Fig. 4: Real-space protection of vortex transport and universal rule for VRS-mediated non-trivial winding.

a, (1, 2) An illustration of two types of vortex mode coupling between two waveguides where \(\kappa\) is the coupling amplitude; the SVMC is not direction dependent (1), while the OVMC features a coupling coefficient \({t}_{{\rm{OV}}}\) dependent on \(\theta\) as plotted in (2). b–d, In a C_n-symmetric disclination structure, all coupling contributions to the central vortex mode can be calculated by sectors as illustrated for \({C}_{3}\) (b, 1–3), \({C}_{4}\) (c, 1–3) and \({C}_{5}\) (d, 1–3) disclination structures, where \({T}_{j}\) is the equivalent coupling for all OVMCs in each sector; (b, 1) depicts the collective OVMC coupling from the three sectors, (b, 2,3) represent real-space winding for charges 1 and 2 vortices in the C₃ disinclination, and (c, 1–3) and (d, 1–3) follow the same layout as (b, 1–3) but are for C₄ and C₅ disclinations, respectively. To guarantee that only a single vortex mode (\(l=1\) in the third row; \(l=2\) in the fourth row) is present at the disclination core, the complex coupling \({T}_{j}\) must have a non-zero winding number (\(w\ne 0)\), as shown. This is better described by the VRS that demands a non-integer value of \(2l/n\) for twofold protection, as summarized in a (3), where blue (orange) indicates protected (unprotected) vortex modes. Taking \(l=2\) as an example, the vortex is protected in the \({C}_{3}\) disclination owing to non-trivial winding \(w=1\) (b, 3), but it is not protected in the \({C}_{4}\) disclination since \(2l/n=1\) is an integer and \(w=0\) in this case (c, 3). A vortex is topologically protected only under non-zero winding conditions. e,f, Experimental results obtained from \({C}_{4}\) (e, 1) and \({C}_{5}\) (f, 1) disclination structures, which show that, as in \({C}_{3}\) disclination (Fig. 2), both \(l=1\) (f, 2) and \(l=2\) (f, 3) vortices are also protected in the \({C}_{5}\) disclination, however, in the C₄ disclination, the l = 1 vortex is protected (e, 2) but the \(l=2\) vortex is not protected (e, 3), in agreement with the winding picture and the relation plotted in a (3).