Time evolution of the entanglement entropy in isolated systems: Page curve and fluctuations

The average entanglement entropy of a random pure state was determined by Page in 1993. Here we consider an isolated quantum system and study the evolution of the entanglement entropy at early and late times, assuming that the system is initially in a non-entangled state. We conduct the analysis for: (i) random-matrix Hamiltonians, which can be used to model the full SYK Hamiltonian, and (ii) random quadratic fermionic Hamiltonians, which are also the SYK-2 Hamiltonians. We show that the entanglement entropy initially grows as $-\alpha t^2 \log(\alpha t^2)$ and determine the scale $\alpha$ in terms of the interaction strength. We also show that, for a random-matrix Hamiltonian, we recover the Page average at late times and estimate the size of the fluctuations around the average. Subsequently, we derive analogous results for random quadratic fermionic systems. Through this analysis, we derive expression for the time-scale at which the entanglement entropy reaches its equilibrium value. The result is of interest for the analysis of systems governed by SYK Hamiltonians and for investigations of information in black hole evaporation.