Fig. 3: Topological phase transition in the synthetic PHOTI lattice by tuning the phase of the modulation.

a Unit cell consisting of two photonic molecules with a phase difference between the modulations of ϕ [Eq. (1)]. b Energy eigenspectrum for a finite lattice with open boundaries in both real dimension x and synthetic frequency dimension m for various phases ϕ, with γ/λ = 0.20. For ϕ = π, there is a bandgap that hosts topologically protected corner modes pinned to zero energy. For ϕ = 0, no such bandgap exists at zero energy, as confirmed by the bulk band structure in d. c and d Bulk band structures for the quadrupole HOTI (ϕ = π) and the 2D SSH model (ϕ = 0), respectively. All energies are in units of λ. In c, both bands are doubly degenerate. e and f Energy eigenspectrum of an ideal finite lattice for ϕ = π and 0, respectively, without disorder in the couplings. Although corner modes exist in both cases, for ϕ = 0, they spectrally overlap with the bulk bands. g and h Energy eigenspectrum with normally distributed random disorder in the couplings with variance σ² = 0.04. Since the lattice with ϕ = π hosts a quantized bulk quadrupole moment, corner modes are visible in the bandgap in g, unlike in h. i and j Steady-state field distribution at the disordered synthetic lattice sites for an excitation with zero detuning Δω = 0 at the lowest frequency mode m = 0 for the leftmost ring, as indicated by the arrows. For the quadrupole PHOTI (ϕ = π) in the top row, the corner modes are still strongly localized in the presence of disorder. For the 2D SSH phase in the bottom row, disorder in the couplings makes the corner excitation not well localized, with leaking into the bulk. Specifically, the excitation preferentially propagates at ~±45° in the lattice³⁹ because the bands in d touch at zero energy along the k_x = k_y and k_x = −k_y lines. In c–j, γ/λ = 0.4