Dissipative dynamics of an interacting spin system with collective damping

The competition between Hamiltonian and Lindblad dynamics in quantum systems give rise to non-equillibrium phenomena with no counter part in conventional condensed matter physics. In this paper, we investigate this interplay of dynamics in infinite range Heisenberg model coupled to a non-Markovian bath and subjected to Lindblad dynamics due to spin flipping at a given site. The spin model is bosonized via Holstein-Primakoff transformations and is shown to be valid for narrow range of parameters in the thermodynamic limit. Using Schwinger-Keldysh technique, we derive mean field solution of the model and observe that the system breaks $\mathcal{Z}_2$-symmetry at the transition point. We calculate effective temperature that has linear dependence on the effective system-bath coupling, and is independent of the dissipation rate and cutoff frequency of the bath spectral density. Furthermore, we study the fluctuations over mean field and show that the dissipative spectrum is modified by ${\rm O}(\frac{1}{N})$ correction term which results change in various physically measurable quantities.