Extended Data Fig. 5: Protected and unprotected high-order vortex modes in a C₅-symmetric disclination lattice.

a Schematic illustration of the chiral-symmetric C₅-disclination lattice. b Calculated eigenvalues for the \({|l|}=5\) vortex modes of the lattice, where two degenerate defect modes appear right at zero energy but with opposite phase vorticity. c1, c2 Two examples showing unprotected high-order vortex modes with topological charge \(l=5\) (c1) and \(l=10\) (c2) after a propagation distance \(Z=40\) (simulation from the tight-binding model). In these cases, the vorticity-coordinated rotational symmetry (VRS) is not satisfied because \(2l/n\) is an integer. d1, d2 Two examples showing protected high-order vortex modes with topological charge \(l=8\) (d1) and \(l=12\) (d2). Here, the VRS is satisfied because \(2l/n\) is not an integer. These results for different high-order vortices in the same disclination structure further validate the real-space topological protection pictured in Fig. 4 of the main text.