The eigenenergies are symmetrical under swap of v ↔ ↔ w , and the dispersion relation is mostly gapped (insulator) except when v = w (metal). By analyzing the energies, the problem is apparently symmetric about v = w , the v > w has the same dispersion as v < w . Nevertheless, not all properties of the system are symmetrical, for example the eigenvectors are very different under swap of v ↔ ↔ w . It can be shown for example that the Berry connection