  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 isotropic
long-range $r^{-\alpha}$ hopping, but not to 3D lattices with dipolar-like
anisotropic long-range hopping. 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 behaviour of eigenstate participation ratios. We use the scaling
behaviour of participation ratios and energy level statistics to show that the
existence of delocalized, non-ergodic extended, or localized states in the
presence of disorder depends on both the exponents $\alpha$ and the isotropy of
the long-range hopping amplitudes.

Quantum Physics;Quantum Gases