Fate of dissipative hierarchy of timescales in the presence of unitary dynamics

Nick D. Hartmann, Jimin L. Li, David J. Luitz

Published 2023-04-18Version 1

The generic behavior of purely dissipative open quantum many-body systems with local dissipation processes can be investigated using random matrix theory, revealing a hierarchy of decay timescales of observables organized by their complexity as shown in [Wang et al., $\href{https://link.aps.org/doi/10.1103/PhysRevLett.124.100604}{Phys. Rev. Lett. \textbf{124}, 100604 (2020)}]$. This hierarchy is reflected in distinct eigenvalue clusters of the Lindbladian. Here, we analyze how this spectrum evolves when unitary dynamics is present, both for the case of strongly and weakly dissipative dynamics. In the strongly dissipative case, the unitary dynamics can be treated perturbatively and it turns out that the locality of the Hamiltonian determines how susceptible the spectrum is to such a perturbation. For the physically most relevant case of (dissipative) two-body interactions, we find that the correction in the first order of the perturbation vanishes, leading to the relative robustness of the spectral features. For weak dissipation, the spectrum flows into clusters with well-separated eigenmodes, which we identify to be the local symmetries of the Hamiltonian.