I have a question which puzzles me for a long time. Usually when people talk about topological phase transition, they usually have a gap-closing picture in their mind. Namely, the phase transition point they mention is actually a quantum critical point, where the (degenerate) ground states touch the continuous spectrum of excited states, leading to gapless excitation and infinite correlation length.
Why don't they discuss first-order topological phase transition? I cannot find any reason why this is forbidden. Is it uninteresting to people? Or is it simply because most of the examples we have are continuous phase transitions?
The only reason I have is that in a free-electron system we have discuss band structure. Imaging that the bands from the lowest band to the nth lowest band are all occupied. The situation where the gap closes is when the n+1 band osculates the nth band. In that case, we have gapless excitation, e.g. emergent massless Dirac fermions. However, the reason I came up with only applies to tight-binding models where we can talk about band structures.
Is there any better reason for that?
In addition, as far as I know, people usually use renormalization group flow to analyze continuous phase transition. I am wondering if we can have a renormalization group flow formulation for first-order topological phase transition. Thanks!