Many-Body Delocalization in Strongly Disordered System with Long-Range Interactions: Finite Size Scaling

The localization in a disordered system of $N$ interacting spins coupled by the long-range anisotropic interaction $1/R^{\alpha}$ is investigated using a finite size scaling in a $d=1$ -dimensional system for $N=8, 10, 12, 14$. The results supports the absence of localization in the infinite system at $\alpha<2d$ and a scaling of a critical energy disordering $W_{c} \propto N^{2d-\alpha}$ in agreement with the analytical theory suggesting the energy delocalization in the subset of interacting resonant pairs of spins as a precursor of the many-body delocalization.The spin relaxation rate $k$ dependence on disordering $k \propto W^{-2}$ has been revealed in the practically interesting case $\alpha=d$. This relaxation mechanism can be responsible for the anomalous relaxation of quantum two level systems in amorphous solids.