Fig. 2: Spin-valley Hall phases realized in 4-fold rotational symmetric square lattices.
From: Winding coupling phase for pseudo-spin-derived topological photonics
a Sketch of a 4-fold rotational symmetric (\({C}_{4}\)) lattice consisting of coupled resonator A (red) and B (blue). The modes supported are labeled by their AM numbers, \(\pm {m}_{A}\) and \(\pm {m}_{B}\). b−h. Band diagrams, and Berry-curvature \(\varOmega ({\bf{k}})\) distributions of valence bands across the Brillouin zone, obtained by tight-binding model (TBM) simulations for various values of \(\mathrm{mod}({W}_{B\to A}^{\uparrow },4)\). When \(\mathrm{mod}\left({W}_{B\to A}^{\uparrow },4\right)=\mathrm{0,2}\), the Berry curvature vanishes and the bands are trivial. When \(\mathrm{mod}\left({W}_{B\to A}^{\uparrow },4\right)=\pm 1\), the special Berry-curvature distributions, together with indicated spin-valley Chern numbers, evidence the SVHP. In the TBM simulations, the resonant frequencies of the \(\pm {m}_{A}\) and \(\pm {m}_{B}\) modes are set to be \(0.2{t}_{0}\) and \(-0.2{t}_{0}\), respectively, and, accordingly the energy difference between the modes is \(\triangle={\omega }_{A}-{\omega }_{B}=0.4{t}_{0}\), where \({t}_{0}\) denotes the coupling strength; only the nearest A−B coupling is considered.
