Note on entropy dynamics in the Brownian SYK model

We study the time evolution of Renyi entropy in a system of two coupled Brownian SYK clusters evolving from an initial product state. The Renyi 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 Renyi entropy, respectively.

Automated summary

The evolution of Renyi entropy in a system of two coupled Brownian SYK clusters was studied, revealing linear growth and saturation of entropy in one cluster. The Page curve was obtained through path integral saddle point analysis and operator dynamics analysis, with a master equation derived from the dynamics' Brownian character controlling the operator dynamics and providing the Page curve for purity. A complementary path integral method identified that replica diagonal and non-diagonal saddles account for the linear growth and saturation of Renyi entropy.