Many-Body Chaos in the Sachdev-Ye-Kitaev Model

Many-body chaos has emerged as a powerful framework for understanding thermalization in strongly interacting quantum systems. While recent analytic advances have sharpened our intuition for many-body chaos in certain large $N$ theories, it has proven challenging to develop precise numerical tools capable of exploring this phenomenon in generic Hamiltonians. To this end, we utilize massively parallel, matrix-free Krylov subspace methods to calculate dynamical correlators in the Sachdev-Ye-Kitaev model for up to $N=60$ Majorana fermions. We begin by showing that numerical results for two-point correlation functions agree at high temperatures with dynamical mean field solutions, while at low temperatures finite-size corrections are quantitatively reproduced by the exactly solvable dynamics of near extremal black holes. Motivated by these results, we develop a novel finite-size rescaling procedure for analyzing the growth of out-of-time-order correlators. Our procedure accurately determines the Lyapunov exponent, $\lambda$, across a wide range in temperatures, including in the regime where $\lambda$ approaches the universal bound, $\lambda =2\pi /\beta$.

Figure 1

Regularized OTOCs in the SYK model, $\tilde{F}(t)\equiv F(t)/F(0)$, as shown for $\beta J=10$ and system sizes $N\in [12,60]$. The early time behavior is characterized by $1-\tilde{F}(t)\sim e^{\lambda t}/N$ and different system sizes are approximately related by a time translation symmetry, $t\to t+1/\lambda \log N$. (b) Applying a finite-size rescaling procedure to the data, we determine $\lambda$ as a function of temperature (points). Our results exhibit excellent agreement with the theoretical predictions of the Schwinger-Dyson (SD) equations (dashed line), including in the regime where $\lambda$ approaches the bound on chaos $2\pi /\beta$ (blue).

Figure 2

Regimes of analytic control for the SYK model as a function of system size, $N$, and inverse temperature, $\beta J$. In the semiclassical limit (red and purple), the model is well described by a dynamical mean-field solution (Schwinger-Dyson equations). At low temperatures, finite-size corrections can be calculated using the Schwarzian action (blue), which is dual to two-dimensional anti–de Sitter gravity. However, at sufficiently small sizes (gray), the dynamics are governed by the discreteness of the energy spectrum and neither effective theory provides a valid description.

Figure 3

Two-point correlation functions in real and imaginary time. (a) Comparison of imaginary-time evolution between our numerics with 40 Majoranas (solid), the large $N$ solution (dotted), and the Schwarzian action (dashed). At high temperatures, we observe quantitative agreement between our numerical results and the large $N$ solution, while at low temperatures our numerics are well described by the Schwarzian action. (b) Analogous comparison for real-time evolution with $\beta J=56$. Our numerics show excellent agreement with the Schwarzian action for $tJ\gtrsim 10$. The disagreement at earlier times is attributed to the difference in high-energy modes, which are cut off at the energy scale $J$ in the SYK model and are unbounded in the effective action. Inset: a salient feature in our real-time numerics is a nonmonotonic trend with respect to temperature, as shown for $tJ=20$. This behavior is captured by the Schwarzian action (dashed) and can be understood as a consequence of the square root edge of the energy spectrum.