Dynamical Transition for a class of integrable models coupled to a bath

We study the dynamics of correlation functions of a class of d−dimensional integrable models coupled linearly to a fermionic or bosonic bath in the presence of a periodic drive. In the absence of the bath, these models exhibit a dynamical phase transition; all correlators decay to their steady state values as n^{−(d+2)/2}[n^{−d/2}] above [below] a critical frequency ω_c, where n_0 is the number of drive cycles. We find that the presence of a linearly coupled fermionic bath which maintains integrability of the system preserves this transition. We provide a semi-analytic expression for the evolution operator for this system and use it to provide a phase diagram showing the different dynamical regimes as a function of the system-bath coupling strength and the bath parameters. In contrast, when such models are coupled to a bosonic bath which breaks integrability of the model, we find exponential decay of the correlators to their steady state. Our numerical analysis shows that this exponential decay sets in above a critical number of drive cycles n_c which depends on the system-bath coupling strength and the amplitude of perturbation. Below n_c, the system retains the power-law behavior identical to that for the closed integrable models and the dynamical transition survives.