Bound states in the continuum of higher-order topological insulators

We show that lattices with higher-order topology can support corner-localized bound states in the continuum (BICs). We propose a method for the direct identification of BICs in condensed matter settings and use it to demonstrate the existence of BICs in a concrete lattice model. Although the onset for these states is given by corner-induced filling anomalies in certain topological crystalline phases, additional symmetries are required to protect the BICs from hybridizing with their degenerate bulk states. We demonstrate the protection mechanism for BICs in this model and show how breaking this mechanism transforms the BICs into higher-order topological resonances. Our work shows that topological states arising from the bulk-boundary correspondence in topological phases are more robust than previously expected, expanding the search space for crystalline topological phases to include those with boundary-localized BICs or resonances.

Figure 1

Characterization of the model lattice with closed and open boundaries. (a) Lattice with Bloch Hamiltonian in Eq. (1). (b) Bulk energy bands along high-symmetry lines of the Brillouin zone for a lattice with periodic boundaries, for $t=0.5$. (c) Density of states when boundaries are open in both directions ($n=20$, $t=0.15$). The lower panels indicate the probability densities per band ($n=5$, $t=0.15$).

Figure 2

Probing the existence of bound states in the continuum by adding the non-Hermitian term, Eq. (2), to the lattice in Fig. 1 in the topological phase. (a) Complex energies. (b) Imaginary component of the energies as a function of system size (the inset shows the shapes of the “system” and “environment” regions in gray and purple, respectively). In (a) and (b), the red open circle is the fourfold degenerate energy of the bound states in the continuum with support at the corners, and the blue solid circles have eigenstates with support in bulk or edges. (c), (d) Probability density function of (c) the BICs and (d) the bulk states at zero real energy. In (c) and (d), the area of the circles is proportional to the amplitude $\left|\psi \right|$ of the states. In (a), (c), and (d), $n=16$ unit cells, $n_s=3$ unit cells. In (b), $n=32$. In all plots, $\kappa =-5\times {10}^{-2}$ and $t=0.25$.

Figure 3

Complex energies in the lattice as a function of the hopping amplitude $t$. Shaded and nonshaded regions correspond to the trivial and topological phases, respectively. (a) Real component of the energies. (b) Imaginary component of the energies. Both plots show the overlapped spectra of three configurations: periodic boundaries in both directions (blue), periodic boundary only along one direction (purple), and open boundaries in both directions (red). Blue spectra are on top of purple spectra, both of which are on top of red spectra. For all plots, $n=16$, $n_s=3$, $\kappa =-5\times {10}^{-2}$.

Figure 4

Breaking the symmetries that protect the BICs. Energy eigenvalues (left panels) and probability densities of the four states whose energies have their imaginary components closest to zero (right panels) under perturbations that preserve certain symmetries: (a) $C_{4v}$ and chiral symmetries, (b) only chiral symmetry, (c) only $C_{4v}$ symmetry, and (d) $C_4$ and chiral symmetries. In the energy plots, the red open circles correspond to the four energies with imaginary components closest to zero (possibly degenerate). Only (a) has BICs; (b), (c), and (d) have corner-localized topological resonances. For all plots, $n=16$, $n_s=3$, $\kappa =-5\times {10}^{-2}$, and $t=0.25$.