Universal Aspects of Operator Thermalization in Hamiltonian Dynamics*

The long-time behavior of observables in thermalizing quantum systems can often be captured through hydrodynamics, involving just a few local quantities. A key question in quantum dynamics is to derive such behavior from the microscopic behavior and predict quantities such as diffusion coefficients or conductivities. We describe a conjectural form for super-operator Green's functions in thermalizing systems at infinite temperature in any dimension. The conjecture is supported by numerical calculations for a broad range of thermalizing many-body models and exact analytic results for large-q SYK models. The universal asymptotic form of the Green's function suggests an efficient numerical technique for extracting diffusion coefficients of operators in strongly interacting systems. Additionally, the conjecture implies some 'universal' behavior of operators under Hamiltonian dynamics.