Extensions to the Su-Schrieffer-Heeger Model: Linear Chains and Their Topological Properties

The Su-Schrieffer-Heeger (SSH) model describes the dynamics of spinless fermions in a one- dimensional lattice, with sublattices 𝑨𝑨 and 𝑩𝑩, and governed by staggered hopping potentials 𝒗𝒗 and 𝒘𝒘 representing the intracell and intercell hopping energies, respectively. In this study, we extend the SSH model into three distinct types – a trimer chain, the generalized trimer chain, and a hexagonal chain. The trimer chain involves three sublattices with intracell and intercell hoppi ng potentials 𝒗𝒗 and 𝒘𝒘, respectively. The generalized trimer chain incorporates the intracell hoppi ng 𝒗𝒗 𝟏𝟏 and 𝒗𝒗 𝟐𝟐 and intercell hopping 𝒘𝒘𝟏𝟏 and 𝒘𝒘𝟐𝟐 to differentiate the hopping energies between different sublattices in the chain. The hexagonal chain is composed of six sublattices with intracell hoppi ng potential 𝒗𝒗 and intercell hopping potential 𝒘𝒘. We utilize exact diagonalization to determine the bulk eigenvalues of the different models. We find that in the trimer and generalized trimer chain, the bulk eigenvalues exhibit conducting characteristics, independent of the hopping parameter, owing to the presence of a flat band situated along the Fermi energy. In the hexagonal chain, the bulk eigenvalues display semi-metallic characteristics in the region 𝒗𝒗 < 𝒘𝒘 and metallic when 𝒗𝒗 = 𝟎𝟎. Furthermore, we investigate the presence of conducting edge states in the finite chains. The trimer and hexagonal chains show the presence of topologically protected edge states which are manifestations of one-dimensional topological insulators. We also established the bulk-boundar y correspondence to calculate the winding number that predicts the existence of localized edge states in the topological nontrivial phase.