Dissipative phase transitions in the fully-connected Ising model with $p$-spin interaction

In this paper, we study the driven-dissipative p-spin models for $p\geq 2$. In thermodynamics limit, the equation of motion is derived by using a semiclassical approach. The long-time asymptotic states are obtained analytically, which exhibit multi-stability in some regions of the parameter space. The steady state is unique as the number of spins is finite. But the thermodynamic limit of the steady-state magnetization displays nonanalytic behavior somewhere inside the semiclassical multi-stable region. We find both the first-order and continuous dissipative phase transitions. As the number of spins increases, both the Liouvillian gap and magnetization variance vanish according to a power law at the continuous transition. At the first-order transition, the gap vanishes exponentially accompanied by a jump of magnetization in thermodynamic limit. The properties of transitions depend on the symmetry and semiclassical multistability, being qualitatively different among $p=2$, odd $p$ ($p\geq 3$) and even $p$ ($p\geq 4$).