Fig. 1: Illustration of doubly protected vortex transport via topological disclinations.

a, A schematic of a vortex travelling through the core of a disclination lattice that has \({C}_{3}\) rotational symmetry and chiral symmetry (CS). The lattice features two sublattices with a single-site core at the centre, where a vortex is transported, guided and topologically protected. b, A Venn diagram of the underlying topology associated with vortex guidance. The light-blue region represents a non-trivial momentum-space winding (illustrated with a non-empty winding loop) as typical for a chiral-symmetric structure. The yellow region represents a non-trivial real-space winding (illustrated with complex coupling vectors winding) when the disclination lattice features a VRS. The overlapping region is where a vortex can have twofold (both real- and momentum-space) topological protection during propagation in a non-trivial disclination structure. c, Numerical simulations showing (2) robust propagation of a doubly protected high-order vortex (\(l=2\)) to a distance L through the disclination core that serves as a TVG, but (1 and 3) the same vortex expands and breaks up during transport when the twofold protection is absent.