When pushes two bands past each other, so the one that should sit on top ends up underneath. The swap cannot be undone without closing the gap, and that is what makes a material rather than ordinary.
Going deeper
Band inversion is the exchange of two bands of opposite character, usually driven by strong spin–orbit coupling. The gap closes and reopens with the order reversed – and because the two orderings cannot be connected without closing the gap, the boundary between them must carry gapless states.
What gets inverted
In an ordinary the comes from one kind of orbital and the valence band from another, usually with opposite parity – s-like above p-like, say. Strong spin–orbit coupling, or the of a thin film, can push these bands past each other. At the crossover the gap closes; increase the coupling further and the gap reopens, but now with the orbital characters exchanged.
The inverted state is not merely a rearranged version of the ordinary one. The topological index computed from the bulk wavefunctions has changed, and it cannot change back without the gap closing again somewhere. Vacuum counts as an ordinary , so any boundary with vacuum is such a place.
The two canonical cases
HgTe quantum wells provided the first: the band ordering of HgTe is already inverted relative to CdTe, and a well thicker than a critical value inherits that inversion. The prediction in 2006 that such wells would be was confirmed in transport the following year, with a conductance close to the quantised value.
The Bi2Se3 family provided the second. Calculations in 2009 showed that spin–orbit coupling inverts the bands at the zone centre, leaving a single on each surface and a bulk gap of about 0.3 eV – large enough that are accessible well above cryogenic temperatures. Both cases follow the same logic: identify an inversion in the bulk, then look for boundary states.
How it is recognised
Calculations diagnose inversion by tracking the orbital character of the bands, or by computing parity eigenvalues at the symmetric points of the Brillouin zone – for a crystal with , their product gives the Z2 invariant directly. Experimentally, can follow the band ordering as composition, thickness or pressure is tuned through the transition, and the signature is a gap that closes and reopens.
Two cautions. Inversion is necessary but not sufficient: the invariant depends on the whole Brillouin zone, and an even number of inversions returns the material to being trivial. And a calculated inversion is only as reliable as the gap: semi-local functionals systematically underestimate gaps, which can invent inversions that better calculations remove.
For specialists
An exchange in the ordering of bands of opposite parity, usually driven by strong spin–orbit coupling, that changes the topological index while the bulk gap stays open elsewhere. Because the index then differs from that of vacuum, the boundary has to carry gapless states: inversion at Γ in HgTe quantum wells and in the Bi2Se3 family is the mechanism behind the quantum spin Hall and three-dimensional topological insulator phases.