Late Time von Neumann Entropy and Measurement-induced Phase Transition

We present our studies on the late-time von Neumann entropy and its transition in Brownian SYK models. Without measurement, we show that the correlations between different replicas account for the Page curve at late time, and a permutation group structure emerges in the calculation. In the presence of measurements, we show that a continuous von Neumann entropy transition from volume-law to area-law occurs at the point of replica symmetry breaking.