Fig. 2: Computed discrete modal spectra of KL chain’s F/WF-phase kinks, including a family of internal modes with smooth variation between sites.

a Phase and kink width diagram of the zero-energy kink in the KL chain as a function of the dimensionless index d¹⁰. b, c Normalized eigenfrequencies of kink states as a function of their normalized widths \({\widetilde{w}}_{0}\) for KL chains with \(\widetilde{r}=1.5\) and N = 280 rotors, where (b) and (c) show results for onsite- (ψ₁₄₀ = 0) and intersite-centered (ψ₁₄₀ = − ψ₁₄₁) kinks, respectively. Markers with black edge indicate internal modes. Insets show zoomed-in, normalized kink states (\(-\sin {\psi }_{n}/\sin \bar{\psi }\)¹⁰). Green color denotes the kink state examined in experiments, which is further used in (d), as well as Figs. 3 and 4, to denote the same geometry. For F- and WF-phase kinks, the lowest eigenfrequency remains zero, indicating barrier-free propagation independent of kink width (i.e., discretization). d In-gap modes of the experimental kink state as a function of the kink’s center rotor angle. Circled marker 1 and 4 indicate onsite- and intersite-centered kinks, respectively, while circled marker 2 and 3 correspond to intermediate kink states between them, which are \({\psi }_{140}/\bar{\psi }=1/4\) and 2/3 (further used in Figs. 3 and 4).