Komoto's paper (ANNALS OF PHYSICS 160, 343-354 (1985)) on the calculation of the Hall conductance provides a clear discussion about how calculate the conductance using the torus shaped magnetic Brillouin zone surface. He shows that the MBZ needs to be broken into two parts to get a non zero Chern number.
Yet, I still have some confusion concerning the surface integral over the magnetic Brillouin zone (MBZ) torus for the Hall conductance. Another reference seems to use Stokes theorem over the MBZ torus to obtain the quantized Hall conductance - see appendix A.1 in S.-Q. Shen, Topological Insulators: Dirac Equation in Condensed Matters. Here, the MBZ is not split into smaller patches......there is no separate area with a boundary inside the 2D MBZ in A.1 which is different from Kohmoto's paper (Fig. 1 Kohmoto). This reference (appendix A.1) has line integrals around the edge of the 2D MBZ and obtains a non-zero conductance.
Komoto's paper discusses the wavefunctions inside the inner area of the MBZ as having a different phase than the wavefunctions in the outer area. A line integral of the Berry curvature around the boundary between these area results in the quantization. It seems that this boundary is required.
In appendix A.1, the wavefunctions have a phase difference between the wavefunctions at kx=0 and kx=2pi as well as between ky=0 and ky=2pi. The line integrals around the edge result in the quantization.
Does this mean that the area outside the Magnetic Brillouin zone in the derivation for Appendix 2A has a boundary with the area inside the MBZ? How does this happen if the MBZ is a torus?
