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In discussions of topological materials, the non-crossing theorem by Wigner and von Neumann is often mention. From the linked PDF:

[It] states that electronic bands (i.e., Bloch states) with the same symmetry cannot be degenerate at a generic point in the Brillouin zone (BZ), which prevents the formation of band crossings. ... However, the non-crossing theorem does not apply to bands with non-trivial wavefunction topology, which can form topologically protected band degeneracies.

Could you explain why a gap appears for bands with the same symmetry, and why this doesn't apply to bands with non-trivial wavefunction topology? I would also like to know how to determine if bands have the same symmetry or not.

I've attached a relevant figure from Burns' book (Introduction to Group Theory With Applications, Fig. 7-5), which might help clarify the concept.

A conceptual diagram of eigenvalues versus perturbation potential V.

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  • $\begingroup$ If it is allowed, I can attached the figure from the book I mentioned. $\endgroup$ Commented Aug 9, 2024 at 3:10
  • $\begingroup$ Yes, you should attach the image. But it is difficult to understand what it is you are trying to ask. The theorem says that you must start with "bands with the same symmetry" for the result to hold; if you have different symmetry, it can cross. Also, the linked pdf says that you may have symmetry-enforced band crossings too. $\endgroup$ Commented Aug 9, 2024 at 3:55
  • $\begingroup$ @naturallyInconsistent Thank you very much. I've attached the figure and rephrased my question. $\endgroup$ Commented Aug 9, 2024 at 11:18

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I would also like to know how to determine if bands have the same symmetry or not.

In the Figure 7.5 from Burns, the two bands that do not cross have symmetry (E) labelled on them. The straight line Pz has symmetry (A) labelled on it.

Could you explain why a gap appears for bands with the same symmetry, and why this doesn't apply to bands with non-trivial wavefunction topology?

Did they not have a proof of the theorem?

In the Figure 7.5 from Burns, the gap is labelled as $2|H_{12}|$, which is clearly the term in the Hamiltonian representing the interaction strength. If you have the proof, you can see how this is obtained.

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  • $\begingroup$ Thank you. Yes, the proof is there. $\endgroup$ Commented Aug 9, 2024 at 11:39

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