The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator

TL;DR

This paper explores the mathematical structure of topological phases in free fermion systems, showing that combining two SSH chains produces a 2D Chern insulator, and unifies topological invariants across dimensions.

Contribution

It introduces a product structure in topological phases via Bott periodicity and demonstrates how SSH chains generate higher-dimensional topological invariants.

Findings

Two SSH chains in independent directions produce a 2D Chern insulator.

Bott periodicity relates phases through a generalized Dirac monopole.

K-theory cohomology captures the classification of topological phases.

Abstract

We provide an extensive look at Bott periodicity in the context of complex gapped topological phases of free fermions. In doing so, we remark on the existence of a product structure in the set of inequivalent phases induced by the external tensor product of vector bundles -- a structure which has not yet been explored in condensed-matter literature. Bott periodicity appears in the form of a generalized Dirac monopole built out of a given phase, which is equivalent to the product of a Dirac monopole phase with that same given phase. The complex K-theory cohomology ring is presented as a natural way to store the information of these phases, with a grading corresponding to the number of Clifford symmetries modulo $2$. The K\"unneth formula allows us to derive the result that, for band insulators, the Su-Schrieffer-Heeger (SSH) chain in one dimension allows one to generate the K-cohomology of the $d$-dimensional Brillouin zone. In particular, we find that the product of two SSH chains in independent momentum directions yields a two-dimensional Chern insulator. The results obtained relate the associated topological phases of charge-conserving band insulators and their topological invariants in all spatial dimensions in a unified way.