Superdiffusion in two dimensional disordered system with ubiquitous long-range hopping

Although it is recognized that Anderson localization always takes place for a dimension d less or equal d = 2, while it is not possible 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) / r−2 is not resolved yet. Following earlier suggestions we show that for the hopping determined by two-dimensional anisotropic dipole-dipole or RKKY interactions 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 crossover between phases is resolved analytically using the extension of scaling theory of localization and verified using an exact numerical diagonalization.

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