Chern Numbers in Discretized Brillouin Zone: Efficient Method of   Computing (Spin) Hall Conductances
  We present a manifestly gauge-invariant description of Chern numbers associated with the Berry connection defined on a discretized Brillouin zone. It provides an efficient method of computing (spin) Hall conductances without specifying gauge-fixing conditions. We demonstrate that it correctly reproduces quantized Hall conductances even on a coarsely discretized Brillouin zone. A gauge-dependent integer-valued field, which plays a key role in the formulation, is evaluated in several gauges. An extension to the non-Abelian Berry connection is also given. 
Takahiro Fukui, Yasuhiro Hatsugai, and Hiroshi Suzuki