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 1/2, modeled by the Lipkin-Meshkov-Glick (LMG) model of a collective spin s = n/2 in the presence of Markovian dissipation processes. Employing semi-classical and variational methods and a systematic Holdstein-Primakov mapping, we determine the phase diagram and the spectral and steady-state properties of the Liouvillian by studying both the thermodynamic limit and 1/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-sate properties.