Scale separation, topology, and time-reversal invariance

I will explain how Born-Oppenheimer-like energy surfaces in systems with slow and fast degrees of freedom can acquire non-trivial topology, and how this is affected by time-reversal invariance. I will then analyze time-reversal invariant two-dimensional energy surfaces and show that, in contrast with topological insulators, these are classified by an integer index, and may have a non-zero Chern number.