Dynamics from Dispersion: A simple predictive tool

We identify emergent hydrodynamics governing charge transport in open quantum systems described by Brownian random time evolution with various symmetries, constraints, and ranges of interactions. This is accomplished via a mapping between the averaged dynamics and the low-energy spectrum of a Lindblad operator, which acts as an effective Hamiltonian in a doubled Hilbert space. By explicitly constructing dispersive excited states of this effective Hamiltonian using a single mode approximation, we provide a comprehensive understanding of diffusive, subdiffusive, and superdiffusive relaxation in many-body systems with conserved multipole moments and variable interaction ranges. Our approach further allows us to identify exotic Krylov-space-resolved diffusive relaxation despite the presence of dipole conservation, which we verify numerically. Therefore, we provide a simple, general, and versatile framework to qualitatively understand the dynamics of conserved operators under random unitary time evolution, and by extension, thermalizing quantum systems.