Topological edge states of 1D chains and index theory

We provide an elementary proof and refinement of a well-known idea from physics: a chiral-symmetric local Hamiltonian on a half-space has the same signed number of edge-localized states with energies in the bulk band gap, as its bulk winding number. The requirement of non-elementary methods to relate generic and non-generic cases is emphasized. Our hands-on approach complements a quick abstract proof based on the classical index theory of Toeplitz operators.