Fig. 4. SSH model and simulations of the band topology of chGNRs.

a Modified Su-Shrieffer-Heeger model to explain the SPT phase transition of chGNRs with the width. b Evolution of the energy gap value and sign with the chGNR width, obtained by fitting the mSSH model in a to frontier bands from 3NN TB simulations. (Right Ordinate) comparison of intra- ($t_i$) and inter-edge (${\mathrm{t}}_{\mathrm{i}}^{\prime }+{\mathrm{t}}_{\mathrm{i}}^{″}$) hopping elements obtained from the fit, indicating that the topological transition occurs when both are balanced. c Color plots of the energy gap value and sign for the family of (z, a, w) chGNRs, showing that the topological transition is a global property of this family of chiral ribbons. Marked squares indicate the ribbons studied in this work. The red/blue color scale refers to the positive/negative sign of E_g, defined as ${\left(-1\right)}^{{\mathbb{Z}}_2}$, where the ${\mathbb{Z}}_2$ invariant is obtained from the Zak phase (Supplementary Note 3).