In the weakly-coupled BCS regime two-dimensional chiral (p+ip) spinless superconductors and superfluids posses a chiral gapless fermionic Majorana state localized near the boundary. This gapless edge mode is a direct manifestation of the topological quantum phase transition present in this system. Indeed the boundary is the line which separates two topologically different phases, it is localized in the region where the gap closes. My question is what happens to the edge states if we tune fine-tune the bulk value of the chemical potential to the critical one, i.e. approach quantum criticality in the bulk. The gapless fermions should penetrate into the bulk, right? What happens to their chiral and Majorana property?
At the critical point, the bulk gap is closed, and there is nothing to prevent the edge state from penetrating into the bulk. So the gapless mode simply merge into the bulk. The modes (of opposite chirality) from both edges will mix. Both the chirality and Majorana property will be "canceled out" by the mixing.