Magnetic impurities along the edge of a quantum spin-Hall insulator: realizing a one-dimensional AIII insulator

In this paper we construct a one-dimensional insulator with an approximate chiral symmetry belonging to the AIII class and discuss its properties. The construction principle is the intentional pollution of the edge of a two-dimensional quantum spin-Hall insulator with magnetic impurities. The resulting bound states hybridize and disperse along the edge. We discuss under which circumstances this chain possesses zero-dimensional boundary modes on the level of an effective low-energy theory. The main appeal of our construction is the independence on details of the impurity lattice: the zero modes are stable against disorder and random lattice configurations. We also show that in the presence of Rashba coupling, which changes the symmetry class to A, one can still expect localized half-integer boundary excess charges protected by mirror symmetry although there is no non-trivial topological index. All of the results are confirmed numerically in a microscopic model.