Page curve like dynamics in interacting quantum systems

We study the dynamics of entanglement in a one-dimensional $XXZ$ spin-$1/2$ chain with and without integrability-breaking interactions and connected to a bath. We start from a state where the system and bath are completely unentangled, and the bath is polarized spin-down. We consider two different initial states for the system: (i) a polarized spin-up state, and (ii) an infinite-temperature state. In the particle representation of the spin chain, the polarized spin-up state corresponds to a filled state, while the polarized spin-down state corresponds to an empty state. Starting from these inhomogeneous quenches, in all the above-mentioned cases, we obtain the Page curve like behavior in the entanglement. We report different power-law behavior in the growth of entanglement for different initial states and different kinds of baths (interacting and noninteracting). In an attempt to explore plausible deep connections between entanglement and Boltzmann entropy, we investigate the latter in both the filled and the infinite-temperature case, for the system and the bath. For the filled case, the Boltzmann entropy of the system has the form of a Page curve but quantitatively deviates from the entanglement. On the other hand, the entropy of the bath keeps increasing. Remarkably, for the infinite-temperature case, we find that the system and bath Boltzmann entropies agree with the entanglement entropy, after and before the Page time, respectively. Our findings are expected to hold for generic interacting quantum systems and could be of relevance to black-hole physics.