  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
divergence of density of states, the band flatness of the edge Majorana modes
is unstable under spontaneously generated spatial variations of Cooper pairing
phases. Staggered current loops appear near the boundary and thus time-reversal
symmetry is spontaneously broken in the orbital channel. This effect can appear
in both condensed matter and ultra-cold atom systems.
