Majorana flat band edge modes of topological gapless phase in 2D Kitaev square lattice

We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes. The degeneracy points are characterized by two vortices in momentum space, with opposite winding numbers, which are not removable unless meet together. We show rigorously that the topological gapless phase always hosts a partial Majorana flat band edge modes in a ribbon geometry, although such a single band model has zero Chern number as a topologically trivial superconductor. The flat band disappears when the gapless phase becomes topologically trivial, associating with the mergence of two vortices. Numerical simulation indicates that the flat band is robust against the disorder. This finding indicates that the bulk-edge correspondence can be extended to superconductors in the topologically trivial regime as recently proposed in Ref. [PRL 118, 147003 (2017)].