Tailoring higher-order topological phases via orbital hybridization

Higher-order topological insulators (HOTIs) have attracted much attention in photonics due to the tightly localized disorder-robust corner and hinge states. Here, we reveal an unconventional HOTI phase with vanishing dipole and quadrupole polarizations. This phase arises in the array of evanescently coupled waveguides hosting degenerate $s$- and $d$-type orbital modes arranged in a square lattice with four waveguides in the unit cell. As we prove, the degeneracy of the modes with the different symmetry gives rise to the nontrivial topological properties rendering the system topologically equivalent to the two copies of the anisotropic two-dimensional Su-Schrieffer-Heeger model rotated by ${90}^{\circ }$ with respect to each other and based on $s\pm d$ hybridized orbitals. We probe the unusual topology of the model by constructing a quantized Fu-Kane-type pseudospin pump. Our results introduce a route to tailor higher-order band topology leveraging both crystalline symmetries and accidental degeneracies of the different orbital modes.

Figure 1

(a) Two-dimensional Su-Schrieffer-Heeger lattice with degenerate monopolar (${\mathrm{TE}}_{0m}$; left) and quadrupolar (${\mathrm{HE}}_{2k}$; right) modes at each site. Arrows indicate the direction of electric field in the respective modes. (b) Bulk band structure with a complete gap highlighted by green; $\pm$ signs indicate inversion eigenvalues at high-symmetry points. The calculation is performed for $\kappa =\gamma ,\mathrm{\Delta }=0.7\kappa , \alpha =0.3$.

Figure 2

Equivalent two-layer model of our system. Each layer corresponds to the anisotropic 2D SSH model with the couplings indicated in the legend for the $s\pm d$ hybrid orbitals with the field profiles depicted schematically on the left of each layer.

Figure 3

(a) Charge distribution at half filling calculated for the finite lattice $14\times 14$ sites with the same parameters as in Fig. 1. Corner charges $Q_{\mathrm{corner}}=\pm 1$ are obtained by the summation over several sites nearest to the corner. In this calculation, on-site energies of the diagonal sites at each unit cell were set to small $\delta =\pm {10}^{-2}\kappa$ as proposed in Ref. [5]. (b) Spectrum of the same finite lattice. The logarithm of inverse participation ratio is color-coded. Zoomed area shows 8-fold degenerate midgap corner states. (c) Real-space profile for one of the corner modes. Two circles in each waveguide indicate the amplitudes of the two modes, and the phase is color-coded. (d) The map of longitudinal component $E_z$ of electric field for the corner states with propagation constants $245\mathrm{rad}/\mathrm{m}$ obtained from the full-wave numerical simulations. Each waveguide [35] hosts ${\mathrm{TM}}_{02}$ and ${\mathrm{HE}}_{22}$ degenerate modes, and has radius $1.26\text{cm}$ and permittivity $\varepsilon =9.9$ corresponding to commercially available ceramic aluminium oxide. The center-to-center nearest-neighbor distance in expanded unit cell is $3.6\text{cm}$, and periods in $x$ and $y$ directions are $6.44\text{cm}$. The calculations are performed for a fixed frequency $f=8.83\text{GHz}$ using COMSOL Multiphysics software package.

Figure 4

Spectrum of the Fu-Kane-like 2D pseudospin pump based on $s-d$ hybridized $50\times 50$ sites lattice versus modulation time. Arrows show the sign of the average pseudospin for the corner-localized states [35]. Color of the arrows encodes the position of the corner mode as illustrated in inset.