The discovery of the quantum spin Hall effect not only opened a new chapter in condensed matter physics^[1–5] but also brought the mathematical concept of topology into physical systems. Topological states have many special features,^[6–10] such as the edge states of unidirectional transmission and backscatter suppression that is insensitive to defects. Some researchers break the time-reversal symmetry of optoelectronic systems by applying a magnetic field to form topological states of unidirectional transmission on the edges of non-trivial and trivial structures.^[11–14] Photonic and phononic crystals (PCs) are highly similar to electron systems in terms of band structures and Bloch’s law, whereby they can also achieve topological states similar to quantum systems.^[15–17] An acoustic system belongs to the Bose subsystem, which is essentially different from an electromagnetic system. Therefore, it is impossible to break the time-reversal symmetry of an acoustic system using a magnetic field.

Accordingly, some researchers have mimicked the symmetry of a magnetic field in acoustic systems by introducing a rotating airflow or an acoustic pseudospin to form acoustic edge states.^[18,19] Some have used the classical graphene model of an electronic system to construct a two-dimensional structure with C_3v symmetry in an acoustic system, thus forming a degenerate Dirac cone at the corner of the Brillouin zone, and then rotated the scatterer. Reducing the symmetry of the structure to C₃ opens the Dirac cone to form a band gap.^[20–26] Lu et al. constructed a three-layer structure similar to a sandwich with multiple single-layer structures with different topological phases, realizing an edge state with topological valley properties.^[20] Xia et al. designed a photonic crystal that simultaneously exhibits the topological states of sound and light using cylindrical spacing of different diameters.^[27] By adjusting the impedance ratio between the cylindrical scatterer and the matrix in the honeycomb crystals, they found a Dirac cone and a double Dirac cone at different degenerate points of the structure. By reducing the symmetry of the structure (from C_6v to C_3v), the authors simultaneously implemented the topological edge transmission of sound and light.

Early studies focused on the corner point or center points of the Brillouin zone. Later, researchers discovered that a Dirac point can also exist at the edge of the Brillouin zone.^[28] For example, researchers found that increasing the ratio of the scatterer of a rectangular PC moves the Dirac point from the midpoint of the edge to the corner. Some have further proposed a doubly degenerate point in a square crystal and found a topological phase transition by adjusting the rotation angle of the scatterer.^[29] Then the author constructed a “back” shape acoustic model and demonstrated its topological edge state and topological angle state. However, no study has been done on the edge state properties of rectangular PCs with double dirac cone.

This paper models a PC structure with rectangular cells placed in water. A rectangular scatterer is placed at each of the four quarter positions of each cell, and the phononic band structure is studied by changing the following variables of each scatterer: its length L, width D, and rotation angle θ around its centroid. We find that when θ = 0, there is always a quadruply degenerate point at the corner M of the irreducible Brillouin zone, but its value changes as L and D change. Changing θ splits the four simple points and causes the system to undergo a topological phase transition. Then, using the finite-element simulation software Comsol, we find that the edge structure constructed by topologically trivial and nontrivial PCs is robust to right-angle, disorder, and cavity defects.

We study a two-dimensional crystal in water with rectangular cells, as shown in Fig. 1. The crystal constant in the x direction is a = 10 mm, and that in the y direction is b = 2a. An identical steel rectangular scatterer is placed at each of the four quarter positions of each unit: (a/4, a/2), (a/4, −a/2), (−a/4, −a/2), and (−a/4, a/2). When changing either L, D, or θ, the mirror symmetry of the entire cell is maintained. The cell basis vectors are and . The acoustic wave equation for propagation in such a system is

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (a) PC consisting of rectangular scatterers surrounded by water positioned in a rectangular lattice. (b) Cell of a rectangular PC. (c) Irreducible Brillouin zone of a cell.

The distance s from the center of the scatterer to its right-angle point is

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (a) Frequency of the quadruply degenerate state at the corner M of the Brillouin zone as a function of scatterer width D and length L. (b) When D = 0.1a, the frequency of M changes with L. (c) When L = 0.48a, the frequency of M changes with D.

Next, we will study two cases with different s of the scatterer, as shown by points A and B in Fig. 2(a). The rotation angle θ of case A is ± 180°, which is called T (transverse) type. The maximum rotation angle θ of case B is ± 31°, which is called V (vertical) type.

In order to study the dispersion relation near the point M, we transform the acoustic wave equation into an eigenvalue problem

Now we change the third scatterer variable θ, and find that the double Dirac cone appears open or closed as θ varies; that is, a topological phase transition of the structure occurs. Figure 3 shows the band structure and topological phases of the V- and T-type structures, revealing the change from non-trivial to trivial as the rotation angle of the scatterer changes. Figure 3(a) is the band structure of a V-type PC with four linear quadruply degenerate points at M for θ = 0°. On the left side, θ = −20°, and M is divided into two doubly degenerate points, where the low frequency is the q state with even parity and the high frequency is the p state with odd parity. On the right side, θ = 20°, and the band structure and band gap frequency are the same as those for θ = −20°, but the sound pressure distribution at M is opposite; that is, the low frequency is the p state while the high frequency is the q state. Figure 3(b) further shows the variation of the q and p states with the rotation angle. When θ increases from −30° to 0°, the bandwidth at M gradually decreases. In this angular range, the q state is always below the p state; the system is in a non-trivial state. At θ = 0, the q and p states coincide and the system is in a quadruply degenerate state. When θ increases from 0° to 30°, the bandwidth at M increases gradually. In this angular range, the q state is always above the p state; the system is in a trivial state. Therefore, figures 3(a) and 3(b) show that the system undergoes a transition from a non-trivial to a trivial phase as θ changes from negative to positive. Figure 3(c) shows the band structure of a T-type PC, which is similar to the V-type one, with a linear quadruply degenerate point at M for θ = 0. When θ = −60° and 60°, the band structures are the same, but the frequencies of the q and p states are opposite. When θ increases from −90° to 0°, the bandwidth at M increases first and then decreases to 0. In this angular range, the q state is always below the p state, so the system is in a non-trivial state. When θ increases from 0° to 90°, the bandwidth at M increases first and then decreases to 0. In this angular range, the q state is always above the p state, so the system is in a trivial state. Therefore, the T-type system also undergoes a phase transition from non-trivial to trivial as θ changes from negative to positive.

Fig. 3.

Figure Option ViewDownloadNew Window

Fig. 3. (a) Band structure of a V-type PC when θ is −20°, 0°, and 20° respectively. (b) Variation of q and p states with rotation angle θ increasing from −30° to 30°, the inset shows the distribution of sound pressure of q and p states when θ = ±20°. (c) Band structure of a T-type PC when θ is −60°, 0°, and 60° respectively. (d) Variation of q and p states with θ increasing from −90° to 90° for the T-type PC, the inset shows the distribution of sound pressure of q and p states when θ = ± 60°.

From the perturbation analysis of the system and the phase diagram of its topology, we find that the two PC types proposed in this paper are similar to a system with a quantum spin Hall effect, so the V and T sound waves should have an acoustic spin Hall effect. Accordingly, we use topologically trivial and non-trivial PCs to splice in the x and y directions. Then, we analyze the projection bands of the spliced structure in the corresponding direction. We splice a 10-layer topologically trivial V-type PC with θ = −20° and a 10-layer topologically non-trivial V-type PC with θ = 20° along the x and y directions. We also splice a 10-layer trivial T-type PC with θ = −60° and a 10-layer non-trivial T-type PC with θ = 60°. The result is shown in Fig. 4. The gray area in Fig. 4(a) represents the bulk state of the spliced structure, the red dotted line represents the edge state existing in the band gap, and the four illustrations represent the sound pressure distribution of the edge state at different positions. The black arrows represent the direction of energy flow. We can see from Fig. 4(a) that when k_x = 0.96π/a, the energy at the low-frequency point flows clockwise downward, and the energy at the high-frequency point flows counterclockwise. When k_x = 1.04π/a, the energy at the low-frequency point flows counterclockwise, and the energy at the high-frequency point flows clockwise downward, opposite to the direction of the energy flow at 0.96π/a. The other three boundary states show the similar phenomenon. The boundary state shapes and bandwidths are slightly different for the different splices of the V- and T-type PCs. Therefore, if the two boundaries are concentrated into one structure, such as the Z-shaped boundary shown in the text below, the frequency range through which sound waves can pass should be the intersection of the two boundary state bands.

Fig. 4.

Figure Option ViewDownloadNew Window

Fig. 4. Projection band structure of V- and T-type PCs. Panels (a) and (b) are projection band structures of a spliced structure of a V-shaped phononic crystal with θ = 20° and −20° (10 layers each) in the x and y directions, respectively. The gray area indicates the bulk state, and red dotted line indicates the edge state. Panels (c) and (d) are projection band structures of a spliced structure of a T-type phononic crystal with θ = 60° and −60° (10 layers each) in the x and y directions, respectively. The illustrations in (a)–(d) show the distributions of sound pressure around the splicing boundary at different wave vectors. The black arrows represent the energy flow directions.

The topological edge is different from an ordinary edge. It is immune to corner, cavity, and disorder defects, and can ensure that the incident sound wave propagates smoothly along the edge without reflection. We simulate a plane wave incident from the left side into a hydroacoustic region consisting of 20 × 15 hybrid PCs with topological edges (Fig. 5). The area is composed of a V-type trivial PC with θ = −20° and a non-trivial PC with θ = 20°, and contains two right-angled Z-shapes, as shown in Fig. 5(a). Figure 5(b) shows the addition of cavity and disorder defects on the basis of Fig. 5(a). We can see from Figs. 5(a) and 5(b) that the sound waves propagate without reflection along the edge, and the sound pressure of the transmitted wave is hardly affected. Figures 5(c) and 5(d) show the Z-shaped edge composed of a T-type trivial PC with θ = 60° and −60°. Injecting an acoustic wave with f = 79.5 kHz into the left side yields a similar phenomenon to that of the V-shaped structure. Therefore, both V-type T-type topological edges have good immunity to right angle, cavity, and disorder, and sound waves can bypass these defects and continue to propagate without reflection.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. (a) and (b) Field pattern of V-type edge simulated at 68 kHz without and with defects, respectively. (c) and (d) Field pattern of T-type edge simulated at 79.5 kHz without and with defects, respectively. The green arrow indicates that the plane wave is incident from the left side.

This study has theoretically constructed a rectangular PC with C_2v symmetry and placed four steel rectangular scatterers at each quarter position inside the crystal. The final composite crystal has two different forms: the V-type, where the distance between the center point of the scatterer and its right-angle point, s, is greater than 0.5a, and the T-type, where s is smaller than 0.5a. We found a quadruply degenerate state when the length L and width D of the scatterer are varied at θ = 0. This quadruply degenerate state consists of two odd-parity degenerate states and two even-parity states at the corner M. With the spatial symmetry of the sound pressure field at the corner M and the sliding symmetry of the structure, the perturbation method was used to calculate the effective Hamiltonian of the structure and the linear degeneracy of the band at the quadruply degenerate point. We also found that when D is fixed, the frequency at M decreases with increasing L; when L is in the range 0.4a–0.8a, the frequency at M first decreases slightly as D increases and then increases. By changing θ, we found that the quadruply degenerate state splits into two doubly degenerate states, and the band structure reverses as θ changes from negative to positive. Then, we spliced the systems with trivial and nontrivial states along the x and y directions and calculated the projection band structures. We found a one-way transmission edge of the “spin–momentum" locking in the frequency range of the body band gap. The finite-element software Comsol was then used to simulate the acoustic reflection of a Z-shaped edge consisting of two right angles composed of 20 × 15 PCs, and we found that the sound waves could propagate without reflection along the edge. Then, we introduced disorder and cavity defects to the edge, and found that the sound wave can still propagate around these defects. This shows the backscattering suppression of the topological edge.

The topological PC composed of steel rectangular scatterers surrounded by water is simple and practical, and can realize stable underwater acoustic wave edge transmission with good performance. This has potential application to underwater acoustic communication.