A ${\mathbb{Z}}_2$ invariant for chiral and particle-hole symmetric topological chains

Abstract

We define a ${\mathbb{Z}}_2$-valued topological and gauge invariant associated to any 1-dimensional, translation-invariant topological insulator which satisfies either particle-hole symmetry or chiral symmetry. The invariant can be computed from the Berry phase associated to a suitable basis of Bloch functions which is compatible with the symmetries. We compute the invariant in the Su-Schrieffer-Heeger model for chiral symmetric insulators, and in the Kitaev model for particle-hole symmetric insulators. We show that in both cases the ${\mathbb{Z}}_2$ invariant predicts the existence of zero-energy boundary states for the corresponding truncated models.

REVTeX class, 20 pages, no figures; to appear in the proceedings volume for the workshop "Learning from Insulators: New Trends in the Study of Conductivity of Metals", 9-13 August 2021, Lorentz Center (Leiden, NL)