Open-system dynamics in local Lindbladians with chaotic spectra

We investigate the physical consequences of having a spectrum that satisfies random matrix theory (RMT) for generic Lindbladians, and compare its implications for spatially local and completely random Lindblad dynamics in one spatial dimension. We find that Lindbladians whose spectrum is described by RMT exhibit quasiuniversal early-time dynamics for quantities nonlinear in the density matrix, in the sense that for generic, highly entangled initial states, the early time evolution is independent of the choice of initial state. We numerically investigate how locality generically imposes constraints on the size-dependence of Lindblad eigenoperators. This size dependence implies that linear observables, such as expectation values of local operators, are highly sensitive to eigenmodes outside the bulk of the spectrum in the thermodynamic limit, and plays a central role in limiting operator growth in the presence of dissipation. We find that when single-site dissipation dominates, an operator's decoherence scales approximately linearly with its Pauli weight, even in the presence of two-site jump operators. When two-site only dissipation dominates, however, this generic trend in operator size can be violated for numerically accessible system sizes, leading to long-lived high Pauli-weight operators.