Fig. 4: Vector fields representing the deviation of state variables in the extended nonlinear SSH model.

a Phase diagram of the linear extended SSH model. a, d, and α are real parameters that determine the strength of intercell, nearest-neighbor intracell, and long-range hoppings, respectively. Each color represents the parameter regions with the same winding number, which is shown by the numbers in the regions. b–d Vector field and the stable manifold. The blue curved arrows represent the vector field at (Ψ(x), Ψ(x + 1)) corresponding to the values of eigenvectors at the sites x and x + 1. The red disks are stable fixed points, the red squares are saddle points, and the green circle is a fully unstable fixed point. The orange curves are eye-guides of the one-dimensional stable manifolds. b If the winding number is one under weak nonlinearity (the blue circle in a), the stable manifold is one-dimensional. The parameters used are a = 0.5, b = d = −1, and α = 0.25. c When the winding number is two under weak nonlinearity (the red filled square in a), the fixed point is stable and thus has a two-dimensional stable manifold. The parameters used are a = 1, b = d = −1, and α = 4. d When the nonlinearity-induced topological phase transition from zero to one occurs (the green arrow in a), nonzero fixed points appear and their stable manifold is one-dimensional. The parameters used are a = 1.75, b = d = −1, and α = 0.25.