Localization of quantum particles with long-range hopping in finite-sized lattices

Non-interacting particles with long-range hopping are known to be delocalized in disordered systems of infinite size. It is thus natural to assume that such particles can traverse any finite-sized lattice. We show that this is not true. Particles with long-range hopping can localize in lattices of \emph{finite} size, even macroscopically finite. This leads to a rather unusual phenomenon: quantum particles can transverse a disordered lattice of size $10A$, but not a lattice of size $A$. As evidence for this, we demonstrate spatial localization in dynamical calculations at long-times, inverse participation ratio distributions characteristic of localized systems, and log-normal fluctuations of the wavefunction. We map out the phase diagram for a particle with long-range hopping in a 3D lattice as a function of on-site disorder strength and filling fraction. Using scaling arguments, we determine the size-dependence of the localization-diffusion crossover line as a function of the system size, which predicts localization in macroscopically finite systems.