  It is widely accepted that topological quantities are useful to describe
quantum liquids in low dimensions. The (spin) Hall conductances are typical
examples. They are expressed by the Chern numbers, which are topological
invariants given by the Berry connections of the ground states. We present a
topological description for the (spin) Hall conductances on a discretized
Brillouin Zone. At the same time, it is quite efficient in practical numerical
calculations for concrete models. We demonstrate its validity in a model with
quantum phase transitions. Topological changes supplemented with the transition
is also described in the present lattice formulation.
