Entanglement distribution in the quantum symmetric simple exclusion process

We study the probability distribution of entanglement in the quantum symmetric simple exclusion process, a model of fermions hopping with random Brownian amplitudes between neighboring sites. We consider a protocol where the system is initialized in a pure product state of M particles, and we focus on the late-time distribution of Rényi-q entropies for a subsystem of size ℓ. By means of a Coulomb gas approach from random matrix theory, we compute analytically the large-deviation function of the entropy in the thermodynamic limit. For q>1, we show that, depending on the value of the ratio ℓ/M, the entropy distribution displays either two or three distinct regimes, ranging from low to high entanglement. These are connected by points where the probability density features singularities in its third derivative, which can be understood in terms of a transition in the corresponding charge density of the Coulomb gas. Our analytic results are supported by numerical Monte Carlo simulations.