Electronic and topological properties of extended two-dimensional Su-Schrieffer-Heeger models and realization of flat edge bands

One-dimensional Su-Schrieffer-Heeger (SSH) chains are one of the simplest topological models and have been studied extensively. The two-dimensional (2D) version has been confirmed to have a nontrivial topology. In this work, we further extend the 2D SSH model by constructing different configurations, including all possible configurations with 4-site unit cells and another two complicated, but typical, configurations with 8- and 36-site unit cells, respectively. We calculate and analyze the electronic structures and topologies of these SSH models in detail and identify several rich and novel properties, such as topologically protected edge states, metallic chains with different shapes, and a bulk–edge separation in a metal system. In particular, a flat-band feature of topological edge states is obtained. By analyzing the spatial distribution of these edge states, we explain the origin of the flat edge bands and show that bonding squares play a crucial role in the formation of flat bands. Our study can be generalized to more configurations and higher dimensions, providing a basis for further theoretical and experimental explorations.