  We consider free fermions living on lattices in arbitrary dimensions, where
hopping amplitudes follow a power-law decay with respect to the distance. We
focus on the regime where this power is larger than the spatial dimension
(i.e., where the single particle energies are guaranteed to be bounded) for
which we provide a comprehensive series of fundamental constraints on their
equilibrium and nonequilibrium properties. First we derive a Lieb-Robinson
bound which is optimal in the spatial tail. This bound then implies a
clustering property with essentially the same power law for the Green's
function, whenever its variable lies outside the energy spectrum. The widely
believed (but yet unproven in this regime) clustering property for the
ground-state correlation function follows as a corollary among other
implications. Finally, we discuss the impact of these results on topological
phases in long-range free-fermion systems: they justify the equivalence between
Hamiltonian and state-based definitions and the extension of the short-range
phase classification to systems with decay power larger than the spatial
dimension. Additionally, we argue that all the short-range topological phases
are unified whenever this power is allowed to be smaller.
