Fig. 3: Zak phases and evolution of edge states.
From: Four-band non-Abelian topological insulator and its experimental realization
a Zak phases for individual bands summarized for each non-Abelian topological charge. For charges \(\pm {q}_{1234}\), two eigenvectors rotate by \(\pi\), while the other two rotate by \(+\pi\) or \(-\pi\) depending on the factorizations (due to the handedness of the subspace). b Edge states of charges \(\pm {q}_{12}\) occur at the bandgap sandwiched by the first and second bands. c Schematic view of charge \({q}_{1234}\) factorizations and their mutual continuous transitions. The double-headed arrows indicate that the paired two factors commute, i.e., \({q}_{12}{q}_{34}={q}_{34}{q}_{12}={q}_{1234}\). The directional arcs define the continuous transitions parametrized by \({\theta }_{m\to n}\), with \(m\) and \(n\) taking values of I, II, and III, corresponding to the factorizations of \({q}_{12}{q}_{34}\), \(-{q}_{13}{q}_{24}\), and \({q}_{14}{q}_{23}\), respectively. d–f Evolution of edge states with varying parameter \({\theta }_{m\to n}\), where lines/dots indicate the numerical/analytical results. g–i Evolution of the extended 2D bands corresponding to (d–f), respectively. Note that we only plot the radial cuts \(E\left({k}_{r}\right)\), as the 2D bands for ideal flat-band models are isotropic in the (\({k}_{1},{k}_{2}\)) plane.
