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 $A$ and $B$, and governed by staggered hopping potentials $v$ and $w$ 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 hopping potentials $v$ and $w$, respectively. The generalized trimer chain incorporates the intracell hopping $v_1$ and $v_2$ and intercell hopping $w_1$ and $w_2$ to differentiate the hopping energies between different sublattices in the chain. The hexagonal chain is composed of six sublattices with intracell hopping potential $v$ and intercell hopping potential $w$. 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 $v<w$ and metallic when $v=0$. 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-boundary correspondence to calculate for the winding number that predicts the existence of localized edge states in the topological nontrivial phase.