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.
The low-frequency response of systems near a many-body localization transition can be dominated by rare regions that are locally critical or ``in the other phase.'' It is known that in one dimension, these rare regions can cause the dc conductivity and diffusion constant to vanish even inside the delocalized thermal phase. Here, we present a general analysis of such Griffiths effects in the thermal phase near the many-body localization transition: we consider both one-dimensional and higher-dimensional systems, subject to quenched randomness, and discuss both linear response (including the frequency- and wave-vector-dependent conductivity) and more general dynamics. In all the regimes we consider, we identify observables that are dominated by rare-region effects. In some cases (one-dimensional systems and Floquet systems with no extensive conserved quantities), essentially all long-time local observables are dominated by rare-region effects; in others, generic observables are instead dominated by hydrodynamic long-time tails throughout the thermal phase, and one must look at specific probes, such as spin echo, to see Griffiths behavior.
The interplay between symmetry and thermalization governs the late-time dynamics of local quantum and classical many-body systems at nonzero temperature. Recently, two parallel frontiers have emerged: the search for robust anomalous hydrodynamics -- such as superdiffusion -- in generic, non-integrable models, and the formal effort to generalize the fundamental concept of global symmetry. In this paper, we bridge these frontiers by demonstrating that quantum group symmetry provides a novel mechanism for anomalous hydrodynamics in chaotic systems. We study the dynamics of local operators carrying
U(1) charge in non-integrable lattice models that also have quantum group symmetry. One example is transverse spin in the XXZ model with integrability breaking deformations. While such excitations are expected to decay very quickly at high temperature because their charge forbids overlap with conventional hydrodynamic densities, we find that protection by the quantum group symmetry makes these modes long-lived, despite the absence of local quantum group charge density or current. Furthermore, the dynamics is superdiffusive across Hamiltonian, Floquet, and classical realizations, and exhibits unusual finite size effects at very late times. We also revisit transverse spin dynamics in the integrable XXZ model.