# Quantum chaos in the Brownian SYK model with large finite $N$: OTOCs and tripartite information

## TL;DR

This paper investigates quantum chaos and information scrambling in the Brownian SYK model with large finite N, using OTOCs and tripartite information, revealing a logarithmic scrambling time and exponential decay of correlators.

## Contribution

It introduces a method to analyze the dynamics of OTOCs and tripartite information in the Brownian SYK model for very large N using a bosonic collective mode approach.

## Key findings

- Scrambling time scales as ln N.
- OTOCs and tripartite information decay exponentially after t*.
- Numerical calculations performed up to N=10^6.

## Abstract

We consider the Brownian SYK model of $N$ interacting Majorana fermions, with random couplings that are taken to vary independently at each time. We study the out-of-time-ordered correlators (OTOCs) of arbitrary observables and the R\'enyi-$2$ tripartite information of the unitary evolution operator, which were proposed as diagnostic tools for quantum chaos and scrambling, respectively. We show that their averaged dynamics can be studied as a quench problem at imaginary times in a model of $N$ qudits, where the Hamiltonian displays site-permutational symmetry. By exploiting a description in terms of bosonic collective modes, we show that for the quantities of interest the dynamics takes place in a subspace of the effective Hilbert space whose dimension grows either linearly or quadratically with $N$, allowing us to perform numerically exact calculations up to $N = 10^6$. We analyze in detail the interesting features of the OTOCs, including their dependence on the chosen observables, and of the tripartite information. We observe explicitly the emergence of a scrambling time $t^\ast\sim \ln N$ controlling the onset of both chaotic and scrambling behavior, after which we characterize the exponential decay of the quantities of interest to the corresponding Haar scrambled values.