Fig. 3: Momentum-space protection of zero-energy vortex modes via topological disclination.

a, (1) Calculated eigenvalues of single-charge vortex modes in a \({C}_{3}\) chiral-symmetric disclination structure, where two degenerate vortex modes (red) appear right at mid-gap but with opposite vorticity (\(l=1\) and \(l=-1\)); (2) corresponding results obtained for a pair of high-order (l = 2 and l = −2) vortex modes in the same C₃ structure. b,c, (1) Intensity (b) and phase (c) distributions of the \(l=1\) vortex mode, showing confinement mostly at the disclination core. Exponentially decaying ‘tails’ distribute only in the same (next-nearest-neighbour) sublattices with a π-phase difference—a characteristic of SSH-type topological states (\({d}_{1}\) and \({d}_{2}\) mark the intra-cell and inter-cell spacing, respectively); (2) corresponding results obtained for a pair of high-order (\(l=2\) and \(l=-2\)) vortex modes in the same \({C}_{3}\) structure. Note in (c, 2) there is a 4π phase circulation for each vortex, and the vortex in the centre again has a π-phase difference compared with those in the ‘tails’. d, Calculated topological invariant. The MCN \(N\), which equals 2 when \({d}_{1}\) is larger than \({d}_{2}\), indicates a topologically non-trivial regime with two zero-energy disclination modes. e, An illustration of multipole moments in the \({C}_{3}\) structure, where \(\widetilde{q}\) and \(\widetilde{p}\) are the differences in dipole and quadrupole moments between sublattices, respectively. We show three sets of coordinates \(({x}_{i},{y}_{i})\) with \(i=\mathrm{1,2,3}\), which correspond to three sectors of the \({C}_{3}\) disclination structure, to generate the multipole operators.