Trivial and Non-trivial topology of band structure I don't understand the meaning of the expression "trivial topology" or "non-trivial topology" for an electronic band structure. Does anybody have a good explanation?
 A: One of the early triumphs of QM (through e.g. Kronig-Penney model) was the explanation of the insulating state of matter. Energy bands (and gaps) appear as the result of hybridization of many atomic orbitals, and for a specific filling you can end up with the top most pair of bands being either entirely filled (valence band) or entirely empty (conduction band). No (small) electric field can perturb them enough to cause motion, and thus you have an insulator. In this trivial insulator, although the bulk is insulating, there is possibility of for example dangling bonds introducing states that lie in an energy gap. These states are localized at the edge; however they are not robust, and as such not particularly useful.
Now if you have a material with a sufficiently strong spin-orbit interaction for example (not essential for the effect, but historically important approach), this can cause the energy bands above and below the gap to swap places. This twisting is protected by time-reversal invariance, and although still an insulator, the resulting phase is topologically different from an ordinary insulator. The twisting of the band structure is what the phrase non-trivial topology is referring to; an analogy would be the way a Mobius strip is a twisted version of an ordinary strip. This manifests itself in the fact that when you put the two in contact, the curled up band structure of the TI must unwind so that the band structure fits the one in the ordinary insulator. This unwinding will will have to close the gap near the edge, hence the topologically protected edge states. This is the interesting part of the topological insulators from the practical standpoint.
So whether the band structure is wound up or not is a topological property, and one can measure it with the topological index, also called a Chern number, defined as
$$
C=\frac{1}{2\pi}\sum_n\oint Fd\mathbf{k}
$$
where the sum is over occupied bands, the integral is over the entire Brillouin zone, and the integrated quantity is the Berry curvature (analogue of the magnetic field in $\mathbf{k}$ space) $F=-i\nabla_{\mathbf{k}}\times\langle u_{n\mathbf{k}}|\nabla_{\mathbf{k}}|u_{n\mathbf{k}}\rangle$, where $u_{n\mathbf{k}}$ are the Bloch eigenvectors. If $C=0$ you have a trivial insulator, and if $C\neq0$ you have a non-trivial or topological insulator.
