Dynamical Phase Transitions in the Two-Dimensional Transverse-Field Ising Model.

We investigate two separate notions of dynamical phase transitions in the two-dimensional nearest-neighbor transverse-field Ising model on a square lattice using matrix product states and a new \emph{hybrid} time-evolving block decimation algorithm. Starting in an ordered ground state, quenches below the dynamical critical point give rise to a ferromagnetic long-time steady state with the Loschmidt return rate exhibiting \emph{anomalous} cusps even when the order parameter never crosses zero. Quenches above the dynamical critical point lead to a paramagnetic long-time steady state with the return rate exhibiting \emph{regular} cusps connected to zero crossings of the order parameter. Additionally, our simulations indicate that quenching slightly above the dynamical critical point leads to a coexistence region where both anomalous and regular cusps appear in the return rate. Quenches from the disordered phase further confirm our main conclusions. Our work supports the recent finding that anomalous cusps arise only when local spin excitations are the energetically dominant quasiparticles. Our results are accessible in modern Rydberg experiments.