Thermalization and hydrodynamic long-time tails in a Floquet system

We systematically investigate whether classical hydrodynamic field theories can predict the long-time dynamics of many-particle quantum systems. We study both numerically and analytically the time evolution of a chain of spins (or qubits) subjected to stroboscopic dynamics. The time evolution is implemented by a sequence of local and nearest-neighbor gates that conserve the total magnetization. The long-time dynamics of such a system is believed to be describable by a hydrodynamic field theory, which, importantly, includes the effect of noise. Based on a field theoretical analysis and symmetry arguments, we map each operator in the spin model to the corresponding fields in hydrodynamics. This allows us to predict which expectation values decay exponentially and which decay with a hydrodynamic long-time tail. We illustrate these findings by studying the time evolution of all 255 Hermitian operators that can be defined on four neighboring sites. All operators not protected by hydrodynamics decay exponentially, while the others show a slow hydrodynamic decay. While most hydrodynamic power laws seem to follow the analytical predictions, we also discuss cases where there is an apparent discrepancy between analytics and the finite-size numerical data.