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, 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.

We study the ballistic-to-diffusive transition induced by the weak breaking of integrability in a boundarydriven XXZ spin chain. Studying the evolution of the spin current density J s as a function of the system size L, we show that, accounting for boundary effects, the transition has a nontrivial universal behavior close to the XX limit. It is controlled by the scattering length L * ∝ V −2 , where V is the strength of the integrability-breaking term. In the XXZ model, the interplay of interactions controls the emergence of a transient "quasiballistic" regime at length scales much shorter than L *. This parametrically large regime is characterized by a strong renormalization of the current which forbids a universal scaling, unlike the XX model. Our results are based on matrix product operator numerical simulations and agree with perturbative analytical calculations.

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