Dynamical phase transitions for the activity biased Ising model in a magnetic field

We consider large deviations of the dynamical activity -- defined as the total number of configuration changes within a time interval -- for mean-field and one-dimensional Ising models, in the presence of a magnetic field. We identify several dynamical phase transitions that appear as singularities in the scaled cumulant generating function of the activity. In particular, we find low-activity ferromagnetic states and a novel high-activity phase, with associated first- and second-order phase transitions. The high-activity phase has a negative susceptibility to the magnetic field. In the mean-field case, we analyse the dynamical phase coexistence that occurs on first-order transition lines, including the optimal-control forces that reproduce the relevant large deviations. In the one-dimensional model, we use exact diagonalisation and cloning methods to perform finite-size scaling of the first-order phase transition at non-zero magnetic field.