Spontaneous breaking of time-reversal symmetry in the orbital channel for the boundary Majorana flat bands

We study the boundary Majorana modes for the single component p-wave weak topological superconductors or superfluids, which form zero energy flat bands protected by time-reversal symmetry in the orbital channel. However, due to the divergency of density of states, the band flatness of the edge Majorana modes is unstable under spontaneously generated spatial variations of Copper pairing phases. Staggered current loops appear near the boundary and thus time-reversal symmetry is spontaneously broke in the orbital channel. This effect can appear in both condensed matter and ultra-cold atom systems.