Robustness of topological corner modes in photonic crystals

We analyze the robustness of corner modes in topological photonic crystals, taking a $C_6$-symmetric breathing honeycomb photonic crystal as an example. First, we employ topological quantum chemistry and Wilson loop calculations to demonstrate that the topological properties of the bulk crystal stem from an obstructed atomic limit phase. We then characterize the topological corner modes emerging within the gapped edge modes employing a semianalytical model, determining the appropriate real-space topological invariants. We provide a detailed account of the effect of long-range interactions on the topological modes in photonic crystals, and we quantify their robustness to perturbations. We conclude that, while photonic long-range interactions inevitably break chiral symmetry, the system is reducible to a chirally symmetric limit and the corner modes are protected by this together with lattice symmetries.

Figure 1

(a) Unit cells of the bulk lattice in the contracted and expanded phases, characterized by a contraction/expansion parameter, $\delta$. The relevant Wyckoff positions are labeled 1a (black circle) and 3c (red star). (b) Band structure of the silicon photonic crystal in the expanded phase for the TM modes. The expansion coefficient $\delta$ is 0.11, and the radius of the cylinders is $0.12a_0$, $a_0$ being the lattice constant of the crystal. (c) Wilson loops for bands 4–6 (Wilson loops for bands 1–3 are similar [38]).

Figure 2

(a) Scheme for the topological particle supercell. (b) Particle lattice, with sublattices $a$ (green) and $b$ (purple). (c), (d) Modes of a photonic crystal particle: Frequency ($\omega$) of topological corner (red), edge (cyan), and bulk (gray) states (c), and displacement field plots, showing $D_z$ (d). (e), (f) Quasistatic model of the topological particle. (e) Frequency of topological corner, edge, and bulk states, for silver nanoparticles with radius 10 nm and height 40 nm. (f) Dipole moments of the six corner eigenmodes. In the color scale used in (d) and (f) red (blue) represents positive (negative) values. In both cases, $\delta =0.11$.

Figure 3

Topological particle eigenvalues. (a) Evolution with increasing unit cell perturbation $\delta$, with topological corner modes (red) well separated from the edge (cyan) and bulk (gray) modes. Other corner modes are shown in magenta. (b) Dependence on interaction length between lattice sites, $\gamma$, for $\delta =0.2$. As interactions go from nearest neighbors $\gamma =0.1$ to long range, chiral symmetry is broken and the spectrum is no longer symmetrical about $E=0$.

Figure 4

Robustness of corner states against defects and disorder in the quasistatic model, for $\delta =0.2$. (a) A $C_6$-symmetry breaking defect in the lattice affects the midgap corner modes. (b) The corner states are robust against another kind of $C_6$-symmetry breaking defect due to the corner modes being close to chiral symmetric. (c) The degeneracy of the corner modes is lifted by random disorder: the position of the lattice sites is shifted randomly up to 5%. (d) Eigenvalue spectrum for increasing positional disorder.