbation and solve for the probe correlation function on the double Keldysh contour. We show how to obtain the OTOC by combining two such solutions for perturbations propagating forward and backward in time. These dynamical perturbations, or scrambling modes, are considered on the thermofield double background and decomposed into a coherent and an incoherent part. For the large-$q$ SYK, we obtain the OTOC in a closed form. We also prove a previously conjectured relation between the Lyapunov exponent and high-frequency behavior of the spectral function.

Figures (9)

• Figure 1: (a) The complex times $\theta_1,\theta_2,\theta_3,\theta_4$ in terms of the variable $e^{i\theta}=e^{-t}e^{i\tau}$. (b) A symmetric case, where the points 1,2,3,4 are placed on a cylinder with the axial coordinate $t$ (going left to right) and connected by a contour such that the contour ordering coincides with the $\tau$ ordering.
• Figure 2: An SYK diagram contributing to the OTOC at late times, such that $\lambda\sim 1$.
• Figure 3: Regions in the $(t_1,t_2)$ plane for equation \ref{['Liouville']}. The green color indicates the region that is affected by the source term. We focus on case (a), the retarded solution.
• Figure 4: Regions for the retarded solution of equation \ref{['brownian eqn']}. The function $W(t_1,t_2)$ has a jump discontinuity at the boundary of the green-colored quadrant. In addition, the normal derivative is discontinuous at the red diagonal line.
• Figure 5: (a) The double Keldysh contour used in the numerics is shown in blue; the red dots represent the insertion of sources. (b) The coordinate system for the Green function $G(l,l')$. The part that we use, $W(t_1,t_2)$, corresponds to the hatched area. (c) The function $f(l)f(l')$, where the white, gray, red, and blue colors represent $1$, $-1$, $i$, and $-i$, respectively.