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 1/2, modeled by the Lipkin-Meshkov-Glick (LMG) model of a collective spin s = N/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/s corrections. Our approach reveals the existence of different kinds of dynamical phases and phase transitions, multistability, and regions where the dynamics is recurrent. We provide a classification of critical and noncritical Liouvillians according to their spectral and steady-state properties.