Fig. 1: Quadrupole topological phase transitions through super-cell PhCs.
a Band degeneracies are created at the center (Γ) and corner (M) of the folded Brilluion zone for a 2 × 2 super-cell PhC (inset). An external magnetic field (B) induces gyromagnetic responses in YIG rods (gray squares), which breaks T but preserves C4. The gyromagnetic response κ = 12.4μ0 is used in the calculations. b, c As the rods simultaneously move inward or outward, the second gap is opened (shaded in yellow), but with different quadrupole phases: inward (outward) displacements with negative (positive) d correspond to trivial (non-trivial) quadrupole phases. d, e Ez mode profiles for the second and third modes at M feature a band inversion between the modes with C4 index of +i (green) and −i (red) in the trivial and non-trivial phases. f Complete topological phase diagram of the PhCs, including quadrupole phases (green), trivial phases (orange), and Chern insulating phases (purple). Systems are gapless along the axes (red). NI normal insulator, QTI quadrupole insulator, CI Chern insulator.
