Quantized Hall conductivity of Bloch electrons: topology and the Dirac   fermion
  We consider the Hall conductivity of two-dimensional non-interacting Bloch electrons when the magnetic flux per unit cell is a rational number $p/q$ where $p$ and $q$ are mutually coprime. We present a counter-example for the naive expectation that the Hall conductivity carried by a band is given by treating gap minima as Dirac fermions. Instead of the above expectation, we show that the {\em change\/} of the Hall conductivity at a gap-closing phenomenon is given by the Dirac fermion argument. Comparing with the Diophantine equation, our result implies that a band-gap closes at $q$ points simultaneously. Furthermore, we show that the dispersion relation is $q$-fold degenerate in the magnetic Brillouin zone. 
Masaki Oshikawa