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

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.

💡 Research Summary

This paper investigates operator growth and scrambling dynamics in a Brownian spin SYK model with all‑to‑all random interactions ranging from two‑body up to L‑body terms. The authors first formulate the full operator‑size distribution (b_w(t)), defined as the summed squared amplitudes of Pauli strings of weight (w), and derive an exact master equation \