One-dimensional noninteracting topological insulators with chiral symmetry

We construct microscopical models of one-dimensional noninteracting topological insulators in all of the chiral universality classes. Specifically, we start with a deformation of the Su-Schrieffer-Heeger (SSH) model that breaks time-reversal symmetry, which is in the AIII class. We then couple this model to its time-reversal counterpart in order to build models in the classes BDI, CII, DIII, and CI. We find that the Z topological index (the winding number) in individual chains is defined only up to a sign. This comes from noticing that changing the sign of the chiral symmetry operator changes the sign of the winding number. This freedom to choose the sign of the chiral symmetry operator on each chain independently allows us to construct two distinct possible chiral symmetry operators when the chains are weakly coupled-in one case, the total winding number is given by the sum of the winding number of individual chains, while in the second case, the difference is taken. We also find that the chiral models that belong to Z classes, AIII, BDI, and CII are topologically equivalent, so they can be adiabatically deformed into one another without the change of topological invariant, so long as the chiral symmetry is preserved. We study the properties of the edge states in the constructed models and prove that topologically protected edge states must all be localized on the same sublattice (on any given edge). We also discuss the role of particle-hole symmetry on the protection of edge states and explain how it manages to protect edge states in Z2 classes, where the integer invariant vanishes and chiral symmetry alone does not protect the edge states anymore. We generalize our results to the case of an arbitrary number of coupled chains, by constructing possible chiral symmetry operators for the multiple chain case, and briefly discuss the applications to any odd number of dimensions.