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

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