Corner states and topological transitions in two-dimensional higher-order topological sonic crystals with inversion symmetry

Macroscopic two-dimensional (2D) sonic crystals with inversion symmetry are studied to reveal higher-order topological physics in classical wave systems. By tuning a single geometry parameter, the band topology of the bulk and the edges can be controlled simultaneously. The bulk band gap forms an acoustic analog of topological crystalline insulators with edge states which are gapped due to symmetry reduction on the edges. In the presence of mirror symmetry, the band topology of the edge states can be characterized by the Zak phase, illustrating the band topology in a hierarchy of dimensions, which is the key feature of higher-order topological phenomena. Moreover, the edge bands undergo topological transitions without closing the bulk band gap, revealing rich topological physics in both the bulk and edges. We characterize the higher-order topological bands using symmetry indicators in the Brillouin zone and link the symmetry indicators with the corner charge—a corner topological index. This approach allows us to identify higher-order topological boundary states through symmetry properties of the Bloch bands. We then generalize the higher-order topological phenomena protected by the inversion symmetry to arch-shaped sonic crystals with nonsymmorphic symmetries where similar but slightly different topological phenomena are revealed. In these systems, the rich, multidimensional topological transitions can be exploited for topological transfer among 0D (corner), 1D (edge), and 2D (bulk) acoustic modes by tuning the geometry of acoustic metamaterials.

Figure 1

(a) Schematic of conventional 2D TCIs with gapless edge states. (b) Schematic of 2D HOTIs as special types of TCIs which support gapped edge states and in-gap corner states. The edge boundaries are represented by the black thick lines, while the edge states are depicted by the wavy lines. The counterpropagating edge states are the consequence of the time-reversal symmetry. Possible dispersions of the gapless and gapped edge states are schematically illustrated in the figures as well. The corner states localized at the corner boundaries (i.e., the intersection points between the edges) are depicted in (b) as solid circles.

Figure 2

(a) Schematic of the unit cell of the airborne SC. The four bar-shaped scatterers (dark blue) are made of epoxy with a tunable rotation angle $\theta$. The unit cell adopted here is not the primitive unit cell. The primitive unit cell is depicted by the magenta dashed rectangle. (b) Acoustic bands and parity eigenvalues ($+$ and $-$, for even and odd parities, respectively) of the bar-shaped SCs with different rotation angles $\theta =-{40}^{\circ }$, $0^{\circ }$, and ${60}^{\circ }$. The red and blue dots represent the even- and odd-parity eigenstates at the $M$ point, respectively. (c) The band edge frequencies at the $M$ point as functions of the rotation angle $\theta$. Insets are the acoustic pressure profiles in the unit cell for $\theta =-{40}^{\circ }$ and ${60}^{\circ }$ for the two topologically distinct phases, $\mathrm{TC}{\mathrm{I}}_{\alpha }$ and $\mathrm{TC}{\mathrm{I}}_{\beta }$, separately.

Figure 3

(a), (b) Calculated spectra of the edge boundaries between the bar-shaped SCs with ${\theta }_1=-{40}^{\circ }$ and ${\theta }_2={60}^{\circ }$ for the edges along the (a) $x$ and (b) $y$ directions, respectively. (c), (d) Wave functions of the edge states at $k_{x/y}=\pi /a$ and the edge states at frequencies $f_1$ and $f_2$. For $k_{x/y}=\pi /a$, the edge states are either even parity or odd parity (the dashed lines indicate the mirror plane). For the edge states at frequencies $f_1$ and $f_2$, the edge states are either pseudospin up or pseudospin down. The yellow arrows in the acoustic pressure phase profiles, $\arg \left(\psi \right)$, indicate the winding directions of the phase vortices (the yellow dots indicate the vortex centers). The green arrows indicate the momentum (“Poynting vector”) distributions of the acoustic edge states. (e) Calculated spectrum for the box-shaped structure (depicted in the inset) with both edge and corner boundaries between the $\mathrm{TC}{\mathrm{I}}_{\alpha }$ with ${\theta }_1=-{40}^{\circ }$ and $\mathrm{TC}{\mathrm{I}}_{\beta }$ with ${\theta }_2={60}^{\circ }$. The spectrum of the four degenerate corner states is indicated by the red dots. (f) Simulated acoustic pressure profile for one of the corner states. The green dashed lines indicate the edge boundaries.

Figure 4

Three different pictures for the understanding of the emergence of topological corner states. (a) Corner states emerging as Jackiw-Rebbi solitons at the corner boundary between massive-Dirac edge states with opposite signs of Dirac mass, e.g., $m_y>0$ and $m_x<0$. (b) Emergence of corner states as indicated by the nontrivial corner charge as from counting the contributions from the edge polarizations, i.e., $p_x^{\mathrm{edge}x}=1/2$ and $p_y^{\mathrm{edge}y}=0$, thus $q_{\mathrm{corner}}=1/2$. (c) Emergence of corner states due to the nontrivial corner topological index. The corner topological index from the symmetry representations for the $\mathrm{TC}{\mathrm{I}}_{\alpha }$ is $Q_c^{\alpha }=1/2$, whereas for the $\mathrm{TC}{\mathrm{I}}_{\beta }$ it is $Q_c^{\beta }=0$. The nontrivial difference, $\mathrm{\Delta }{Q}_c=Q_c^{\alpha }-Q_c^{\beta }$, indicates the emergence of topological corner states.

Figure 5

Robustness of the corner states between the bar-shaped SCs with ${\theta }_1=-{40}^{\circ }$ and ${\theta }_2={60}^{\circ }$. Wave functions (hot color) and eigenfrequencies (inset) of the corner states when the bar-shaped epoxy block in the lower left corner of the unit cell marked by a green square is replaced by (a) an epoxy cylinder (dashed circle), (b) a shifted epoxy block (dashed block), and (c) a rotated epoxy block (dashed block). The insets at the upper left show the detailed structure of the deformed unit cell (magenta) and the undeformed unit cell (white).

Figure 6

Calculated topological transitions in a hierarchy of dimensions when ${\theta }_1=-{40}^{\circ }$. (a) Topological “phase diagram” for the edge states propagating along the $x$ direction as represented by the frequencies of the odd and even modes at $k_x=\pi /a$. The transition occurs at the parity inversion (i.e., gap closing) point where the odd and even modes become degenerate. (b) Illustration of the glide symmetry at the edge boundary along the $x$ direction when ${\theta }_2=-{\theta }_1$. (c) Gap-closing edge states along the $x$ direction for the case with ${\theta }_2=-{\theta }_1={40}^{\circ }$. (d) Topological “phase diagram” for the edge states propagating along the $y$ direction as represented by the frequencies of the odd and even modes at $k_y=\pi /a$. (e) Illustration of the glide symmetry at the edge boundary along the $y$ direction when ${\theta }_2=-{\theta }_1$. (f) Gap-closing edge states along the $y$ direction for the case with ${\theta }_2=-{\theta }_1={40}^{\circ }$.

Figure 7

(a) Illustration of the structure with edge and corner boundaries between $\mathrm{TC}{\mathrm{I}}_{\alpha }$ with ${\theta }_1=-{40}^{\circ }$ and $\mathrm{TC}{\mathrm{I}}_{\beta }$ with $0^{\circ }<{\theta }_2<{90}^{\circ }$. Different colors represent different regions for the calculation of the local density of states (DOS). The DOS for the corner, edge, and bulk are obtained by integration of local DOS over the corner, edge, and bulk regions, respectively. There are four corner (edge) regions. Each corner region contains $2\times 2$ unit cells, while each edge region contains $2\times 10$ unit cells. (b)–(d) Local DOS for the corner, edge, and bulk regions for ${\theta }_2={26}^{\circ },{40}^{\circ },$ and ${70}^{\circ }$, respectively. The corner resonances are indicated by the black arrows.

Figure 8

Schematics of the unit cell of the arch-shaped SCs. There are four arch-shaped epoxy blocks in each unit cell. The geometry parameters are $l=0.35a,w=0.1a,$ and $a=2$ cm, and the tunable rotation angle is $\theta$. (b) Bulk energy bands of arch-shaped SCs with $\theta =-{40}^{\circ }$, $0^{\circ },$ and ${65}^{\circ }$. The red symbols $+$ and $\text{–}$ denote, respectively, the eigenvalues 1 and −1 of the $C_2$ symmetry at the HSPs of the Brillouin zone. (c) Band edge frequencies of the first four bulk bands at the M point vs the rotation angle $\theta$ in the regime from $-{90}^{\circ }$ to ${90}^{\circ }$. The insets in the blue and red regions indicate the acoustic pressure profiles at the M point for the cases with $\theta =-{40}^{\circ }$ and ${65}^{\circ }$, respectively. (d) Evolution of the band edge frequencies of the first four bulk bands at the M point with the rotation angle $\theta$ in the regime from ${90}^{\circ }$ to ${270}^{\circ }$.

Figure 9

(a), (b) Simulated spectra of the edge boundaries between the arch-shaped SCs with ${\theta }_1=-{40}^{\circ }$ and ${\theta }_2={65}^{\circ }$ for the edges along the (a) $x$ and (b) $y$ directions, respectively. (c), (d) Phase and Poynting vectors (green arrows) of the edge states at frequencies $f_3$ and $f_4$. The yellow arrows indicate the winding directions of the phase vortices (the yellow dots indicate the vortex centers). (e) Simulated spectrum for the finite-sized structure (depicted in the inset) with both edge and corner boundaries between the SCs with ${\theta }_1=-{40}^{\circ }$ and ${\theta }_2={65}^{\circ }$. The spectrum of the four corner states is indicated by the red dots. (f) Simulated acoustic pressure profile of one corner state. The green dashed lines indicate the edge boundaries.

Figure 10

Robustness of the corner state between the arch-shaped SCs with ${\theta }_1=-{40}^{\circ }$ and ${\theta }_2={65}^{\circ }$ against deformations. Wave functions (hot color) and eigenfrequencies (inset) of the corner state when the arch-shaped epoxy block in the lower-left corner of the unit cell marked by the green square is replaced by (a) a circle epoxy block, (b) a shifted epoxy block, and (c) a rotated epoxy block (rotation angle ${50}^{\circ }$). The insets at the upper left show the structure of the deformed unit cell (magenta) and the undeformed unit cell (white).

Figure 11

(a) Complete band gap for the edge states propagating along the $x$ direction as represented by the frequencies of the minimum of the upper edge band and maximum of the lower edge band. (b) Band structure of the edge boundaries along the $x$ direction for the arch-shaped SCs when ${\theta }_2={22}^{\circ },{40}^{\circ },$ and ${55}^{\circ }$. (c) Illustration of the absence of the glide symmetry at the edge boundary along the $x$ direction when ${\theta }_2=-{\theta }_1$. (d) Complete band gap for the edge states propagating along the $y$ direction as represented by the frequencies of the minimum of the upper edge band and maximum of the lower edge band. (e) Dispersions of the edge boundaries along the $y$ direction for the arch-shaped SCs when ${\theta }_2={22}^{\circ },{40}^{\circ },$ and ${55}^{\circ }$. (f) Illustration of the glide symmetry at the edge boundary along the $y$ direction when ${\theta }_2=-{\theta }_1$.

Figure 12

(a) Illustration of the finite-sized structure with edge and corner boundaries between the arch-shaped SCs with ${\theta }_1=-{40}^{\circ }$ and $0^{\circ }<{\theta }_2<{90}^{\circ }$. Different colors represent different regions for the calculation of the local density of states (DOS). The DOS for the corner, edge, and bulk are obtained by integration of local DOS over the corner, edge, and bulk regions, respectively. The finite-sized structure consists of $12\times 12\mathrm{TC}{\mathrm{I}}_{\beta }$ unit cells inside and four layers of $\mathrm{TC}{\mathrm{I}}_{\alpha }$ unit cells outside. There are four corner (edge) regions. Each corner region contains $2\times 2$ unit cells, while each edge region contains $2\times 10$ unit cells. In the insets on the left side, the white blocks represent the location of the epoxy blocks in arch-shaped SCs. (b)–(d) Local DOS for the corner, edge, and bulk regions for ${\theta }_2={22}^{\circ },{40}^{\circ },$ and ${55}^{\circ }$, respectively. The corner resonances are indicated by the black arrows.