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The Born-Markov approximation is widely used to study dynamics of openquantum systems coupled to external baths. Using Keldysh formalism, we showthat the dynamics of a system of bosons (fermions) linearly coupled tonon-interacting bosonic (fermionic) bath falls outside this paradigm if thebath spectral function has non-analyticities as a function of frequency. Inthis case, we show that the dissipative and noise kernels governing thedynamics have distinct power law tails. The Green's functions show a short time"quasi" Markovian exponential decay before crossing over to a power law tailgoverned by the non-analyticity of the spectral function. We study a system ofbosons (fermions) hopping on a one dimensional lattice, where each site iscoupled linearly to an independent bath of non-interacting bosons (fermions).We obtain exact expressions for the Green's functions of this system which showpower law decay $\sim |t-t'|^{-3/2}$. We use these to calculate density andcurrent profile, as well as unequal time current-current correlators. While thedensity and current profiles show interesting quantitative deviations fromMarkovian results, the current-current correlators show qualitatively distinctlong time power law tails $|t-t'|^{-3}$ characteristic of non-Markoviandynamics. We show that the power law decays survive in presence ofinter-particle interaction in the system, but the cross-over time scale isshifted to larger values with increasing interaction strength.

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