Fig. 3.

Non-trivial winding in the light–matter interaction. a Dependence of the magnitude of the light–matter coupling constant $\mid {\Lambda }_{\mathbf{q}\mathrm{0}\tau }\mid \propto \mid {\mathrm{e}}^{\mathrm{i}{\phi }_{\mathbf{q}}}+\tau \mid$ on the direction of q from Γ−M to Γ−K(K′) (see inset), for the upper (red line) and lower (blue line) quasistatic bands. Plots obtained with |q| = |K|/2. b–e Schematics of the bright (↑↑) and dark (↑↓) configurations of the two sublattices interacting with the photonic mode which is indicated by the field profile. Panels b and d represent the crossings labeled ‘1’ and ‘3’ in Fig. 2 along the Γ−K(K′) directions, respectively, while c and e represent the crossings labeled ‘2’ and ‘4’ along the Γ−M directions, respectively. Crucially, the light–matter interaction strength for the dark mode vanishes (|Λ_q0−| = 0) along the Γ−K(K′) directions due to the complete destructive interference between the two sublattices (see d), leading to the emergence of six inequivalent type-II Dirac points in the polariton spectrum