# What are some types of topologically-relevant band degeneracies in contemporary 2D condensed matter research?

In the study of 2D condensed matter systems, I have seen several kinds of band degeneracies. I call 'bands' the eigenvalues to the time-independent Schrodinger equation, solved over 2D k-space. By 'degenerate', I refer to when the eigenvalues to one eigenvector are the same as the eigenvalues of a different eigenvector at the same k-space point.

I have seen bands that are:

1. degenerate at certain high symmetry points (such as graphene);
2. 'fully gapped' (with no degeneracies whatsoever);
3. completely degenerate (such as the 4-band Kane-Mele model, where each pair of bands is degenerate everywhere over k-space);
4. degenerate at non-high-symmetry points (degenerate in such ways that make them 'accidental degeneracies'?)
5. degenerate at certain high symmetry areas (similar to 4., but also includes lines and patches of k-space)

### My question is:

Which of the above are relevant to topological effects (or coincide with those)? I have seen 1-3 above pretty often, but am having trouble understanding 4 (and 5). You only need to list one to post an answer, see . If there's another type of degeneracy relevant to topological effects that I didn't list above, you can explain that one too.

Consider the attached picture, which is a random band diagram for $$MoS_2$$ that I found online. To my understanding, people do not care about degenercies of the kind I have circled in red (i.e. of type 4 in the list above). This is because they tend to care about the physics around the 'valleys' indicated by red arrows (i.e. fully gapped, no degeneracies). They usually do this by working with an effective Hamiltonian focusing on a subspace of the complete Hilbert space. Then, for topological reasons, they might treat the entire 'bundle' of wavefunctions above and below the Fermi level (indicated by red arrows), regardless of the degeneracies circled in red.

However, it seems suspicious to me that the 'degeneracies' circled are assumed to not play a role in topological properties. To paint a naive/vague picture, consider Haldane's interpretation of topologically significant degeneracies of a 2-level model as Dirac strings/wormholes that connect two otherwise-separate genus-1 tori into a genus-3 torus (second picture, taken from these slides). So, I am looking for consolation/clarification on why we can simply ignore degeneracies that occur outside the regions of interest, in general topological concerns. Or if anyone sees my overall confusion, I would appreciate any guiding remarks.

Note: I am more concerned about the type of 2D topology that deals with Berry-like quantities (Berry phase, curvature, Chern / valley Chern, etc).

• +1 But you seem to be asking 2 questions in 1 post which is usually discouraged: For the second question, are you only asking for examples that are relevant to topological effects? Then this could be acceptable as in the case where I wrote "any others you may wish to add" at the bottom of this: mattermodeling.stackexchange.com/q/1594/5. But if you're asking for other types of degeneracies in general, maybe that could be left for a separate one-topic-per-answer question, in which each answer is a new type of degeneracy? May 2 at 4:41
• Cross posting (asking the same question on multiple sites) is discouraged because it makes it harder to see whether a question has already been answered on another site. You should either delete the question on one of the sites or include a link at the top of each post to the version of the question on the other site.
– Tyberius
May 2 at 17:21
• Thanks for adding the link. When a question fits on both sites, I think it can be a good idea to have multiple posts, but it's just good practice to keep everyone aware of multiple posts to avoid duplicating effort.
– Tyberius
May 2 at 18:06
• @TribalChief Great, now it's just one question :) It can stay the way it is :) May 3 at 14:39
• +1 A few ideas: graphene also has doubly-degenerate bands everywhere (and the Dirac point is 4-fold degenerate). Any system with both inversion and time-reversal symmetry will have all bands doubly degenerate. As to the question about band crossings away from the Fermi level, they are usually physically irrelevant because you cannot access them -- you are typically limited to electrons within a few 100 meV of the Fermi level. Jun 30 at 20:56