Stimulated by my some recent calculations on edge states(ES) for time-reversal invariant(TRI) topological superconductors(TS) as well as many questions concerning the "edge states" in Physics StackExchange(e.g. Is edge state of topological insulator really robust？, Chiral edge state as topological properity of bulk state, What is the mathematical reason for topological edge states?), I get some questions about the gapless ES for TS and I'm puzzled by these problems.
I studied two examples, say A and B, of TRI triplet p-wave supercondutors with full gap both on a 2D lattice, and my calculations of the $Z_2$ bulk invarints show that they are both odd numbers, by definition, both A and B are TRI TS. Then I numerically calculated the ES for A and B both on stripy geometries with open boundary conditions(OBC) along one direction, now here come the puzzles:
For A, no matter how I change the width of the strip, no matter how I change the shapes of the two edges, I just can not find the gapless ES "crossing" at the TRI momentum points( $k=0,\pi$ in my calculations), so my first question is: Is the existence of gapless ES "crossing" at the TRI momentum points really a necessary condition for TRI TS ? Now I personally tend to think it's just a sufficient not necessary condition for TRI TS. Moreover, the so called bulk-boundary correspondence can not be proved very generally(Counterexamples to the bulk-boundary correspondence (topological insulators)), so I think there may exist some special examples violating this.
Regarding B, I fortunately found one pair of gapless ES "crossing" at the TRI momentum $k=\pi$(or $-\pi$). But according to the numerical results, the dispersion near $k=\pi$ seems more like quadratic$(\propto k^2)$ rather than the usual linear one . So my second question is: Is there any possibilty of the quadratic dispersion of gapless ES for TRI TS ?
Thanks in advance.