Computing thermalization times and hydrodynamic modes from microscopic quantum dynamics

Computing quantum dynamics of an interacting system presents a fundamental challenge. The Heisenberg time evolution of a local operator generates superpositions with an exponentially growing number of terms, each representing increasingly non local operators. To address this problem, we construct the graph of basis operators (Pauli string in spin-half systems) spanned by applying the Liouvillian n times; n serves as a truncation cutoff. Integrating out the outward flow of operators across the truncation boundary toward "large operators" gives rise to an imaginary self energy term that acts as an absorbing boundary condition. The resulting non-unitary dynamics respects all conservation laws related to local operators, e.g. energy, charge and spin. The approximation scheme depends on a single free rate parameter, which we determine numerically by requiring the results to become cutoff independent asymptotically. For a broad class of ergodic models this scheme allows to identify slowly decaying operators, and compute the diffusion constant of hydrodynamic modes. Furthermore this scheme captures the non diffusive dynamics in certain integrable models.