Stochastic field effects in a two-state system: Symmetry breaking and symmetry restoring

We study the Ising model under a time-varying, but spatially homogeneous, Gaussian random magnetic field. In the Monte Carlo simulations, we go beyond the standard analysis of the order parameter by measuring the magnetization probability distribution as a function of temperature and field strength, and by computing the time required for the system to escape from a completely ordered state of the magnetization. We identify three distinct phases: a broad-paramagnetic phase, a broad-ferromagnetic phase, and a bona fide ferromagnetic phase. These broad phases display wide magnetization distributions that tend toward limiting forms that remain finite in both height and width in the thermodynamic limit. The transition between the broad-paramagnetic and broad-ferromagnetic phases is noise-induced and, for small field amplitudes, occurs at the critical temperature of the field-free Ising model. The transition from the broad-ferromagnetic to the ferromagnetic phase occurs at lower temperatures and is discontinuous, yet it does not fall into the conventional first-order class. Instead, it is characterized by a diverging escape time from an ordered magnetization state.