I am under the impression that topological insulators have a distinguishing characteristic where they have an odd number of Dirac points that intersect band gaps at the Fermi energy.

However, this seems to violate the Nielsen-Ninomiya theorem which states there is an even number of states of opposite chirality. I take this to mean there should be an even number of Dirac fermions.

a topological insulator could violate this theorem if it did not have translation symmetry, but I understand that they do. How do they manage to obtain an odd number of Dirac points then?

To respond to the comment below, First, I'm using the term point and fermion (referring to the emergent quasiparticle that crosses that point) interchangeably, is that accurate?

Secondly, the comment seems to imply that odd numbers of Dirac fermions do not violate the NN theorem because it has a net chirality of zero. My confusion lingers because topological insulators do have net transport in their edge states which after further reading, seems to be a result of your namesake, a chiral anomaly.

The Adler-Bell- Jackiw anomaly is mentioned in section of 1.3 and 2.6 of Witten's notes here. Are topological insulators a realization of this anomaly?


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