Topological Subspace-Induced Bound State in the Continuum

We propose and experimentally realize a new kind of bound states in the continuum (BICs) in a class of systems constructed by coupling multiple identical one-dimensional chains, each with inversion symmetry. In such systems, a specific separation of the Hilbert space into a topological and a nontopological subspace exists. Bulk-boundary correspondence in the topological subspace guarantees the existence of a localized interface state which can lie in the continuum of extended states in the nontopological subspace, forming a BIC. Such a topological BIC is observed experimentally in a system consisting of coupled acoustic resonators.

Figure 1

(a) Two identical SSH chains coupled by a coupler atom per unit cell. The hopping parameters and indices of the atom orbitals are labeled. The on-site energies are ${\epsilon }_1={\epsilon }_2={\epsilon }_3={\epsilon }_4=0$ and ${\epsilon }_5$. (b) A unit cell consisting of coupled acoustic cavities. (c) Photos of the coupled acoustic cavities with $\beta =0$ for the measurement of the topological interface state in the midgap. The regions encircled by a red rectangle are the interface. (d) and (e) show the virtual lattices in the topological and the nontopological subspace, respectively.

Figure 2

Solid curves (left $y$ axis) in (a)–(c) show the band inversion and quantized Zak phases for the 2nd and 4th bands (blue) when hopping parameters are changed from (a) $|t|>|s|$ with $t=-55.7$, $s=-13.6$, through (b) the transition point with $t=s=-34$, to (c) $|t|<|s|$ with $t=-13.6$, $s=-55.7$. Zak phases for the three bands labeled by red are not quantized. The same values are chosen for other parameters with ${ϵ}_5=110.2$, $\beta =0$, and $r_1=r_2=-33.7$. (d) Solid curves show the band structure with parameters $t=-13.6$, $s=-56.9$, $r_1=r_2=-26.5$, ${ϵ}_5=-22$, and $\beta =13.5$. All the parameters above are obtained by fitting to comsol calculated bands marked by circles (right $y$ axis).

Figure 3

Interface structures and their eigenspectra for $\beta =0$ [(a) and (b)] and $\beta \ne 0$ [(c) and (d)]. There are three sites at the interface and ten cells on each side in (a) and (c). The odd (even) interface modes are marked by blue (red) dots in (c) and (d). The topological (odd) interface mode is inside the gap in (c) and inside the continuum in (d).

Figure 4

(a),(b) The response spectra with an (a) antiphase and (b) an in-phase excitation. The green arrow in (a) marks the topological BIC at 2190 Hz. The gray shaded region in (a) denotes the pressure response without an interface. (c),(d) The field distributions of the (c) odd interface state and (d) the even extended state at 2190 Hz. The green line in (c) shows the tight-binding fitting result. The insets in (c) and (d) show the phase difference of pressure responses on the two SSH chains, which demonstrates the odd and the even parity of the excited modes, respectively.