Sharp Page transitions in generic Hamiltonian dynamics

We consider the entanglement dynamics of a subsystem initialized in a pure state at high energy density (corresponding to negative temperature) and coupled to a cold bath. The subsystem's Renyi entropies S_α first rise as the subsystem gets entangled with the bath and then falls as the subsystem is cooled. We find that the peak of the min-entropy, lim_α→∞ S_α sharpens to a cusp in the thermodynamic limit, at a well-defined time we call the Page time. We construct a hydrodynamic ansatz for the evolution of the entanglement Hamiltonian, which accounts for the transition as well as the complex dynamics of the entanglement spectrum before the Page time. Based on this ansatz, we conjecture that S_α at the Page time is also sharp for some range of α > 1, in higher-dimensional systems or systems with long-range interactions. Our results hold both when the bath has the same Hamiltonian as the system and when the bath is taken to be Markovian.