Why do people always talk about continuous topological phase transition?

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!

• The first order transition is definitely possible. The reason is that it is uninteresting. Unlike second order transitions, first order transitions have no gapless excitations emerging at the transition point, and therefore no long-wave-length limit, no field theory description, no renormalization group analysis. The first order transition just happens by switching from one phase to another. Studying the transition itself can not tell us about the adjacent phases (whereas by studying the continuous transition, we can gain a lot of knowledge about the adjacent phases). – Everett You Dec 12 '16 at 6:20

As you mention, for interacting phases (like the fractional quantum Hall effect), there can be first order phase transitions between competing topological phases. For example, at the $\nu = \tfrac{2}{3}$ plateau of GaAs, the system can either form a spin-polarized $\nu = 1 - \tfrac{1}{3}$ Laughlin state, or the spin-unpolarized (SO(3) symmetric) $113$-state. In a clean system the strength of the Zeeman field tunes a first order transition between them, and a spin-polarization transition at $\nu = \tfrac{2}{3}$ is indeed seen in GaAs if one adds an in-plane component to the field. The reason they are less discussed is both experimental and theoretical. Experimentally, first order transitions can be rounded out by disorder - presumably that's what actually happens for the spin-polarization transition above - so they often appear continuous. Theoretically, there is no universality at a first order transition (no scaling exponents, etc.), so there really isn't much to say.