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 $$M(x)$$:

$$\tag{1} H(k,M)=-t \sum_{\delta} [\cos(k\cdot\delta)\sigma_x-\sin(k\cdot\delta)\sigma_y+M(x)\sigma_z],$$

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.

• +1. But evidently this question has turned out to be a bit difficult for this community, and hasn't received any answer or even comments in over 6 months. I'd recommend to ask just one of the three questions, to make it simpler and easier to answer. If you get an answer, you can ask the other two questions in a follow-up comment or in a separate answer. I'm not trying to say it always has to be that way, but in this case, for various reasons I think that would be the best course of action. Nov 23 '21 at 21:29
• @NikeDattani, thank you for bringing this up. I posted an attempt at the answer after a (hopefully) improved understanding I developed over the months since. Nov 23 '21 at 22:32
• Excellent, thanks for doing that! I'd recommend removing the green checkmark, so that others are still encouraged to answer. Nov 23 '21 at 22:56

1. The idea of phases in this context is related to the Chern number $$c$$, in the context of topological/Chern insulators. The answer to this question would require computing the Chern number and associating with it some physical meaning (ex: $$c=0$$ trivial, $$c=\pm 1$$ non-trivial).
2. The confusion here is that I am considering two different contexts. A regular gap closing may be thought of in the context of a band diagram in a Brillouin zone, whereas edge states are associated with the interface between two different kinds of materials (or phases, using the language in 1. above). In this case, we may consider a domain wall, associated with gap closings (such as when $$M(x)=0$$ at the point where $$M(x)<0$$ transitions to $$M(x)>0$$, in this model). If the Chern number of each region in 1. above is different in these two regions, we may see topologically protected edge states.
My main confusion was that, in my simulations, I plotted the band diagrams using $$2\times 2$$ matrices for varying values of $$M(x)$$: I calculated the eigenvalues for each iteration of $$M(x)$$ and plotted them all together at the end. Instead, for this domain wall problem, I should have chosen some real-space geometry, and used the tight-binding formalism to construct a huge Hamiltonian which then yields the band structure once diagonalized.