Steady-state properties of quantum non-Hermitian lattice models

The physics of non-Hermitian systems is an active area of research. In a quantum setting, it is tempting to directly define the system’s state (i.e. its density matrix) from the Hamiltonian. While various prescriptions have been proposed (e.g. exponentiating the non-Hermitian Hamiltonian, occupying right eigenvectors), these are largely problematic. Further, a number of natural questions about non-Hermitian steady-states, such as the role of right and left eigenvectors, particle statistics or the sensitivity to boundary conditions analogous to the non-Hermitian skin effect have not been fully studied. Here, we address these issues in a more physical manner, using the fact that quantum non-Hermitian dynamics almost always requires a coupling to external dissipative environments. We study quantum versions of the paradigmatic Hatano-Nelson model (i.e. a non-Hermitian, non-reciprocal 1D tight binding model) that are realized using engineered dissipation, both for bosons and fermions. Our analysis reveals a number of basic and generic insights. In particular, we highlight the role of an emergent momentum-dependent temperature, and the fact that the non-Hermitian skin effect alone does not determine the steady-state sensitivity to boundary conditions.