  Recent works unraveled an intriguing finite-time dynamical phase transition
in the thermal relaxation of the mean field Curie-Weiss model. The phase
transition reflects a sudden switch in the dynamics. Its existence in systems
with a finite range of interaction, however, remained unclear. Here we
demonstrate the dynamical phase transition for nearest-neighbor Ising systems
on the square and Bethe lattices through extensive computer simulations and by
analytical results. Combining large-deviation techniques and Bethe-Guggenheim
theory we prove the existence of the dynamical phase transition for arbitrary
quenches, including those within the two-phase region. Strikingly, for any
given initial condition we prove and explain the existence of non-trivial speed
limits for the dynamical phase transition and the relaxation of magnetization,
which are fully corroborated by simulations of the microscopic Ising model but
are absent in the mean field setting. Pair correlations, which are neglected in
mean field theory and trivial in the Curie-Weiss model, account for kinetic
constraints due to frustrated local configurations that give rise to a global
speed limit.
