Fig. 1: Coupling winding number: definition, pseudo-spin-contrasting characteristics, and k-space vortices.

a Coupling winding number \({W}_{B\to A}\), a real-space topological invariant, defined by tracing the evolution of the coupling coefficient along a closed path S (parameterized by the relative angle \(\theta\)) where one resonator (B, blue) winds counterclockwise around the other (A, red). Consider a pair of coupled resonators with continuous rotational symmetry that support modes with angular-momentum quantum numbers \(\pm {m}_{A}\) and \(\pm {m}_{B}\), respectively, within the frequency range of interest. The coupling winding numbers assume a simple form and take opposite values for the opposite pseudo-spins (\(\uparrow\), \(\downarrow\), see Eq. (2) for the definition). b–d Vortices of the k-space coupling coefficient at the \({\boldsymbol{\Gamma }}\) point in the Brillouin zone of \({\left.{C}_{N}\right|}_{N=\mathrm{3,4,6}}\) crystal, controlled by the coupling winding number. The k-space coupling term \({f}_{B\to A}^{s}\) is computed by summarizing up the nearest \(B\to A\) couplings: \({f}_{B\to A}^{s}\left({\bf{k}}\right)=\mathop{\sum }\nolimits_{n=1}^{N}{t}_{B\to A}^{s}({\theta }_{n}){e}^{i{\bf{k}}\cdot {{\bf{a}}}_{n}}\) with \({{\bf{a}}}_{n}=a\cos ({\theta }_{n})\hat{x}+a\sin \left({\theta }_{n}\right)\hat{y}\), \({\theta }_{n}=2\pi \left(n-1\right)/N\), and \({t}_{B\to A}^{s}\left({\theta }_{n}\right)={t}_{0}{e}^{i{W}_{B\to A}^{s}({\theta }_{n}-{\theta }_{1})}\) with \({t}_{0}\) being a positive real number. Note that once \({W}_{B\to A}^{\uparrow }\) is known, \({W}_{B\to A}^{\downarrow }\) can be inferred from \({W}_{B\to A}^{\downarrow }=-{W}_{B\to A}^{\uparrow }\), so only \({W}_{B\to A}^{\uparrow }\) is indicated in (b–d).