By plotting the spectrum as a function of v for a fixed value of w , the spectrum is divided into two insulating regions divided by the metallic intersection at w = v . The spectrum would be gapped in both insulating regions, but one of the regions would show zero energy eigenstates and the other region would not, corresponding to the dimerized cases. The existence of edge states in one region and not in the other demonstrate the difference between insulating phases and it is this sharp transition at w = v that correspond to a topological phase transition.