# Gapless modes at the boundary between topological insulator and normal insulator

I am currently learning about topology in condensed matter physics. I think I understand most of how topological indeces come about and differences between Z and Z2 indeces and the symmetries that play a role. Though, I still have one (as it seems) very basic question. In a lot of articles and books I read I repeatedly came across statements like:

"At the interface between such a standard insulator and a topological insulator, it is not possible for the « band structure » to interpolate continuously between a topological insulator and the vacuum without closing the gap. This forces the gap to close at this interface leading to metallic states of topological origin." (M. Fruchart, D. Carpentier, arXiv: 1310.0255)

My question is just: Why?

Why does the gap need to close or the other way around: Why can I have a topological phase transition if the gap closes at the interface? And furthermore, looking e.g. at a 3D Z2 TI: Why are the emerging states at such an interface linearly dispersed and spin helical?

• Is this question about interfaces or bulk transitions? – Ryan Thorngren Jan 16 at 14:15
• hm..kind of both, isn't it? The topological invariant is a bulk property. As soon as I bring two bulk materials together I have to have a transition from one bulk invariant to the other. Though, this happens at the interface and only there the gap therefore needs to close. – Thomas Jan 16 at 14:24
• @RyanThorngren Well, that's the thing: At least the folk idea is that this is the same. – Norbert Schuch Jan 16 at 14:24
• @Norbert Schuch It's completely possible that I "only" have a folk understanding of the matter here. – Thomas Jan 16 at 14:26
• They're not quite the same. There are interfaces that can be gapped (but carry non-trivial topological degrees of freedom), but at a phase transition the gap always has to close (but can be first order or continuous). – Ryan Thorngren Jan 16 at 14:34