Note on entropy dynamics in the Brownian SYK model

TL;DR

This paper investigates the evolution of R'enyi entropy in coupled Brownian SYK models, revealing a Page curve through path integral and operator dynamics methods, highlighting the roles of different saddle points.

Contribution

It introduces a detailed analysis of entropy dynamics in Brownian SYK systems using novel path integral and operator techniques, providing new insights into their thermalization process.

Findings

R'enyi entropy exhibits linear growth followed by saturation.

Path integral and operator methods yield consistent Page curves.

Different saddle points govern entropy growth and saturation.

Abstract

We study the time evolution of R\'enyi entropy in a system of two coupled Brownian SYK clusters evolving from an initial product state. The R\'enyi entropy of one cluster grows linearly and then saturates to the coarse grained entropy. This Page curve is obtained by two different methods, a path integral saddle point analysis and an operator dynamics analysis. Using the Brownian character of the dynamics, we derive a master equation which controls the operator dynamics and gives the Page curve for purity. Insight into the physics of this complicated master equation is provided by a complementary path integral method: replica diagonal and non-diagonal saddles are responsible for the linear growth and saturation of R\'enyi entropy, respectively.