As a condition for a topological phase transition it seems that there must be an energy gap that closes and reopens. I have seen this many places, but never an intuitive, easy explanation. Can someone give that?
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There's a lot I don't know about the topic but I guess the simple intuitive explanation must be as follows.
Topological peculiarities arise only with shapes that are not simply-connected. For instance, a torus as opposed to a sphere. There have to be two points, with two ways of connecting them, and the two paths not continuously transformable one to the other. The clearest illustration of this in QM is the Aharanov-Bohm solenoid effect. So in terms of the energy landscape there must be a "hole" or gap for topological properties to come into play. I don't know why such gap must "close and reopen" - not even sure that's true. But one way or another I think you'll find it depends on the space being not simply-connected, because of that gap.
In terms of band theory, a topologically trivial state is connected to the vacuum adiabatically.This means if you deform it without closing the gap it is still trivial. So if you want to go from a trivial to nontrivial phase vice versa, you must go through a phase transition where the gap closes. For this reason, one you cross a Landau level, the hall conductivity jumps by one unit. This is found by integrating the Berry curvature over the filled bands.
In the Haldane model, this phase transition happens when you tune the Dirac mass term proportional to sigma z (one for each valley in graphene). Once you get to the critical value, the gap closes and changes the relative signs of the Dirac mass terms.You'll have a chiral edge state and a nonzero hall conductance in the nontrivial regime, which turns out to be related to the sign of the mass term.