# Help with understanding topologically-protected edge states in domain wall systems

Let's say that I have a simple domain wall system for the following Hamiltonian with added on-site potential $$M(x)$$:

$$H(k,M)=-t \sum_{\delta} [\cos(k\cdot\delta)\sigma_x-\sin(k\cdot\delta)\sigma_y+M(x)\sigma_z],$$

For the on-site potential $$M(x)=0$$, the above is the Hamiltonian for graphene. Otherwise, $$M(x)$$ gaps the gapless graphene system.

In a domain-wall like setup comprising multiple unit cells, $$M(x)$$ can be made to vary between, say $$-0.5\leq M(x)\leq 0.5$$. In this question, I will focus on the energy spectra around only one Dirac point, as shown in the figure at the end (y-axis is energy). So, I am deliberately avoiding considering the bulk / entire Brillouin zone. In terms of Berry-phase related phenomena, we know that the sign of $$M(x)$$ (i.e. $$\pm$$) directly affects the Berry phase (and topology).

I broke my overall question down into three consecutive parts:

1. Do the $$M(x)<0$$ and $$M(x)>0$$ bands in this setup correspond to different 'phases'? I think of this solely due to the $$M(x)<0$$ and $$M(x)>0$$ bands having different Berry phases being separated by a graphene phase. But I am not sure, as I am not comparing the Chern number everywhere.
2. Does the gap closing for $$M(x)=0$$ mean that there is an edge state connecting $$M(x)<0$$ and $$M(x)>0$$ bands? I guess the confusion is distinguishing an edge state from a gapless point in the dispersion. Again, I am not too sure because I am not looking at spectral flow analytically (i.e. number of edge states = difference in Chern numbers on either side of domain).
3. If this is an edge state, is it necessarily topologically protected? Besides the Chern number again, I am not sure why this would be topologically protected.

I apologize there are sub-questions here, but I thought this was better than asking three separate questions because they all are about my confusion on topologically protected edge states.