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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.

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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.

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