Nielsen–Ninomiya theorem states that in a lattice system one can not have just one chiral fermion. Fermions necessarily come in pairs of opposite chirality. I am wondering if one can "explain" this theorem using the following argument:

Since a Weyl crossing is a monopole of Berry curvature, following Dirac we should attach and unobservable solenoid to it (Dirac string), which necessarily ends on an antimonopole. Usually the end of the string can be put to infinity, but since the Brillouin zone is compact, we can not get rid of the antimonopole.

Do you think this is a correct way how to understand the Nielsen-Ninomiya theorem?

  • $\begingroup$ What is a "Weyl crossing"? (Google gives me exactly three hits for the phrase, please explain uncommon terminology) $\endgroup$
    – ACuriousMind
    Nov 10, 2015 at 22:20
  • $\begingroup$ By Weyl crossings I meant a point in the Brillouin zone, where two bands touch each other. Near this point the energy dispersion is linear and the system can be described by the Weyl equation. $\endgroup$ Nov 11, 2015 at 0:35

1 Answer 1


A proof of the Nielsen-Ninomiya theorm, basically based on your quite ingenious suggestion is given in section 7 of Elias Kiritsis article on the topological properties of the Berry's phase.


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