Cooperative shielding in three-dimensional lattices

Cooperative shielding is the phenomenon that can make quantum systems with long-range interactions behave effectively as those with short-ranged interactions. Cooperative shielding has been previously demonstrated for both single-particle and many-body systems in \emph{one}-dimensional (1D) lattices. We demonstrate that cooperative shielding extends to single-particle systems with \emph{isotropic} long-range hopping in \emph{three}-dimensional (3D) lattices. We analytically diagonalize a Hamiltonian containing isotropic long-range hopping terms of the form $r^{-\alpha}$ for a 3D lattice, under periodic boundary conditions, and where $\alpha$ is an arbitrary, real constant. We find that the obtained energy level structure is analogous to that observed in 1D. We also find that, for the 3D system of sidelength $N$, the shielding gap responsible for cooperative shielding diverges as $\Delta \propto N^3$, in contrast to the 1D case where $\Delta \propto N$. We further demonstrate, via numerical diagonalization, that cooperative shielding also extends to 3D systems with open boundary conditions.