Slow diffusion and Thouless localization criterion in modulated spin chains

In recent years the ergodicity of disordered spin chains has been investigated via extensive numerical studies of the level statistics or the transport properties. However, a clear relationship between these results has yet to be established. We present the relation between the diffusion constant and the energy-level structure, which leads to the Thouless localization criterion. Together with the exponential-like dependence of the diffusion constant on the strength of quasiperiodic or random fields, the Thouless criterion explains the nearly linear drift with the system size of the crossover/transition to the nonergodic regime. Moreover, we show that the Heisenberg spin chain in the presence of the quasiperiodic fields can be well approached via a sequence of simple periodic systems, where diffusion remains finite even at large fields