  Two-dimensional atomic arrays exhibit a number of intriguing quantum optical
phenomena, including subradiance, nearly perfect reflection of radiation and
long-lived topological edge states. Studies of emission and scattering of
photons in such lattices require complete treatment of the radiation pattern
from individual atoms, including long-range interactions. We describe a
systematic approach to perform the calculations of collective energy shifts and
decay rates in the presence of such long-range interactions for arbitrary
two-dimensional atomic lattices. As applications of our method, we investigate
the topological properties of atomic lattices both in free-space and near
plasmonic surfaces.
