Topological properties of tetratomic Su-Schrieffer-Heeger chain with hierarchical long-range hoppings


 A new generalized Su-Schrieffer-Heeger model with hierarchical long-range hoppings based on the one-dimensional tetratomic chain is proposed. The properties of the topological states and phase transition, depending on the cointeraction of the intracell and intercell hoppings, are investigated by the phase diagram of the winding number. It is found that topological states with large positive/negative winding numbers can be readily generated in the system. The properties of the topological states can be verified by the ring-type structures in the trajectory diagram of the complex plane. The topological phase transition is strongly related with the opening up or closure of the energy bandgap at the center/boundaries of the Brillouin zone. Finally, the feature of the non-zeroenergy edge states at the ends of the finite system is revealed and matched with the bulk-boundary correspondence.