Topological invariant and the quantization of the Hall conductance

The topological aspects of wavefunctions for electrons in a two dimensional periodic potential with a magnetic field are discussed. Special attention is paid to the linear response formula for the Hall conductance σxy. It is shown that the quantized value of σxy is related to the number of zeros of wavefunctions in the magnetic Brillouin zone. A phase of wavefunctions cannot be determined in a unique and smooth way over the entire magnetic Brillouin zone unless the magnetic subband carries no Hall current.