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We study the zero-temperature phase diagram of a two dimensional squarelattice loaded by spinless fermions, with nearest neighbor hopping andalgebraically decaying pairing. We find that for sufficiently long-rangepairing, new phases, not continuously connected with any short-range phase,occur, signaled by the violation of the area law for the Von Neumann entropy,by semi-integer Chern numbers, and by edge modes with nonzero mass. The latterfeature results in the absence of single-fermion edge conductivity, presentinstead in the short- range limit. The definition of a topology in the bulk andthe presence of a bulk-boundary correspondence is still suggested for thelong-range phases. Recent experimental proposals and advances open thestimulating possibility to probe the described long-range effects innext-future realistic set-ups.

Edge insulating topological phases in a two-dimensional long-range superconductor