The Lipkin-Meshkov-Glick model with Markovian dissipation -- A description of a collective spin on a metallic surface

Motivated by recent prototypes of engineered atomic spin devices, we study a fully connected system of $N$ spins $1\mathrm{/}2$, modeled by the Lipkin-Meshkov-Glick (LMG) model of a collective spin $s=N\mathrm{/}2$ in the presence of Markovian dissipation processes. We determine and classify the different phases of the dissipative LMG model with Markovian dissipation, including the properties of the steady-state and the dynamic behavior in the asymptotic long-time regime. Employing variational methods and a systematic approach based on the Holstein-Primakoff mapping, we determine the phase diagram and the spectral and steady-state properties of the Liouvillian by studying both the infinite-$s$ limit and $1\mathrm{/}s$ corrections. Our approach reveals the existence of different kinds of dynamical phases and phase transitions, multi-stability and regions where the dynamics is recurrent. We provide a classification of critical and non-critical Liouvillians according to their spectral and steady-state properties.