Why the dissipative gap fails to predict relaxation times in dissipative lattice models

Quantum systems exhibiting effective non-Hermitian dynamics are the subject of considerable recent interest. Among their most striking features is the non-Hermitian skin effect (NHSE), where a small change in boundary conditions can completely change the spectrum and localization of eigenmodes. Recent work suggests that the NHSE directly also yields surprising dynamics: anomalously long relaxation times that are not simply given by the dissipative gap of the system's Liouvillian, but instead grow with system size and depend on the NHSE localization length ξ_loc. Despite the simplicity of the underlying argument, we show here that the situation is often more complex: a subtle interference effect makes the extreme localization of modes largely irrelevant in determining relaxation times. Focusing on a fully quantum open-systems implementation of the paradigmatic Hatano-Nelson model [1], we show why this intuitive interference effect must occur, how it helps determine anomalous relaxation times, and how this physics is ultimately a consequence of locality. We also show that there is a crucial sensitivity to particle statistics (i.e. fermionic versus bosonic models), and that similar effects occur in other non-Hermitian quantum lattice models.

[1] A. McDonald, R. Hanai, and A. A. Clerk, Phys. Rev. B 105, 064302 (2022)