Power law tails and non Markovian dynamics in open quantum systems: An exact solution from Keldysh field theory

The Born-Markov approximation is widely used to study dynamics of open quantum systems coupled to external baths. Using Keldysh formalism, we show that the dynamics of a system of bosons (fermions) linearly coupled to non-interacting bosonic (fermionic) bath falls outside this paradigm if the bath spectral function has non-analyticities as a function of frequency. In this case, we show that the dissipative and noise kernels governing the dynamics have distinct power law tails. The Green's functions show a short time ``quasi'' Markovian exponential decay before crossing over to a power law tail, governed by the non-analyticity of the spectral function. We study a system of bosons (fermions) hopping on a one dimensional lattice, where each site is coupled linearly to an independent bath of non-interacting bosons (fermions). While the density and current profiles show interesting quantitative deviations from Markovian results, the current-current correlators show qualitatively distinct long time power law tails |t-t'|^-3 characteristic of non-Markovian dynamics. We show that the power law decays survive in presence of inter-particle interaction in the system, but the cross-over time scale is shifted to larger values with increasing interaction strength.