A (semantic?) question of the usage of "band gap closing" in the context of topological insulators

Question

Consider for simplicity a two-band system in one spatial dimension, if by tuning some parameters of the system, the band structure changes as follows(vertical axis as energy, horizontal axis as lattice momentum)

Then does it count as a "band gap closing"? Normally I would say yes, because the spectrum (range of permissible energies) changes from two disjoint intervals to one interval. 

Now what confuses me is that, in topological-insulator context I've seen, people often say "Chern numbers cannot change without closing some band gap", but in the above illustrated band-closing situation, Chern numbers seem to be well-defined all the time since the two band curves don't intersect, and so they must stay constant. So is it a slight abuse of language of "Chern number cannot change without closing the band gap" where they actually mean “Chern numbers cannot change without some band curves intersecting”, or is there something I misunderstood so that the example I showed does not count as bang gap closing?

Answer 1

Well, the sentence "Chern numbers cannot change without closing some band gap" is still true: It's a necessary condition for Chern numbers to change, not a sufficient condition. It's fine for Chern number to stay constant even when a band gap is closed.

The second picture you mention is still coming from a Hamiltionian without band gap. In this case, translation symmetry allows you to distinguish some eigenstates with equal energy, and you can still draw energy bands. But when you add disorder, the Brillouin Zone ceases to exist, and the topological invariant may change.

Comments

Thanks for the reply, so it is mostly semantics. I understood the necessity vs. sufficiency issue in the question, I guess I'm puzzled by the expression "Chern numbers cannot change without closing some band gap", because it can be easily made more precise by saying  “Chern numbers cannot change without some band curves intersecting”.

I don't quite get the point your second paragraph, where you mentioned disorder issue while specifically referring to the second picture. If there is disorder, shouldn't the Brillouin zone disappear also in the first picture?

---

When disorder is turned on adiabatically, the Brillouin zone would disappear in the first example as well, but it would not be able to change the topological class of the material (as long as the band gap remains open). The key point of the topological classification is that it is robust under adiabatic perturbations (disorder and others). However, in the second example, disorder would indeed be able to change the topological class.

From this perspective, the refinement “Chern numbers cannot change without some band curves intersecting” is correct, but unimportant, because it doesn't say much about the topological class of the material.

---

Interesting. What's the justification of the claim that disorder cannot change topological class of the first while it can for the second? Do you have any reference in mind, or is there some simple reasoning you can write down here?

In fact although it is clear that topological class still makes sense in presence of disorders, it is not clear to me how to assign an integer to the topology in this case, since Brilouin zone is destroyed(I heard it could be done by twisted boundary condition somehow?)

---

It's a tautology: by definition, two systems are in the same topological class if and only if they can be connected adiabatically without closing the band gap. In the first example, adiabatically adding disorder doesn't close the band gap, so they have to be in the same class. (Of course, this is only valid as long as the disorder is weak enough to not close the band gap. In the second example, the band gap has been close anyway, so who knows what happens (topologically).

From this perspective, it's kind of amazing that you can predict the topological class by calculating some integral over the Brillouin zone. In many cases, the response to twisted boundary conditions, i.e. to magnetic flux, is also helpful for calculating the class. However, in the general case, the only technique available is to adiabatically connect it to a system without disorder and calculate the integer of the latter.

---

Thanks! This clears a lot of doubts of mine.