I read in a review that there are two Dirac points in graphene, where the conduction band and valence band touch each other. Near these points electrons obey a linear dispersion relation. Breaking of time symmetry leads to a quantum hall state. The solution of the Schrödinger equation gives rise to a single band connecting the valence and conduction band, which represents the edge states, and the slope of that single band gives the chirality of edge states. Why do electronic states exist between valence and conduction band? Also can someone explain to me why Dirac points occur in graphene?
Topologically, a non-trivial state also occur when degeneracy is broken away from the Dirac points because of the spin orbit interaction. In that situation, when the band connecting the degenerate points cross the Fermi level an odd number of times, that leads to a topologically protected state and the even crossings don't. I didn't understand why.