Let’s think, what could be the simplest topological system protected by reflection symmetry.

We need \(d=2\), since the only possible reflection symmetry is broken by the boundary in \(d=1\), and we can once again consider coupled Majorana nanowires.

We can put two nanowires in a unit cell of the lattice and make their parameters different. In this way, the weak topological invariant is trivial (there is an even number of Majoranas per unit cell).

On the other hand, if the hopping between the nanowires is reflection invariant, there will be a reflection symmetry axis passing through each nanowire, like this:

If we do everything right (this does require some trial and error in searching for the hopping that actually can couple the two Majoranas from the edge), we get a painfully familiar dispersion:

In a similar way, we can also construct a tight-binding model with a mirror Chern number. The only difference with the Majorana wires that we need to worry about is that Chern number is a \(\mathbb{Z}\) invariant instead of \(\mathbb{Z}_2\).

This means that the Chern number of the alternating layers has to have opposite signs, or otherwise the surface would just have surface states going in a single direction.

Once again, coupling the layers we get a familiar Dirac cone on the surface:

Again, the dispersion of the edge states looks exactly like what we saw already because the edge state dispersion in any topological insulator is just given by the Dirac equation.