Quantized Hall conductivity of Bloch electrons: Topology and the Dirac fermion

We consider the Hall conductivity of two-dimensional noninteracting 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 counterexample 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 expectations, we show that the 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.