Effect of the anisotropy of long-range 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 one-dimensional (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 isotropic long-range r(-alpha) hopping, but not to 3D cubic lattices with anisotropic long-range hopping. The specific anisotropy we consider corresponds to the interaction between dipoles aligned along one of the principal axes of the lattice. We demonstrate the presence of localization in 3D lattices with uniform (alpha = 0) isotropic long-range hopping and the absence of localization with uniform anisotropic long-range hopping by using the scaling behavior of eigenstate participation ratios. We use the scaling behavior of participation ratios and energy-level statistics to showthat the existence of delocalized, extended nonergodic, or localized states in the presence of disorder depends on both the exponents a and the anisotropy of the long-range hopping amplitudes.