Square-root topological insulator with high winding number

As a distinctive scheme towards topological phases with unconventional properties, square-root topological insulators have garnered great interest in recent years. Here, we investigate the topological phases of square-root manipulations on an extended Su-Schrieffer-Heeger (SSH) model with long-range hoppings. By calculating the energy spectra, topological boundary states, and winding number, we demonstrate that the square-root operation allows for the symmetric distribution of quadruple degenerate boundary states in the doubled bulk gaps. These states are inherited from the original extended SSH model and can be characterized by a high winding number $w=2$. Finally, we propose a feasible scheme for realizing our model in an electric circuit system. Our work provides a scheme for the discovery and systematic construction of new topological states.