Quantum non-Markovian Hatano-Nelson model

While considering non-Hermitian Hamiltonians arising in the presence of dissipation, in most cases, the dissipation is taken to be frequency independent. However, this idealization may not always be applicable in experimental settings, where dissipation can be frequency-dependent. Such frequency-dependent dissipation leads to non-Markovian behavior. In this work, we demonstrate how a non-Markovian generalization of the Hatano-Nelson model, a paradigmatic non-Hermitian system with nonreciprocal hopping, arises microscopically in a quasi-one-dimensional dissipative lattice. This is achieved using non-equilibrium Green's functions without requiring any approximation like weak system-bath coupling or a time-scale separation, which would have been necessary for a Markovian treatment. The resulting effective system exhibits nonreciprocal hopping, as well as uniform dissipation, both of which are frequency-dependent. This holds for both bosonic and fermionic settings. We find solely non-Markovian nonreciprocal features like unidirectional frequency blocking in bosonic setting, and a non-equilibrium dissipative quantum phase transition in fermionic setting, that cannot be captured in a Markovian theory, nor have any analog in reciprocal systems. Our results lay the groundwork for describing and engineering non-Markovian nonreciprocal quantum lattices.