Keldysh approach to dissipative dynamics of interacting spins with long range interactions.

The interplay of coherent dynamics and dissipation in many body sysems give rise to a rich class of non-equilibrium phenomena with the emergence of non-trivial phases. In this paper, we investigate this interplay of dynamics in a model of interacting spins with infinite range interactions described by Heisenberg interaction. Using Holestien-Primarkoff transformations, the spin model is bosonized to a collective mode with self interaction at $\frac{1}{N}$ order. Employing Keldysh field theoretic technique with saddle point approximation, we see that the system breaks $\mathcal{Z}_2$-symmetry at the transition point. An effective temperature arises due to dissipation which depends linearly on effective system-bath coupling, and is independent of the dissipation rate and cutoff frequency of the bath with Drude-Lorentz cutoff in wide class of bath spectral densities. Furthermore, fluctuations over mean field are studied and is shown that the dissipative spectrum is modified by ${\rm O}(\frac{1}{N})$ correction term which results change in various physically measurable quantities.