The effect of anisotropy of dipolar hopping on localization in three-dimensional lattices

It has become widely accepted that particles with long-range hopping do not undergo Anderson localization. However, several recent studies demonstrated localization of particles with long-range hopping. In particular, it was recently shown that the effect of long-range hopping in 1D lattices can be mitigated by cooperative shielding, which makes the system behave effectively as one with short-range hopping. Here, we show that cooperative shielding, demonstrated previously for 1D lattices, extends to 3D lattices with \emph{isotropic} long-range $r^{-3}$ hopping, but not to 3D lattices with dipolar \emph{anisotropic} hopping. Since cooperative shielding enables localization, our results suggest (though do not prove) that localization in 3D lattices is possible for particles with isotropic long-range hopping, but not anisotropic long-range hopping. We show that the anisotropy of the dipolar long-range hopping qualitatively changes the energy level statistics, the scaling with the lattice size and the diffusion dynamics of wave packets in disordered 3D lattices.