Superdiffusion in random two dimensional system with time-reversal symmetry and long-range hopping

Although it is recognized that Anderson localization takes place for all states at a dimension $d$ less or equal $2$, while delocalization is expected for hopping $V(r)$ decreasing with the distance slower or as $r^{-d}$, the localization problem in the crossover regime for the dimension $d=2$ and hopping $V(r) \propto r^{-2}$ is not resolved yet. Following earlier suggestions we show that for the hopping determined by two-dimensional anisotropic dipole-dipole interactions in the presence of time-reversal symmetry there exist two distinguishable phases at weak and strong disorder. The first phase is characterized by ergodic dynamics and superdiffusive transport, while the second phase is characterized by diffusive transport and delocalized eigenstates with fractal dimension less than $2$. The transition between phases is resolved analytically using the extension of scaling theory of localization and verified numerically using an exact numerical diagonalization.