:Understanding Topology in CMP:

Introduction:
Topological phase transition is one of the pro founding discovery in first half of the twenty fast century.in Condensed matter physics.

Here is figure from prof Cane (Penn State Unive.) presentation.

The  band of normal insulator and the landau level of  2D electron gas under high magnetic field is similar. But what makes them difference is the "Topology" of the band structure. It was first pointed out by David Thouless.