Method for constructing acoustic three-dimensional dirac super-material based on positive and negative coupling and application thereof

Abstract

This invention discloses a method and application for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling, belonging to the field of acoustic metamaterials. This invention utilizes positive and negative coupling terms between structural units to construct three-dimensional Dirac metamaterials. The construction process is no longer limited by specific spatial group structures, making acoustic topological three-dimensional Dirac metamaterials easier to prepare. The acoustic topological metamaterial constructed by this invention possesses a doubly degenerate dispersive band and a quadruple degeneracy point, and the metamaterial exhibits topologically protected boundary states. Based on the acoustic three-dimensional Dirac metamaterial, the topologically protected boundary states are realized, and a robust acoustic waveguide system is achieved using these boundary states. Specifically, the topologically protected boundary states prevent the acoustic waveguide system from being affected by structural defects; even when defects occur in several units within the metamaterial array, the transmittance of the acoustic waveguide can still maintain a high level.

Description

Technical Field

[0001] This invention belongs to the field of acoustic metamaterials and relates to a method for constructing three-dimensional Dirac metamaterials by controlling the coupling coefficient. Background Technology

[0002] Three-dimensional Dirac metamaterials possess the advantage of immunity to local defects in their structure, thus showing broad application prospects as carriers for novel acoustic waveguide devices. Due to their unique physical phenomena, the topological phases of matter play a crucial role in condensed matter physics, such as the quantum Hall effect, the quantum spin Hall effect, topologically protected surface states, and Dirac and Weyl degeneracy. In recent years, research on the topological properties of matter has expanded to optical and acoustic fields. Photonic crystals made from gyromagnetic materials can achieve photonic simulation of Chern insulators because their time-reversal symmetry is broken under an external magnetic field. The quantum spin Hall effect in two-dimensional photonic crystals has also been verified by establishing a pseudotime-reversal operator. Various two-dimensional photonic materials with nontrivial topological properties have been designed to realize topologically protected edge states. Meanwhile, 3D topological materials with topologically protected surface states have also attracted considerable attention. 3D topological systems have the potential to achieve topologically protected transmission in all directions. Weyl systems possess Weyl degeneracy points, which always appear in pairs, leading to helical surface states. A Dirac point can be viewed as the superposition of two Weyl points with opposite chirality. Under certain special symmetries, the intersection of spiral surface states can be topologically nontrivial.

[0003] 3D photonic Dirac points in metamaterials can be designed under electromagnetic duality symmetry. Electromagnetic duality achieved through a fixed ratio between all dielectric constants and permeability tensor elements requires carefully designed structural units with the desired electromagnetic resonance. Recently, 3D Dirac band structures protected by glide symmetry and time-reversal symmetry have been theoretically and experimentally verified. Pseudo-antiunitary symmetry can be obtained by combining glide reflection and time-reversal symmetry. Pseudo-antiunitary symmetry leads to a doubly degenerate band along a high symmetry line, analogous to Kramer degeneracy in spin-orbit coupled electron materials. In photonic or acoustic systems, constructing a… Pseudo-time reversal symmetry is crucial for ensuring bidegeneracy and for realizing 3D Dirac points.

[0004] Three-dimensional Dirac metamaterials have the advantage of being immune to local defects in their structure, thus they have broad application prospects as carriers for novel acoustic waveguide devices; however, existing three-dimensional Dirac metamaterials must be constructed according to specific space group structures, which is difficult to form and not convenient for subsequent technology applications. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the formation of three-dimensional Dirac metamaterials requires specific spatial group structures, which makes formation difficult and hinders subsequent technological applications. This invention provides a method for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling. This method utilizes positive and negative coupling terms between structural units to construct the three-dimensional Dirac metamaterial, eliminating the limitation of specific spatial group structures during construction and making the acoustic topological three-dimensional Dirac metamaterial easier to prepare. The acoustic topological metamaterial constructed by this invention possesses doubly degenerate dispersive bands and a quadruple degeneracy point (three-dimensional Dirac point), and the metamaterial exhibits topologically protected boundary states.

[0006] Based on the method for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling disclosed in this invention, this invention also provides an acoustic waveguide system realized based on the three-dimensional Dirac metamaterial. The acoustic three-dimensional Dirac metamaterial realizes topology-protected boundary states, and the topology-protected boundary states are used to realize an acoustic waveguide system with robust transmission performance. That is, the topology-protected boundary states are used to make the acoustic waveguide system unaffected by structural defects. When defects occur in several units in the metamaterial array, the transmittance of the acoustic waveguide can still maintain a high level.

[0007] The objective of this invention is achieved through the following technical solution:

[0008] This invention discloses a method for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling, comprising the following steps:

[0009] Step 1: First, staggered positive and negative interlayer coupling terms are introduced into the tight-binding model of stacked hexagonal lattices to construct a unit cell with 3D Dirac points, where only nearest-neighbor coupling between units is considered. The unit cell of the proposed structure consists of two hexagonal lattice layers. The Hamiltonian of this tight-binding model is expressed as:

[0010]

[0011] a(b) and These are the annihilation production operators for unit A and unit B, respectively, where ε is the potential energy difference and t is the annihilation production operator. n t1 and t2 are intra-layer unit couplings, while t1 and t2 are inter-layer unit couplings along the z-direction. The subscript (i, k) represents the i-th lattice in the k-th layer. Only nearest-neighbor couplings between units are considered. The second term of the Hamiltonian represents intra-layer coupling, while the third term represents inter-layer coupling. The Bloch-Hamiltonian corresponding to the k-space of the tight-bound model is written as:

[0012] H(k)=d1τ0σ1+d2τ0σ2+d3τ1σ0+d4τ1σ3+d5τ2σ0+d6τ2σ3+ετ3σ3 (2)

[0013] τ and σ represent the Pauli matrices for different degrees of freedom within the element, a is the period in the xy plane, and h is the period in the z direction. The parameter d... i＝1，2... for:

[0014]

[0015]

[0016] d3=(t1+t2cos(k z h)+t2+t1cos(k z h)) / 2

[0017] d4=(t1+t2cos(k z h)-t2-t1cos(k z h)) / 2

[0018] d5=-(t1sin(k z h)+t2sin(k z h)) / 2

[0019] d6=(t1sin(k z h)-t2sin(k z h)) / 2 (3)

[0020] When the interlayer coupling in the z-direction has staggered positive and negative coupling, the four units in a unit cell form a π flux loop, which enables the gauge transformation of the inversion symmetry operator, thereby generating a double degenerate band and a three-dimensional Dirac point in the tight-binding model.

[0021] The physical model constructed in step 1, which has 3D Dirac points, is the physical basis for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling.

[0022] Step 2: To achieve opposite coupling terms along the z-direction, additional non-resonant units are inserted between adjacent layers. By designing the in-situ energy and nearest-neighbor coupling strength of the additional units, the sign and strength of the effective interlayer coupling terms can be adjusted to give it the same band dispersion and nontrivial topological properties as the system with staggered positive and negative interlayer coupling. Achieving staggered coupling with equal strength but opposite signs along the z-direction is key to constructing a double degenerate band structure. However, achieving opposite-sign coupling in a passive system is not easy. It would be convenient to construct a tight-binding model with only negative or positive coupling to simulate a tight-binding model with staggered positive and negative coupling. To construct a tight-binding model with effective negative coupling, the negative coupling is replaced by inserting additional sites. The four-band tight-binding model becomes a six-band tight-binding model with additional units. The corresponding Hamiltonian of the new six-band tight-binding model is expressed as:

[0023]

[0024] Where δ represents the energy mismatch between sites A and B, and ε c t3 is the in-situ energy of the additional site, and t3 is its nearest coupling with the adjacent A or B site. By designing the nearest coupling strength and characteristic frequency of the additional site, effective negative coupling between A and B sites in different layers of the unit cell can be obtained. To ensure that the first four bands of the new Hamiltonian have the same dispersion and characteristic mode distribution as the previous one, it is necessary to set... The six-band tight-binding model with additional units has four band degeneracy points that are also 3D Dirac points. This six-band tight-binding model exhibits double-helix band crossings, forming topologically protected surface states with topological structure effects, unaffected by structural defects. Furthermore, the bottom-up construction method of the tight-binding model eliminates the constraints of specific space group structures during the construction of three-dimensional Dirac metamaterials, making the design, fabrication, and subsequent application of acoustic topological metamaterials easier.

[0025] Step 3: In a real acoustic system, the designed tight-binding lattice with positive and negative coupling terms is realized through a periodic array of acoustic resonant cavities connected by thin tubes. In the design of the acoustic three-dimensional Dirac metamaterial, each cylindrical acoustic resonant cavity is equivalent to an atomic unit of the six-band tight-binding model in Step 2, where the resonant cavities representing units A and B have the same size, and the radius of the connecting tubes between units is much smaller than the diameter of the acoustic resonant cavities. Units A and B are arranged in a hexagonal lattice, forming a layer of metamaterial with sixfold rotational symmetry. Acoustic resonant cavities representing unit C are inserted between each layer, connecting several layers of metamaterial. This achieves the construction of the acoustic three-dimensional Dirac metamaterial based on positive and negative coupling. The constructed acoustic topological metamaterial has a doubly degenerate dispersive band and a quadruple degeneracy point (three-dimensional Dirac point), possessing topologically protected boundary states.

[0026] This invention also discloses an acoustic waveguide system based on the aforementioned three-dimensional Dirac metamaterial. The method for constructing an acoustic three-dimensional Dirac metamaterial based on positive and negative coupling disclosed in this invention is used to construct the acoustic three-dimensional Dirac metamaterial. By periodically arranging the metamaterial lattice designed in step 2, and terminating the array of detuned acoustic cavities with radii and heights approximately half that of units A and B in each layer of the metamaterial, the chiral symmetry of the acoustic three-dimensional Dirac metamaterial is ensured to simulate the open boundary conditions in the tight-binding model. Based on the aforementioned acoustic three-dimensional Dirac metamaterial, topologically protected boundary states are realized. When a sound wave of the response frequency is incident from the designed acoustic Dirac metamaterial boundary, the metamaterial can act as an acoustic waveguide, with the sound wave propagating along the surface units of the metamaterial. Because of the existence of Dirac points, the acoustic waveguide based on this metamaterial has a topological protection effect. The topologically protected boundary states are used to realize an acoustic waveguide system with robust transmission performance. That is, the topologically protected boundary states are used to make the acoustic waveguide system unaffected by structural defects. Even when defects occur in some units in the metamaterial array, the transmittance of the acoustic waveguide can still maintain a high level.

[0027] Preferably, the acoustic three-dimensional Dirac metamaterial is composed of cylindrical acoustic resonant cavities of different sizes. By changing the size of the resonant cavity, the response frequency of the acoustic topology material can be arbitrarily controlled, thereby enabling the construction of highly robust acoustic waveguide systems with different operating frequencies according to actual conditions.

[0028] Beneficial effects:

[0029] 1. This invention discloses a method for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling. The method utilizes the interleaved positive and negative coupling terms between structural units to form π flux loops, thereby forming a double degenerate band and a topology-protected 3D Dirac point. This results in an acoustic topological metamaterial with a double degenerate dispersive band and a quadruple degenerate point (three-dimensional Dirac point), as well as topology-protected boundary states. Furthermore, the construction of the three-dimensional Dirac metamaterial is no longer restricted by a specific spatial group structure, making the acoustic topological 3D Dirac metamaterial easier to prepare.

[0030] 2. Based on the method for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling disclosed in this invention, this invention also provides an acoustic waveguide system realized based on the three-dimensional Dirac metamaterial. The acoustic three-dimensional Dirac metamaterial realizes topology-protected boundary states, and the topology-protected boundary states are used to realize an acoustic waveguide system with robust transmission performance. That is, the topology-protected boundary states are used to make the acoustic waveguide system unaffected by structural defects. When defects occur in several units in the metamaterial array, the transmittance of the acoustic waveguide can still maintain a high level.

[0031] 3. The present invention discloses a method for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling. The acoustic metamaterials are composed of cylindrical acoustic resonant cavities of different sizes. By changing the size of the resonant cavity, the response frequency of the acoustic topology material can be arbitrarily adjusted, thereby enabling the construction of highly robust acoustic waveguide systems with different operating frequencies according to actual conditions. Attached Figure Description

[0032] Figure 1 This is a flowchart of an acoustic three-dimensional Dirac metamaterial constructed based on positive and negative coupling according to the present invention.

[0033] Figure 2 These are structural schematics. (a) Schematic diagram of the unit cell structure of a tight-binding model of an acoustic metamaterial with 3D Dirac points. Black and gray spheres represent different unit cells. Blue lines represent intralayer coupling. Solid black lines and dashed black lines represent interlayer coupling. (b) Half of the first Brillouin zone.

[0034] Figure 3 This is the calculated band structure. (a) and (c) are k at t1=t2=-1(a) and t1=-t2=1(c), respectively. z Band structure of the high-symmetry line in the =0 plane. (b and d) When t1 = t2 = -1 (b) and t1 = -t2 = 1 (d), k x =4π / 3a and k y =0 (along the KH line) along the z-direction with dispersion. (e) The non-Abelberg phase calculated at the polar angle of the sphere enclosing the Dirac point at t1 = -t2 = 1.

[0035] Figure 4 This is the band structure calculated after adding unit cell C. (a) Schematic diagram of the unit cell with additional units to achieve effective coupling. Green spheres are additional non-resonant site units, and gray lines are nearest-neighbor couplings of the additional units. (b) With additional non-resonant units, k z (c) Band structure along the high symmetry line in the 0 plane. (d) Band dispersion along the KH line with additional non-resonant units.

[0036] Figure 5 These are the calculated boundary state band structures. (a) A schematic diagram of the unit cell stripes of the designed additional unit cells, with the calculated projected band structure. (b) k xz (c) Half of the Brillouin zone projected onto the plane. (d) Projected band structure along the high symmetry line of the Brillouin zone. (e) Helical surface state of the entire Brillouin zone. (f) Gapless surface state along a circle centered on the Dirac point in the Brillouin zone. Gray areas represent bulk bands.

[0037] Figure 6 This explains the change of the Dirac point after deviating from the ideal parameters. (a) When the near coupling of the additional non-resonant potential unit and the potential energy are t3 = -10 and ε c When k = -90, z =0 plane along the high symmetry line. (b) Non-Abelberg phase at the polar angle of the sphere around the nodal line.

[0038] Figure 7 It is a designed acoustic metamaterial. (a) Top and side views of the unit cell of the designed phononic crystal. (b) The designed acoustic crystal at k... z =0 plane band structure along the high symmetry line. (c) Designed phononic crystal band structure along the z-direction k x =4π / 3a and k y =0. (d) Projected band structure of the designed phonon crystal. The inset is a schematic diagram of the structure in the simulation of periodic projected bands along the x and z directions. (e) Figure 5 (d) Sound field distribution of the star points. Detailed Implementation

[0039] To better illustrate the purpose and advantages of the present invention, the invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0040] like Figure 1 As shown in the figure, this embodiment discloses a method for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling. The specific implementation method is as follows:

[0041] Step 1: First, introduce staggered positive and negative interlayer coupling terms into the tightly bound model of the stacked hexagonal lattice to construct a unit cell with 3D Dirac points. Only nearest-neighbor coupling between units is considered. The specific structure is as follows: Figure 2 As shown in (a), the Hamiltonian of the tight-binding model is expressed as:

[0042]

[0043] a(b) and These are the annihilation production operators for unit A and unit B, respectively, where ε is the potential energy difference and t is the annihilation production operator. n t1 and t2 are intra-layer unit couplings, while t1 and t2 are inter-layer unit couplings along the z-direction. The subscript (i,k) represents the i-th lattice in the k-th layer. Here, we only consider nearest-neighbor couplings. The second term of the Hamiltonian represents intra-layer coupling, while the third term represents inter-layer coupling. Figure 2 (a) The first Brillouin zone of the structure is as follows Figure 2 As shown in (b), the corresponding Bloch-Hamiltonian in momentum space can be written as:

[0044] H(k)=d1τ0σ l+d2τ0σ2+d3τ1σ0+d4τ1σ3+d5τ2σ0+d6τ2σ3+ετ3σ3 (2)

[0045] τ and σ represent the Pauli matrices for different degrees of freedom within the element, a is the period in the xy plane, and h is the period in the z direction. The parameter d... 1,2...6 for:

[0046]

[0047]

[0048] d3=(t1+t2cos(k z h)+t2+t1cos(k z h)) / 2

[0049] d4=(t1+t2cos(k z h)-t2-t1cos(k2h)) / 2

[0050] d5=-(t1sin(k z h)+t2sin(k z h)) / 2

[0051] d6=(t1sin(k z h)-t2sin(k z h)) / 2 (3)

[0052] When t1 = t2 = -1, the calculated band structure is as follows: Figure 3 As shown in (a) and (b), the calculated band structure is as follows when t1 = -t2 = 1. Figure 3 As shown in (c) and (d), it clearly demonstrates that the band structure of the tight-binded model becomes doubly degenerate when the interlayer coupling t1 changes from -1 to 1. When t1 = t2 = -1, the tight-binded model follows inversion symmetry PH(k)P. -1 =H(-k), However, when t1 = -t2 = 1, the four units in a unit cell form a π flux loop, causing a gauge transformation of the inversion symmetry operator. As a result, the inversion symmetry operator becomes... It is obvious that Capable of generating double degenerate bands, such as Figure 3 As shown in (c) and (d). Calculate the non-Abelberg phase of the two lower energy bands on the sphere surrounding the Dirac point, as follows: Figure 3 As shown in (e). The gapless non-Abelberg phase in the figure indicates that the four-band crossover point is a nontrivial 3D Dirac point with a topological charge number Z2 = 1.

[0053] Step 2: To achieve opposite coupling terms along the z-direction, additional non-resonant units are inserted between adjacent layers, as shown in Figure 4(a). By designing the in-situ energy and nearest-neighbor coupling strength of the additional units, the sign and strength of the effective interlayer coupling terms can be adjusted to give it the same band dispersion and nontrivial topological properties as the system with staggered positive and negative interlayer coupling. Achieving staggered coupling with equal strength but opposite signs along the z-direction is key to constructing a double degenerate band structure. However, achieving opposite-sign coupling in a passive system is not easy. It would be very convenient to construct a tight-binding model with only negative or positive coupling to simulate a tight-binding model with staggered positive and negative coupling. To construct a tight-binding model with effective negative coupling, the negative coupling is replaced by inserting additional sites. The four-band tight-binding model becomes a six-band tight-binding model with additional units. The corresponding Hamiltonian of the new six-band tight-binding model is expressed as:

[0054]

[0055] Where δ represents the energy mismatch between sites A and B, and ε c δ is the in-situ energy of the additional unit, and t3 is its nearest neighbor coupling with adjacent unit A or unit B. By designing the nearest neighbor coupling strength and characteristic frequency of the additional unit, effective negative coupling between unit A and unit B in different layers of the unit cell can be obtained. To ensure that the first four bands of the new Hamiltonian have the same dispersion and characteristic mode distribution as the tightly bound model in step 1, δ = -1, t3 = -10, and t3 = -10 can be set. The calculation of the four energy bands above, such as Figure 4 As shown in (b) and (c). Figure 4 (b) and (c) with Figure 3 The band dispersions in (c) and (d) are almost identical. Figure 4 (d) shows the non-Abelberg phase at the degeneracy point in the calculated six-band tight-binding model. The evolution of the non-Abelberg phase is not affected by the additional unit. Therefore, the degeneracy point of the six-band tight-binding model with the additional unit is also the 3D Dirac point.

[0056] The most prominent feature of metamaterials with 3D Dirac points is the nontrivial crossover of helical surface states, which are topologically protected by symmetry. To calculate the projected band structure of tightly bound lattice stripes, a tightly bound model lattice is repeated with a finite number of lattice elements in the y-direction and an infinite number of lattice elements in the x and z directions, such as... Figure 5 As shown in (a). The projection of the Brillouin zone is as follows. Figure 5 As shown in (b), the projection zone structure along the high symmetry line of the Brillouin zone on the surface is as follows: Figure 5 As shown in (c), the surface states in the band gap are two helices whose intersection is protected by the system's hidden symmetry. Figure 5(d) shows the surface condition of the entire Brillouin zone, where the intersection of the double helical surfaces is clearly visible. Meanwhile, the surface condition along the circle centered on the projected Dirac point is seamless, as shown... Figure 5 As shown in (e), this also indicates that it is a Z2 monopole.

[0057] Figure 4 The unit inversion and six-fold spiral symmetry of the six-band tight-binding model structure in (a) force degeneracy of the high-symmetry lines. Typically, degenerate nodal lines appear in such structures. However, when the condition t1 = -t2 is fully satisfied, Figure 4 (b) and (c) can be considered accidental degeneracy, leading to double-degenerate band dispersion. When the in-situ energy of the additional unit deviates... At this time, the effective coupling amplitudes are not exactly equal, which transforms 3D Dirac points into nodal lines, such as... Figure 6 As shown in (a), it should be noted that the topological charge of the degenerate nodal line is still Z2 = 1, from Figure 6 (b) The non-Abelberg phase at the polar angle of the sphere enclosing the degenerate nodal line can be seen.

[0058] Step 3: In a real acoustic system, the designed tight-binding lattice with positive and negative coupling terms is realized through a periodic array of acoustic resonant cavities connected together by thin tubes, such as... Figure 7 As shown in (a), in the design process of the acoustic three-dimensional Dirac metamaterial, each cylindrical acoustic resonator is equivalent to an atomic unit of the six-band tight-binding model in step two, where the resonator representing unit A and unit B has the same size, and the radius of the connecting tube between units is much smaller than the diameter of the acoustic resonator. Units A and B are arranged in a hexagonal lattice to form a layer of metamaterial with six-fold rotational symmetry. Acoustic resonators representing unit C are inserted between each layer to connect several layers of metamaterial, thus realizing the construction of the acoustic three-dimensional Dirac metamaterial based on positive and negative coupling. The constructed acoustic topological metamaterial has a doubly degenerate dispersive band and a quadruple degeneracy point (three-dimensional Dirac point), and has topologically protected boundary states. Since each layer of the designed metamaterial has C6 symmetry in its hexagonal lattice, the resonators representing unit A and unit B have the same size and equal height h. a =h b =0.7a, radius r a =r b =0.2a. The radius of the connecting pipe is r. t =0.06a. The radius and height of the element C between the insertion layers are r. c =0.2a and h c =0.12a. The pale yellow surface of the structure is a hard boundary. At the end of the connecting pipe, a periodic boundary condition is applied, and the interior of the designed structure is filled with air.

[0059] Figure 6 The band structure in (b) was obtained using the finite element method. The lower frequency bands are generated by element C, while the four upper bands are generated by elements A and B. Notably, the four upper bands are essentially doubly degenerate, consistent with the tight-binding model. The four upper bands intersect at point K. Figure 7 As shown in (b) and (c), the band dispersion of the designed phononic crystal is linear in all directions around point K. To verify the topologically protected cross-section of the double helix, the projected band structure of the band composed of honeycomb cells can be simulated, as shown in... Figure 7 As shown in (d), the band is periodic in the x and z directions and has a finite unit cell in the y direction (see [reference]). Figure 7 (Illustration d). Here, acoustic resonators with a radius of 0.2a and a height of 0.35a are used to terminate the array to simulate the open boundary conditions in the tight-binding model. In the case of acoustic coupling, these boundary units can maintain the chiral symmetry at the ends of the structure. The intersection of the simulated projection bands on the surface Brillouin zone can be clearly observed, which is generated by the 3D Dirac points of the acoustic crystal. Figure 7 (e) shows Figure 7 (d) shows the sound field distribution at the star point. The field is located on both surfaces of the structure, indicating the existence of surface states induced by the 3D Dirac point.

[0060] This embodiment also discloses an acoustic waveguide system based on the aforementioned three-dimensional Dirac metamaterial (see...). Figure 7 (Illustration d) Based on the method disclosed in this embodiment for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling, an acoustic three-dimensional Dirac metamaterial is constructed. By periodically arranging the metamaterial lattice designed in step 2, and terminating the array of each layer of metamaterial with a detuned acoustic cavity whose radius and height are approximately half that of units A and B, the chiral symmetry of the acoustic three-dimensional Dirac metamaterial is ensured to simulate the open boundary conditions in the tight-binding model. Based on the acoustic three-dimensional Dirac metamaterial, topologically protected boundary states are realized. When a sound wave of the response frequency is incident from the boundary of the designed acoustic Dirac metamaterial, the metamaterial can act as an acoustic waveguide, and the sound wave propagates along the surface units of the metamaterial. Because of the existence of Dirac points, the acoustic waveguide based on this metamaterial has a topological protection effect. The topologically protected boundary states are used to realize a robust acoustic waveguide system, that is, the topologically protected boundary states are used to make the acoustic waveguide system unaffected by structural defects. Even when defects occur in several units in the metamaterial array, the transmittance of the acoustic waveguide can still maintain a high level.

[0061] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling, characterized in that: Includes the following steps, Step 1: First, staggered positive and negative interlayer coupling terms are introduced into the tight-binding model of stacked hexagonal lattices to construct a unit cell with 3DDirac points, where only nearest-neighbor coupling between units is considered; the unit cell of the proposed structure consists of two hexagonal lattice layers; the Hamiltonian of this tight-binding model is expressed as: a i,k and b i,k These are the annihilation operators for cell A and cell B, respectively. and These are the production operators for unit A and unit B, respectively. ε is the potential energy difference, and t... n t1 and t2 are intra-layer unit couplings, while t1 and t2 are inter-layer unit couplings along the z-direction; the subscript (i,k) represents the i-th lattice of the k-th layer; only nearest-neighbor couplings between units are considered; the second term of the Hamiltonian represents intra-layer coupling, while the third term represents inter-layer coupling; the Bloch-Hamiltonian corresponding to the k-space of the tight-bound model is written as: H1(k)=d1τ0σ1+d2τ0σ2+d3τ1σ0+d4τ1σ3+d5τ2σ0+d6τ2σ3+ετ3σ3 (2) τ and σ represent the Pauli matrices for different degrees of freedom within the element, a is the period in the xy plane, and h is the period in the z direction; parameter d i＝1,2… for: When the interlayer coupling in the z-direction has staggered positive and negative coupling, the four units in a unit cell form a π flux loop, which enables the gauge transformation of the inversion symmetry operator, thereby generating a double degenerate band and a three-dimensional Dirac point in the tight-binding model. The physical model with 3D Dirac points constructed in step 1 is the physical basis for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling. Step 2: To achieve opposite coupling terms along the z-direction, additional non-resonant units are inserted between adjacent layers. By designing the in-situ energy and nearest-neighbor coupling strength of the additional units, the sign and strength of the effective interlayer coupling terms can be adjusted to give it the same band dispersion and nontrivial topological properties as the system with staggered positive and negative interlayer coupling. Achieving staggered coupling with equal strength but opposite signs along the z-direction is key to constructing a double degenerate band structure. However, achieving opposite-sign coupling in a passive system is not easy. Constructing a tight-binding model with only negative or positive coupling to simulate a tight-binding model with staggered positive and negative coupling would be very convenient. To construct a tight-binding model with effective negative coupling, additional sites are inserted to replace the negative coupling. The four-band tight-binding model becomes a six-band tight-binding model with additional units. The corresponding Hamiltonian of the new six-band tight-binding model is expressed as: Where δ represents the energy mismatch between sites A and B, and ε c t3 is the in-situ energy of the additional site, and t3 is its nearest coupling with the adjacent A or B site. By designing the nearest coupling strength and characteristic frequency of the additional site, effective negative coupling between A and B sites in different layers of the unit cell can be obtained. In order to make the first four bands of the new Hamiltonian have the same dispersion and characteristic mode distribution as the previous one, it is necessary to set... The six-band tight-binding model with additional units has four band degeneracy points that are also 3D Dirac points; the bands of this six-band tight-binding model have the intersection of double-helix bands, forming topologically protected surface states, which have topological structure effects and are not affected by structural defects; at the same time, the construction method of the tight-binding model is from bottom to top. Step 3: In a real acoustic system, the designed tight-binding lattice with positive and negative coupling terms is realized through a periodic array of acoustic resonant cavities connected by thin tubes. In the design of the acoustic three-dimensional Dirac metamaterial, each cylindrical acoustic resonant cavity is equivalent to an atomic unit of the six-band tight-binding model in Step 2, where the resonant cavities representing unit A and unit B have the same size, and the radius of the connecting tube between units is much smaller than the diameter of the acoustic resonant cavity. Units A and B are arranged in a hexagonal lattice to form a metamaterial with six-fold rotational symmetry. An acoustic resonant cavity representing unit C is inserted between each layer to connect several layers of metamaterial, thus realizing the construction of the acoustic three-dimensional Dirac metamaterial based on positive and negative coupling. The constructed acoustic topological metamaterial has a doubly degenerate dispersive band and a quadruple degeneracy point, and has topologically protected boundary states.

2. The method for constructing acoustic three-dimensional Dirac metamaterials based on positive and negative coupling as described in claim 1, characterized in that: The acoustic three-dimensional Dirac metamaterial is composed of cylindrical acoustic resonant cavities of different sizes. By changing the size of the resonant cavity, the response frequency of the acoustic topology material can be arbitrarily controlled, thereby enabling the construction of highly robust acoustic waveguide systems with different operating frequencies according to actual conditions.

3. An acoustic waveguide system based on the aforementioned three-dimensional Dirac metamaterial, wherein the acoustic three-dimensional Dirac metamaterial is constructed based on a method for constructing an acoustic three-dimensional Dirac metamaterial based on positive and negative coupling as described in claim 1 or 2, characterized in that: By periodically arranging the lattice of the metamaterial designed in step 2, and terminating the array with a detuned acoustic cavity in each layer of the metamaterial, the chiral symmetry of the acoustic three-dimensional Dirac metamaterial is ensured to simulate the open boundary conditions in the tight-binding model. Based on the acoustic three-dimensional Dirac metamaterial, topologically protected boundary states are realized. When the acoustic wave of the response frequency is incident from the boundary of the designed acoustic Dirac metamaterial, the metamaterial can act as an acoustic waveguide, and the acoustic wave propagates along the surface unit of the metamaterial. Because of the existence of Dirac points, the acoustic waveguide based on this metamaterial has a topological protection effect. The topologically protected boundary states are used to realize an acoustic waveguide system with robust transmission performance, that is, the topologically protected boundary states are used to make the acoustic waveguide system unaffected by structural defects.

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