Fine-grained dynamics of entanglement in non-integrable quenches far across the Ising quantum critical point

The task of exploring and understanding various aspects of far-from-equilibrium dynamics of closed and generic quantum many-body systems has received a thrust of attention in recent years, driven partly by remarkable advances in ultracold experimental technologies. In this work, for the paradigmatic Ising spin chain with transverse and longitudinal fields and partly motivated by the practice of site-resolved control in contemporary ultracold experiments, we present numerical observations of several $\textit{fine-grained}$ (small-subsystem level) features of far-from-equilibrium dynamics from a quantum informational point of view, induced by quantum quenches far across the Ising critical point between states deep inside the para- and ferro-magnetic regimes. Rather featureless dynamics is seen for ferromagnetic to paramagnetic quenches, but paramagnetic to ferromagnetic quenches exhibit rich behaviour, including recurrences of an approximately Page-like dynamics of entanglement entropies of one- and two-spin subsystems, periodic but short-lived occurrences of approximately $1-$uniform states, a series of sudden deaths and revivals of entanglement between two spins in the system's bulk, non-analytic cusps in single-copy entanglement entropy for three-spin and bigger subsystems, insufficient mixedness and a series of scrambling-$\textit{un}$scrambling of local mutual information between neighboring spins. Moreover, essentially indistinguishable dynamics is seen at very early times between the integrable limit (zero longitudinal field) and non-integrable cases, with the former eventually showing signatures of better mixing and faster approach to equilibration than the latter. These features are expected to hold for quench dynamics across Ising quantum critical points in more complicated systems.