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?