Fig. 4: Topologically protected edge states by the invariant Z1Z3.
a A loop circulating the nondefective intersection point (NIP), as the Brillouin zone of the 1D lattice model in Eq. (6), is partitioned into four paths, with α and α‘ residing in exact phases. b Sample designs of the lattice model under periodic boundary condition [PBC: panel (i), terminal unit cells are connected with hoppings] and open boundary condition [OBC: panel (ii), terminal unit cells are disconnected]. Here the black circles denote unit cells and the green bonds denote the hopping matrices connecting adjacent unit cells. The dashed blocks encircle two unit cells, and the structure inside the block is shown in panel c. c Realization of the lattice model. The dashed block shows the internal structure of unit cells and the hoppings (labeled in panel b with dashed blocks). The hopping parameters \({t}_{{{{{\mathrm{1,2}}}}}}^{11}\), \({t}_{{{{{\mathrm{1,2}}}}}}^{12}\), \({t}_{{{{{\mathrm{1,2}}}}}}^{21}\) and \({t}_{{{{{\mathrm{1,2}}}}}}^{22}\) are the entries of the hopping matrices \({\hat{t}}_{1}\) or \({\hat{t}}_{2}\) in Eq. (7). d Eigenvalue dispersions (real part) of the model of Eq. (7) in the 1D Brillouin zone. Since the Brillouin zone cuts through exceptional lines (ELs) four times, the band structure experience gap closing four times. e Joining the trajectories of two bands on the path α forms a loop in Re(E)-f2-f3 space lα, along which the Berry phase is π. This quantized Berry phase is equal to the relative rotation angle between the two eigenstates resulting from frame deformation along α. For the path α‘, joining the two bands forms the loop lα‘, along which the Berry phase is –π. This is because from α to α‘ the two eigenstates swap due to band inversion at NIP. The relative rotation angle between the eigenstates changes sign. f Plots of projection bands of the 1D lattice model under open boundary condition (OBC, black dots) and periodic boundary condition (PBC, red dots). There exists a pair of edge modes in the line gap for eigenstates along the loops lα and lα‘ in panel e. g Field distribution of one edge mode. The lattice model with OBC has 300 periods (600 lattice sites, denoted by Ns). Inset: zoom-in view showing the field distribution near the left edge.
