Bulk-Edge Correspondence for Chern Topological Phases: A Viewpoint from a Generalized Index Theorem

We explore the bulk-edge correspondence for topological insulators (superconductors) without time-reversal symmetry from the point of view of the index theorem for open spaces. We assume generic Hamiltonians not only with a linear dispersion but also with higher order derivatives arising from generic band structures. Using a generalized index theorem valid for such systems, we show the equivalence between the spectral flow of the edge states and the Chern numbers specifying the bulk systems.