In the Haldane model we break both the inversion symmetry and time reversal symmetry, as a consequence I didn't not have any expectations when it comes to symmetry of the energy bands. However, to my big surprise, the energy bands for the armchair edge is symmetric under the $k$ -> $-k$ exchange. How can we explain this symmetry and why are the symmetry of the two cases different?

[This image shows the energy bands for the armchair edge.[1]

As expected this plot for the zigzag edge does not posess this symmetry


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