Higher-Order Corrections to Scrambling Dynamics in Brownian Spin SYK Models

TL;DR

This paper develops a detailed theoretical framework for understanding operator growth and scrambling in Brownian spin SYK models, including higher-order corrections and their impact on quantum chaos.

Contribution

It introduces a systematic $1/N$ expansion to incorporate higher-order effects in operator dynamics, extending previous leading-order analyses.

Findings

Exact solutions for operator-size distribution in large-$N$ limit

Higher-order corrections significantly influence operator scrambling

Full operator-size distribution offers a refined chaos probe

Abstract

We investigate operator growth in a Brownian spin Sachdev--Ye--Kitaev (SYK) model with random all-to-all interactions, focusing on the full operator-size distribution. For Hamiltonians containing interactions of order two up to $L$, we derive a closed master equation for the Pauli-string expansion coefficients and recast their dynamics into a generating-function formulation suitable for the large-$N$ limit. This approach allows us to diagonalize the leading-order evolution operator explicitly and obtain exact solutions for arbitrary initial operator distributions, including the effects of decoherence. Going beyond leading order, we develop a systematic $1/N$ expansion that captures higher-order corrections to the operator-size dynamics and the late-time behavior. Our results demonstrate that higher-order effects play a crucial role in operator scrambling and that the full operator-size distribution provides a more refined probe of quantum chaos in Brownian and open quantum systems.