Logarithmic growth of entanglement entropy in out-of-equilibrium long-range systems

In this work, we derive the analytical relation between bipartite entanglement entropy and collective spin-squeezing in long-range spin systems in and out of equilibrium, and use it to elucidate the mechanism responsible for the logarithmic growth in time of entanglement entropy after a quench, which has been numerically observed in a number of recent studies. We further discuss the special cases of quenches to dynamical critical points, in which entanglement entropy increases linearly in time rather than logarithmically, and relate these behaviors to the structure of the underlying semiclassical trajectories. All our analytical results agree with exact numerical computations, and extend to systems with algebraically-decaying interactions with exponent $\alpha \le d$, where $d$ is the system dimensionality. Our findings also provide immediate access to experimental measurements of entanglement entropy in this class of systems.