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

TL;DR

This work tackles operator scrambling in Brownian spin SYK models by deriving a closed master equation for the Pauli-string size distribution and recasting the dynamics into a generating-function PDE amenable to a systematic $1/N$ expansion. In the dilute, large-$N$ limit, the leading-order spectrum is exactly solvable, yielding explicit expressions for the generating function and exact late-time plateaus for arbitrary initial operators, including decoherence. The authors then develop first- and second-order $1/N$ corrections, demonstrating that higher-order effects crucially affect late-time operator growth and enable mixing between operator-weight sectors; these corrections are validated against numerical simulations. For two- and three-body interactions, parity constraints and metastable states emerge, with even/odd initial weights leading to distinct asymptotic plateaus. Overall, the generating-function method provides a transparent framework to study scrambling in Brownian circuits with imperfections and opens avenues toward understanding Krylov dynamics in open quantum systems.

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.

Figures (4)

• Figure 1: An illustration of the overlap patterns in case $p=1,m=2$ and case $p=3,m=0$.
• Figure 2: Perturbative predictions versus numerical simulations for different initial operator weights $w_0$. Parameters: $N=100$, $\kappa=0.5$, $r=1$, $a_2=1$, and $a_{j>2}=0$. All four cases converge to the same universal late-time plateau.
• Figure 3: Comparison between perturbation theory and numerical simulation for an initial operator of weight $w_0 = 3$. Parameters: $N=100$, $\kappa=0.5$, $r=1$, $a_2=1$, and $a_{j>2}=0$.
• Figure 4: Comparison of perturbative results with numerical simulations of Eq. \ref{['eq:L3-bw']} for three-body interactions. Parameters are set to $N=100$, $\kappa=0.5$, $r=1$, $a_3=1$, and $a_{j \neq 3}=0$.