Note on Green Function Formalism and Topological Invariants

It has been discovered previously that the topological order parameter could be identified from the topological data of the Green function, namely the (generalized) TKNN invariant in general dimensions, for both non-interacting and interacting systems. In this note, we show that this phenomena has a clear geometric derivation. This proposal could be regarded as an alternative proof for the identification of the corresponding topological invariant and topological order parameter.