Topological properties of a class of generalized Su–Schrieffer–Heeger networks: Chains and meshes

Abstract

We analyze the topological properties of a family of generalized Su-Schrieffer-Heeger (SSH) chains and mesh geometries. In both the geometries the usual staggering in the distribution of the two overlap integrals is delayed (in space) by the inclusion of a third (additional) hopping term. A tight-binding Hamiltonian is used to unravel the topological phases, characterized by a topological invariant. While in the linear chains, the topological invariant (the Zak phase) always appears to be quantized, in the quasi-one dimensional strip geometries and the generalized SSH mesh patterns the quantization of the Zak phase is sensitive to the strength of the additional interaction (the ‘extra’ hopping integral). We study its influence thoroughly and explore the edge states and their robustness against disorder in the cross-linked generalized SSH mesh geometries. The systems considered here can be taken to model (though crudely) two-dimensional polymers where the cross-linking brings in non-trivial modification of the energy bands and transport properties. In addition to the topological features studied, we provide a prescription to unravel any flat, non-dispersive energy bands in the mesh geometries, along with the structure and distribution of the compact localized eigenstates. Our results are analytically exact.