  We study the phase diagram and edge states of a two-dimensional p-wave
superconductor with long-range hopping and pairing amplitudes. New topological
phases and quasiparticles different from the usual short-range model are
obtained. When both hopping and pairing terms decay with the same exponent, one
of the topological chiral phases with propagating Majorana edge states gets
significantly enhanced by long-range couplings. On the other hand, when the
long-range pairing amplitude decays more slowly than the hopping, we discover
new topological phases where propagating Majorana fermions at each edge pair
nonlocally and become gapped even in the thermodynamic limit. Remarkably, these
nonlocal edge states are still robust, remain separated from the bulk, and are
localized at both edges at the same time. The inclusion of long-range effects
is potentially applicable to recent experiments with magnetic impurities and
islands in 2D superconductors.
