Non-Hermitian Sonic Second-Order Topological Insulator

Topological phases of matter that have been recently extended to topological phases of sound can confine acoustic energy at the corners of higher-order topological insulators. We broaden this concept by incorporating parity-time symmetry and show new topologically protected confinement rules that are dictated by the geometrical arrangement of gain and loss units. Particularly, our findings reveal how sound trapping occurs at all corners when parity-time symmetry is intact, beyond the exceptional point within the broken phase; however, opposite corners sustain either sink- or sourcelike states that could lead to novel non-Hermitian guides for sound.

Figure 1

(a) Schematic illustration of an unperturbed fluid SC. (b) Dispersion relation for a structure with ratio $D/a=0.5$. Colored curves display the results calculated with the FEM and the circles represent the bands obtained by the PWE method. Inset: the first Brillouin zone (BZ). Band inversions induced by tuning the ratio from (c) $D/a<0.5$ (shrunk metamolecule) to (d) $D/a>0.5$ (expanded metamolecule). The “$+/-$” signs represent the even and odd parity at the high symmetry points before and after the band inversions. Insets: eigenmodes at the $X$ point from the lowest to the highest band.

Figure 2

Simulated (a) real and (b) imaginary eigenfrequencies of the Hermitian CSC. The crystal is made of a $8a\times 8a$ metamolecule array defining the nontrivial region with $D/a=0.714$, which is enclosed by a trivial region ($D/a=0.280$) of thickness $14a$. Black dots, circles, blue dots, and red dots denote the bulk, edge, pseudohinge and corner states, respectively. Acoustic eigenmode profiles of the (c) pseudohinge and (d) corner states.

Figure 3

CSC with parallel non-Hermiticity. (a) Imaginary band diagram of the sonic modes in the presence of parallel non-Hermiticity. Colored curves display the results calculated with the FEM and the circles represent the bands obtained by the PWE method. Inset: the metamolecule with parallel arrangement of gain and loss components. (b) Zoom of the CSC displaying the arrangement of the trivial and nontrivial regions. Simulated (c) real and (d) imaginary eigenfrequencies for the parallel non-Hermiticity CSC. The eigenfrequencies of the second (${\mathrm{pH}}_2$) and third (${\mathrm{pH}}_3$) pseudohinge states and all four corner states ($C_1$, $C_2$, $C_3$, and $C_4$) become complex conjugate pairs. Acoustic eigenmodes profiles of (e) pseudohinge and (f) corner states.

Figure 4

CSC with diagonal non-Hermiticity. (a) Imaginary dispersion relations. (b) Enlargement of the CSC displaying the arrangement of the trivial and nontrivial regions. Simulated (c) real and (d) imaginary eigenfrequencies for the diagonal non-Hermiticity crystal. The eigenfrequencies of four corner states ($C_1$, $C_2$, $C_3$, and $C_4$) become complex conjugate. (e) Acoustic eigenmode profile of corner states.

Figure 5

$\mathcal{P}\mathcal{T}$-symmetry phase diagram. The complex (a) pseudohinge and (b) corner states with parallel non-Hermiticity, and (c) the corner states with diagonal non-Hermiticity are calculated with varying $\beta$. (d) Enlarged view of the $\mathcal{P}\mathcal{T}$-symmetry breaking transition of $C_3$ and $C_4$ within the black dashed frame in (c).