Fig. 1

The SSH model with second-nearest neighbor hopping. (a) Schematic diagram of hopping terms. (b) Wave function rotation in pseudo-spin space. The red point is the topological singular point and \(\mathcal {W}\) is corresponding winding number. Topological phase diagram by calculating \(\mathcal {W}\) with (c) \(J_{11}=0.1,J_1=1\), (d) \(J_{11}=1,J_1=0.1\). (e), (f) Particle distribution of a typical bulk state (diamond) and edge states (five-pointed star and inverted triangle) for \(\Delta _{J_3(J_{33})-J_1}=0.75\), which are along the \(J_3 (J_{33})\) axis in (c), respectively. Inset: The corresponding energy spectrum for a 20-site chain. Here, we define that \(\Delta _{J_3-J_1}=(J_3-J_1)/2\), \(\Delta _{J_{33}-J_1}=(J_{33}-J_1)/2\) and \(\Delta _{J_{33}-J_{11}}=(J_{33}-J_{11})/2\).