Stochastic Two-temperature Nonequilibrium Ising model

We investigate the nonequilibrium stationary state (NESS) of the two-dimensional Ising model under a stochastic dichotomous modulation of temperature, which alternates between $T_c \pm \delta$ around the critical temperature $T_c$ at a rate $\gamma$. Both magnetization and energy exhibit non-monotonic dependence on $\gamma$, explained by a renewal approach in the slow-switching limit, while for small $\delta$ dynamical response theory quantitatively captures the $\gamma$-dependence of the observables. In the fast-switching regime, the NESS appears Boltzmann-like with a $\gamma$-dependent effective temperature. However, a finite energy current flowing through the system from hot to cold reservoir confirms the intrinsic nonequilibrium nature of the dynamics.