  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.
