The edge spectrum of Chern insulators with rough boundaries

Chern insulators are periodic band insulators with the property that their projector onto the occupied bands has a nonzero Chern number. From numerical calculations, it is known that a Chern insulator with a homogeneous boundary displays a continuum spectrum that fills the entire insulating gap. The local density of states corresponding to this part of the spectrum is localized near the boundary, hence the terminology edge spectrum. An interesting question arises, namely, if a rough boundary, which can be seen as a strong random potential acting on these quasi-one-dimensional states, would destroy the continuum edge spectrum. Numerical simulations seem to indicate that the answer is no. The present paper shows how the question can be answered analytically by connecting the expectation value of the charge edge current to the index of a Fredholm operator, which remains invariant under arbitrary deformations of the boundary.