Spectral gaps of local quantum channels in the weak-dissipation limit

We consider the dynamics of generic chaotic quantum many-body systems with no conservation laws, subject to weak bulk dissipation. It was recently observed [T. Mori, Phys. Rev. B 109, 064311 (2024)2469-995010.1103/PhysRevB.109.064311] that the generator of these dissipative dynamics, a quantum channel E, retains a nonzero gap as the dissipation strength γ→0 if the thermodynamic limit is taken first. We use a hydrodynamic description of operator spreading in the presence of dissipation to estimate the gap of E as γ→0, to calculate the operator-size distribution of the slowest eigenmodes of E, and to relate the gap to the long-time decay rates of autocorrelation functions under unitary Floquet dynamics. We provide a microscopic derivation of this hydrodynamic perspective for random unitary circuits. We argue that the gap in the γ→0 limit can change nonanalytically as one tunes the parameters of the unitary dynamics.