Generic Chern Numbers for a Degenerate Multiplet: For a Characterization of Topological Orders

Chern numbers for a multiplet which is a set of low lying states near the ground states are defined and an explicit expression is obtained in a gauge dependent form. We allow intrinsic degeneracies within the multiplet where a well-known standard procedure does not work. As an example, we give expressions for a spin Hall conductance for unitary superconductors with equal spin pairing. Generic topological orders will be treated in this manner particularly with nontrivial topological degeneracies.\footnote{Y. Hatsugai, J. Phys. Soc. Jpn, 2604, (2004), cond-mat/0405551}. It can be useful for a characterizaton of some class of low dimensional quantum (spin) liquids with topological degeneracies. \footnote{Y. Hatsugai, preprint}