Fig. 4. Topological transition induced by changing the system’s size.

a Sketch of the H_CNHSE-SSH model with cross inter-chain non-reciprocal couplings  ±δ_ab. b Open-boundary spectra (black dots) at N = 20, 30, and 40 unit cells and coupling δ_ab = 0.5 × 10⁻³. The main part of the spectrum in each case behaves similarly as the model in Fig. 2b, except for a pair of topological edge states blue emerging within the point gap at zero energy. The open-boundary E_∞ spectrum is given by green and red colors in the decoupled and coupled thermodynamic limits respectively. Other parameters are δ_a = −δ_b = 0.5, t₁ = 0.75, and V = 1.2. c κ solutions (red, blue, green, and yellow surfaces) of f(z, E) = 0 as a function of the complex energy, with the same parameters in b. Intersecting regions (green and red dotted lines) give the open-boundary skin solutions of the system in the thermodynamic limit. Among them, green lines correspond to the skin solutions of two decoupled chains at δ_ab = 0. The solutions of red curves emerge at a small but nonzero δ_ab, and the skin solutions of the weakly coupled system are given by the intersecting regions with the smallest ∣κ∣, i.e., the red loop in the center and green lines at the two ends with large and small $\mathrm{Re}\left[E\right]$. d Emergence of in-gap degenerate modes as a function of δ_ab/δ_a and N with δ_a = − δ_b = 0.5, t₁ = 0.75, and V = 1.2, with the plotted boundary scaling logarithmically with N.