Twist-projected two-dimensional acoustic topological insulators

Acoustic analogs of electronic or photonic topological insulators provide unique approaches to manipulate sound wave propagation. Inspired by twist-induced topological photonic insulators, here we propose a type of two-dimensional acoustic topological insulator (TI) via projecting a section of a three-dimensional twisting structure to a plane, assembling the projected meta-atoms into metamolecules, and arranging the metamolecules into unit cells to form a honeycomb lattice. It follows that in this acoustic TI, topological phases mimic pseudospin-up and pseudospin-down states, and the pseudospin-orbital couplings are tuned via changing the rotation angles of the meta-atoms, which eventually leads to band inversion. By calculating acoustic band structures, pressure field distributions, and spin Chern numbers of bands, we verify that the topological phase transition occurs around the double Dirac cone and present the topological phase diagram as a function of the rotation angle of the meta-atoms. Once the coupling between adjacent metamolecules is sufficiently strong, mode inversion of topological states emerges. Furthermore, we numerically demonstrate the existence of topologically protected edge states. It is shown that robust pseudospin-dependent one-way transmission is immune to defects at the edge of topological distinct regions, which can be applied to acoustic wave transmissions and communications. Our approach in acoustic systems provides a strategy to explore abundant topological states in two-dimensional systems.

Figure 1

(a) Schematic of the twisted wire bundle and its projections to $xy$ plane. By projecting a bundle of six helical wires with a finite height to the $xy$ plane, we obtain six meta-atoms with $C_6$ symmetry, which are assembled into a metamolecule. Obviously, the rotation angle $\theta$ of meta-atoms changes in the metamolecule if the center of the projected wire section moves on the $z$ axis, as marked in the plot. For example, the rotation angles of meta-atoms become distinct in two different regions. (b) Schematic of the metamolecules assembled into a honeycomb lattice. The meta-atoms shown in (a) are simplified to the elliptical shapes, and the metamolecule with six meta-atoms is used as a unit cell in a 2D honeycomb lattice. The lattice constant is $a$ with unit vectors ${\vec{a}}_1$ and ${\vec{a}}_2$. The separation between centers of neighboring meta-atoms in the two adjacent metamolecules is $d=a/3$, and the nearest separation between these two meta-atoms along the $x$ axis is $l$. The inset shows an enlarged view of the metamolecule and the meta-atom, where $\theta$ is the rotation angle of the meta-atoms in the metamolecule. The semimajor axis and the semiminor axis are $b_1$ and $b_2$.

Figure 2

The calculated acoustic band structure of the 2D metasystems with different rotation angles $\theta$ of meta-atoms. (a) The band structure of the metastructure type A ($\theta ={90}^{\circ }$), where a trivial band gap occurs at the $\mathrm{\Gamma }$ point. (b) The band structure of the metastructure type C ($\theta =44.3^{\circ }$), where a double Dirac cone emerges at the $\mathrm{\Gamma }$ point. (c) The band structure of the metastructure type B ($\theta =0^{\circ }$), where an inverted nontrivial band gap appears at the $\mathrm{\Gamma }$ point. The inset shows the corresponding unit cell and the state occupation below and above the gap at the $\mathrm{\Gamma }$ point. In all three metastructures, the air meta-atoms embed in a water background, and the geometrical parameters are taken to be $a=3\text{cm}, b_1=0.35\text{cm}, b_2=0.23\text{cm}$.

Figure 3

The pressure field distributions at the $\mathrm{\Gamma }$ point are shown for scenarios of type A and type B in Figs. 2 and 2. In all cases, the pressure field possesses $p_x\left(p_y\right)$ and $d_{xy}\left(d_{x^2-y^2}\right)$ orbitals, which are separated by a bulk band gap. For the scenario of type A (left, with $\theta ={90}^{\circ }$), the two doubly degenerate acoustic states denoted by $d_{xy}\left(d_{x^2-y^2}\right)$ and $p_x\left(p_y\right)$ orbitals appear at (a) the higher frequency and (c) the lower frequency, respectively; for the scenario of type B (right, with $\theta =0^{\circ }$), two inverted doubly degenerate acoustic states appear at (b) higher frequency and (d) lower frequency, corresponding to $p_x\left(p_y\right)$ and $d_{xy}\left(d_{x^2-y^2}\right)$ orbitals, respectively. This feature demonstrates clearly the mode inversion in the system.

Figure 4

The acoustic energy flow at the $\mathrm{\Gamma }$ point for type A and type B in Figs. 2 and 2. For the scenario of type A (left, with $\theta ={90}^{\circ }$), the pseudospin states denoted by the pressure field distributions with positive and negative angular momenta $d_{\pm }=(d_{x^2-y^2}\pm i{d}_{xy})$ and $p_{\pm }=(p_x\pm i{p}_y)$ appear at (a) the higher frequency and (c) the lower frequency, respectively. For the scenario of type B (right, with $\theta =0^{\circ }$), the pseudospin states denoted by the pressure field distributions with $p_{\pm }=(p_x\pm i{p}_y)$ and $d_{\pm }=(d_{x^2-y^2}\pm i{d}_{xy})$ appear at (b) the higher frequency and (d) the lower frequency, respectively.

Figure 5

The calculated Berry curvatures of the pseudospin states $p_{\pm }$ and $d_{\pm }$ around the $\mathrm{\Gamma }$ point for (a) type A, and (b) type B. By integrating Berry curvature around the $\mathrm{\Gamma }$ point in the first Brillouin zone, we obtain the spin Chern numbers $C_s$ for the $p_{\pm }$ and $d_{\pm }$ states. It is shown that $C_s=0$ for those $p_{\pm }$ and $d_{\pm }$ states in type A, indicating the topological trivial phase, while $C_s=+1$ for both $p_{+}$ and $d_{+}$ states and $C_s=-1$ for both $p_{-}$ and $d_{-}$ states in type B, indicating the topological nontrivial phase.

Figure 6

(a) The topological phase diagram of the metastructures as a function of the rotation angle $\theta$ of the meta-atoms, where the blue (red) curve represents the frequency of the $d_{\pm } \left(p_{\pm }\right)$ states at the $\mathrm{\Gamma }$ point. The trivial and nontrivial regions are marked with different colors. (b) The rotation angle $\theta$ of the meta-atoms corresponding to the double Dirac cone as a function of ellipticity $\gamma$ of the elliptic meta-atom. (c) The frequencies corresponding to the $p_{\pm }$ or $d_{\pm }$ states at the $\mathrm{\Gamma }$ point as a function of the ellipticity $\gamma$ of the meta-atom for the scenarios of type A ($\theta ={90}^{\circ }$, blue curves) and type B ($\theta =0^{\circ }$, red curves).

Figure 7

(a) The schematic of a ribbon-shaped supercell, which is composed of 12 metamolecules for the topological trivial regions (type A) on the left and the right and 15 metamolecules for the topological nontrivial region (type B) in the middle. (b) The acoustic band structure of the metastructure with the ribbon-shaped supercells. The gray regions represent the bulk modes, whereas the doubly degenerate topological edge modes appear in the bulk band gap, as marked by the blue and red curves, respectively. The right panel shows an enlarged view of the acoustic band structure around the bulk band gap. (c) The real-space distributions of the pressure field and the energy flow in the metastructure with the supercells at typical momenta of points A and B in (b), respectively. The rainbow colors represent the amplitude of the pressure field, and the arrows show the direction and amplitude of the energy flow. The upper graph shows the typical momentum of point A in (b); the left (right) edge supports an upward edge mode with pseudospin down (pseudospin up). The lower graph shows the typical momentum of point B in (b); the left (right) edge supports a downward edge mode with pseudospin up (pseudospin down). The middle graph schematically shows four sound transmission channels at the left and right edges, where blue (red) arrows represent the pseudospin up (pseudospin down) and the transport direction of the corresponding pseudospin.

Figure 8

(a) The schematic of a pseudospin splitter for an acoustic wave. Black dashed lines denote the cross edges in between the topological trivial regions (type A, in gray, $15a\times 15a$ for each region) and the topological nontrivial regions (type B, in purple, $15a\times 15a$ for each region). There are four edges in total, i.e., the left, right, up, and down edges, which are marked L, R, U, and D, respectively. (b) The calculated pressure intensity distribution when the pseudospin-up source of 920 Hz is placed at the splitter center. It is shown that the pseudospin-up propagates only along the up and down edges. (c) The calculated pressure intensity distribution when the pseudospin-down source of 920 Hz is placed at the splitter center. It is shown that the pseudospin-down propagates only along the left and right edges. (d)–(g) The acoustic one-way transmission of pseudospin down and the robustness against defects in the metastructures, with a $15a\times 24a$ topological trivial region (type A, gray) beside a $15a\times 24a$ topological nontrivial region (type B, purple). The left panels show only the region near the edge between these two regions; the right panels show the calculated pressure intensity distribution when the pseudospin-down source is placed at the bottom of the edge. The metastructure is shown (d) without any defects, (e) with meta-atomic disorder, (f) with local cavity, and (g) with bending. The pressure intensity distributions in (e)–(g) show unambiguous robust sound transport against meta-atomic disorder, local cavity, and bending, respectively. The green (orange) star represents the pseudospin-up (pseudospin-down) source.

Figure 9

The simulated transmission spectra of the sound waves along different edges in the metastructures shown in Figs. 8–8. The black curves correspond to the scenario without any defects, while (a) the red, (b) blue, and (c) green curves indicate the scenario with local cavity, meta-atomic disorder, and bends, respectively. The shaded regions represent the bulk bands, and the white regions represent the bulk band gaps. It is obvious that the transmissions through the edges are relatively high within the bulk band gap, and they are nearly unaffected by the defects, especially around the center of the band gap.