Generalized Hydrodynamic Description of the Page Curve–like Dynamics of a Freely Expanding Fermionic Gas

We consider an analytically tractable model that exhibits the main features of the Page curve characterizing the evolution of entanglement entropy during evaporation of a black hole. Our model is a gas of noninteracting fermions on a lattice that is released from a box into the vacuum. More precisely, our Hamiltonian is a tight-binding model with a defect at the junction between the filled box and the vacuum. In addition to the entanglement entropy we consider several other observables, such as the spatial density profile and current, and show that the semiclassical approach of generalized hydrodynamics provides a remarkably accurate description of the quantum dynamics including that of the entanglement entropy at all times t. Our hydrodynamic results agree closely with those obtained via exact microscopic numerics. We find that the growth of entanglement with time is linear and universal, i.e, independent of the details of the defect. The decay shows 1/t scaling for conformal defect while for nonconformal defects, it is slower. Our analytical results from hydrodynamics enables us to make predictions for long time behaviors of observables that are otherwise highly inaccessible computationally by exact microscopic numerics. Our study shows the power of the semiclassical approach and could be relevant for discussions on the resolution of the black hole information paradox.