CN108615521A - A kind of sound topological insulator - Google Patents

Abstract

Sound topological insulator structure 1 is made of regular hexagon tablet 2 and regular triangular prism 3.Regular triangular prism 3 presses triangular crystal lattice periodic arrangement regular hexagon tablet 2.Regular triangular prism 2 divides for four parts, and upper left 7 and lower right-most portion 8 rotate clockwise 30 degree, bottom left section 9 and 30 degree of the rotation counterclockwise of upper right portion 10.The regular triangular prism 3 of sound topological insulator of the present invention is from left-handed 30 degree to 0 degree again to 30 degree of dextrorotation, the band gap of dual degeneracy energy interband first tapers into, and dual degeneracy Dirac points are finally merged into, then dual degeneracy Dirac points are opened, and as becoming larger for rotation angle becomes larger.The dual degeneracy energy band of sound topological insulator of the present invention completes the reversion of physical efficiency band and topological phase transition, realizes counterfeit rotation effect from closure is opened to again to opening.Sound topological insulator can realize the steady one-way transmission of sound topology, the antetype devices 1 such as design acoustics splitter using counterfeit rotation effect.

Description

An Acoustic Topological Insulator

technical field

The invention relates to the accidental degenerate double Dirac point of the phonon crystal, the pseudo-rotation effect of the phonon crystal, sound propagation control and noise reduction and sound insulation technology, and in particular to an acoustic topological insulator based on a triangular lattice.

Background technique

The nature of vibration and noise is not only an important index to measure the comfort of people's daily work and living environment, but also an important technical index to measure the comprehensive performance of aircraft, passenger cars, machine tools and other large mechanical equipment. It is also important in the field of military defense. meaning. The study of phononic crystals opens a new chapter for artificially manipulating wave propagation in elastic media and structures. According to the Bloch theory of periodic structures ω(π/a)=ω(-π/a), the energy bands of ordinary acoustic metamaterials are continuous at the boundary of the Brillouin zone, and each frequency in the same energy band has two modes. The two modes have group velocities (dω/dk) of opposite signs, corresponding to counterpropagating sound waves, respectively. Therefore, ordinary acoustic metamaterials propagate sound waves forward and backward at the same time, do not support a one-way propagation mode, and are prone to scattering when encountering defects and impurities. In electronic crystals, quantum Hall edge states allow electrons to propagate in one direction. When a vertical magnetic field is applied in the two-dimensional electron gas plane, electrons will move directionally along the boundary and are insensitive to the scattering of defects and impurities. Inspired by this, the topological properties of the energy bands of acoustic metamaterials can be realized by air flow circulation or gyroscope rotation. The topological state has the characteristic of one-way acoustic wave transmission that is not affected by impurities or external disturbances, which embodies the non-trivial topological properties brought about by the time-reversal symmetry breaking of acoustic metamaterials. However, no matter it is the air flow circulation or the gyroscope rotation, it is difficult to control the synchronous and coordinated work. Any potential external disturbance may cause changes in the air flow state and rotational motion, thereby destroying the one-way edge state effect of the acoustic metamaterial. . In addition, the air flow and the movement of the gyroscope will generate inherent loss and noise during the sound wave propagation process, which will seriously affect its engineering application value.

Contents of the invention

The present invention designs an acoustic topological insulator based on the pseudo-rotation effect of the accidental degenerate double Dirac point of the triangular lattice phononic crystal, which enables the acoustic wave to propagate in one direction along the topological boundary, and has good robustness.

In order to solve the above technical problems, the present invention provides an acoustic topological insulator. The acoustic topological insulator structure is composed of regular hexagonal plates and regular triangular prisms. The regular triangular prisms are periodically arranged on the regular hexagonal plate according to the triangular lattice. The regular triangular prism is divided into four parts, the upper left part and the lower right part are rotated 30 degrees in the clockwise direction, and the lower left part and the upper right part are rotated 30 degrees in the counterclockwise direction. The domain walls of the four-part regular triangular prism form the cross acoustic topological boundary.

As an improvement of the acoustic topological insulator of the present invention: the bottom plate of the acoustic topological insulator is a regular hexagonal flat plate.

As a further improvement of the acoustic topological insulator of the present invention: the regular hexagonal plate of the acoustic topological insulator is evenly arranged with 400 circular holes in a triangular lattice.

As a further improvement of the acoustic topological insulator of the present invention: the lower part of the regular triangular prism of the acoustic topological insulator is a bolt structure.

As a further improvement of the acoustic topological insulator of the present invention: the regular triangular prisms of the acoustic topological insulator are periodically fixed on the regular hexagonal flat plate according to the triangular lattice with nuts.

As a further improvement of the acoustic topological insulator of the present invention: the regular triangular prism of the acoustic topological insulator is divided into four parts, the upper left part and the lower right part rotate 30 degrees in the clockwise direction, and the lower left part and the upper right part rotate 30 degrees in the counterclockwise direction.

As a further improvement of the acoustic topological insulator of the present invention: the domain walls of the four-part regular triangular prism of the acoustic topological insulator form cross acoustic topological boundaries.

As a further improvement of the acoustic topological insulator of the present invention: the cross acoustic topological boundary of the acoustic topological insulator is divided into four sections.

Compared with the background technology, the present invention has beneficial effects as follows:

The acoustic topological insulator structure is processed by materials with high rigidity (such as steel and other materials), and has relatively low production cost. When the rotation angle of the regular triangular prism of the acoustic topological insulator of the present invention is 0 degrees, an accidental degenerate double Dirac point is generated in the center of the first irreducible Brillouin zone. When the regular triangular prism of the acoustic topological insulator of the present invention is left-handed at 30 degrees, there is a band gap between the double degenerate energy bands of the acoustic topological insulator. When the regular triangular prism of the acoustic topological insulator of the present invention rotates from 30 degrees to 0 degrees from the left, the band gap between the double degenerate energy bands of the acoustic topological insulator gradually becomes smaller, and finally merges into a double degenerate Dirac point. When the regular triangular prism of the acoustic topological insulator of the present invention rotates from 0 to 30 degrees to the right, the double Dirac point formed by the accidental degeneracy of the double degenerate energy bands of the acoustic topological insulator will open to generate a band gap, and with the change of the rotation angle Larger, the band gap gradually widens. In the process of the double degenerate energy band of the acoustic topological insulator of the present invention from opening to closing and then opening again, the body energy band inversion and topological phase conversion are completed near the double degenerate Dirac point, and the pseudo-rotation effect is realized. The pseudo-rotation effect of the left-handed 30-degree regular triangular prism and the right-handed 30-degree regular triangular prism makes the domain wall between them have the property of topological unidirectional transport.

The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

Description of drawings

Fig. 1 is an acoustic topological insulator structure of the present invention.

Fig. 2 is a direct lattice and an inverted lattice diagram of a Bravais regular hexagonal lattice of an acoustic topological insulator unit cell of the present invention.

Fig. 3 is the energy band structure of the unit cell when the triangular prism of an acoustic topological insulator of the present invention is rotated at 0 degrees.

Fig. 4 is the energy band structure of the unit cell when the triangular prism of an acoustic topological insulator of the present invention is rotated by 30 degrees.

Fig. 5 is an edge state energy band structure of an acoustic topological insulator structure of the present invention.

Fig. 6 is a pseudo-rotation effect diagram of an acoustic topological insulator structure of the present invention.

Detailed ways

Figure 1 shows an acoustic topological insulator structure. Acoustic topological insulator 1 is composed of regular hexagonal plate 2 and regular triangular prism 3 . The square plate 2 is evenly arranged with 400 circular holes 4 in a triangular lattice. The lower part of the regular triangular prism 3 is a bolt structure 6 . The regular triangular prism 3 is periodically fixed on the regular hexagonal 2 flat plate with a nut 7 according to the triangular lattice. The regular triangular prism is divided into four parts, the upper left part 7 and the lower right part 8 rotate 30 degrees in the clockwise direction, and the lower left part 9 and the upper right part 10 rotate 30 degrees in the counterclockwise direction. The domain walls of the four-part regular triangular prism form the cross acoustic topological boundary. The boundary is divided into four sections, namely topological boundaries 11 , 12 , 13 and 14 .

The working principle of the acoustic topological insulator structure of the present invention is as follows:

(1) The side length and height of the regular hexagonal plate of this acoustic topological insulator are l=100mm and d=5mm respectively, the side length and height of the regular triangular prism are respectively b=6.8mm and h=25mm, the lattice constant of the regular triangular prism It is a=8mm.

(2) The base vector of the triangular lattice is a=(a ₁ ,a ₂ ). The response for grid point r is u(r). Since the triangular lattice is periodic, its response u(r) is periodic:

u(r)=u(r+R) (1)

Where R is the lattice vector of the lattice, which can be expressed as R=b ₁ a ₁ +b ₂ a ₂ . b1 and _b2 are _integers .

The Fourier series expansion of the periodic function u(r) is:

Bring (1) into (2) to get

G _j R=2πk (3)

Where G _j is the periodic lattice of Fourier space, that is, the reciprocal lattice of the lattice. The base vector is expressed as

(3) Calculate the band structure diagram of the structure by using the finite element method. For linear elastic, isotropic, and inhomogeneous media elastic waves, the wave equation can be expressed as

ρ(r) ^-1 {μ▽ ² u(r)+▽[(λ+μ)(▽ u(r))]}＝-ω ² u(r) (4)

Where ρ is the density of the medium, u(r) is the response vector, λ and μ are the elastic constants of the medium, and ω is the characteristic frequency of the elastic wave.

According to Bloch's principle, the response vector u(r) can be expressed as

u(r)＝e ^{ik r} u _k (r) (5)

where k=(k _x , _ky ) is the wave loss in the first Brillouin zone. u _k (r) is a periodic response vector. Applying the Bloch-Floquet condition on the periodic boundary of the unit cell, the energy band structure of the Mie resonance type acoustic metamaterial can be calculated by the finite element method

Ku-ω ² Mu＝0 (6)

Among them, the stiffness matrix and mass matrix of K and M unit cells can be expressed as

K＝∫B ^T C(r)BdV _e (7)

M=∫ρ(r)N ^T NdV _e (8)

Among them, B is the strain matrix, C(r) is the elastic tensor, and _Ve is the unit cell area.

In order to obtain the complete energy band structure of the unit cell, it is necessary to calculate all the modal frequencies of the wave loss k theoretically. In the Bloch theory, the wave loss k is symmetric and periodic in the reciprocal lattice loss, so the wave loss k can be constrained in the first incommensurable Brillouin zone.

(4) The first condition for the realization of acoustic topological insulators is to generate accidental degenerate double Dirac points in the bandgap structure of phononic crystals. By using the finite element analysis software ComsolMultiphysics analysis, it is found that the triangular prism phononic crystal with C _3v symmetry (as shown in Figure 2) has an accidental degenerate double Dirac point in the center Γ of the first irreducible Brillouin zone (as shown in Figure 3). There are two types of phonon modes near the accidental degenerate double Dirac point, ie, symmetric (Symmetric) mode and anti-symmetric (Anti-Symmetric) mode. By simulating the electronic spin state of the Quantum Spin Hall Effect (Quantum Spin Hall Effect), frequency matching of symmetric and antisymmetric modes across the bandgap enables polarization and spin degeneration.

(5) The second condition for the realization of acoustic topological insulators is that the accidental degeneracy Dirac is opened and the topological phase transition occurs. When the triangular prism rotates left or right by 30 degrees, a synthetic scalar field (Synthetic Gauge Field) that simulates rotation-orbit coupling can be generated. The degeneration of the inversion symmetry of the synthetic scalar field will lead to the degeneration of the symmetric mode and the antisymmetric mode, and make the original occasional degenerate double Dirac point reopen, and generate a new band gap [15158Hz, 19230Hz] (as shown in Fig. 4 ). In detail, the degradation of the inversion symmetry will lead to a pair of couplings between the original Dirac energy bands: the upper symmetry (S _u ) mode and the lower inversion symmetry (A _l ) mode coupling, and the upper inversion The symmetric (A _u ) mode and the lower symmetric (S _l ) mode are coupled. For all frequencies near the original Dirac point, the eigenstates of the system are polarized, which can simulate pseudo-up-rotation and pseudo-down-rotation. Based on the First Principe Finite Element Method, the spin-Chern number (Spin-Chern number) C _s of the four energy bands of the topological bandgap near the double degenerate Dirac point can be directly obtained, from bottom to top are - 1, +1, -1 and +1. When the triangular prism rotates from 30 degrees left to 30 degrees right, the signs of the four rotation Chern numbers will be reversed, which guarantees the topological phase conversion.

(6) The third condition for the realization of an acoustic topological insulator is to generate a helical edge state (Helical Edge State) without causing the coupling of the two rotational states. For this purpose, a boundary with opposite topological rotation Chern numbers on both sides can be designed. As shown in Figure 5, this boundary is the domain wall (Domain Wall) of two types of triangular prisms, in which the triangular prism on the upper side of the domain wall is 30 degrees left-handed, and the triangular prism on the lower side is 30 degrees right-handed. In this case, the phononic crystals on the upper and lower sides of the domain wall have opposite spin Chern numbers. According to the principle of bulk boundary correspondence, a pair of boundary states appear in the bulk band gaps of two different types of phononic crystal coincidences. The topological boundary mode has a topological helical state (Topological Helical State) that can be compared to the asymmetric spin-polarized one-way propagation (Unidirectional Spin-Polarized One-Way Propagation) in condensed matter physics. Therefore, although the symmetry inversion produces a bulk bandgap that prevents acoustic propagation, acoustic topologically robust unidirectional transport can be achieved at the topological boundaries of the two types of phononic crystals (as shown in Fig. 5).

(7) In general, it is considered difficult to observe pseudospin phenomena under experimental conditions. We exploit a designed acoustic topological insulator (Fig. 1) to study phonon pseudospin transport with high fidelity. The acoustic insulator shown in FIG. 6 has four criss-cross boundaries 11 , 12 , 13 and 14 , and the outer sides of each boundary can be used as input and output terminals. The triangular prisms on either side of each boundary rotate in opposite directions. The gradient of the Chern number for different steering triangular prisms defines the group velocity at the topological boundary. Based on the reverse slope of the group velocity in the Brillouin zone, the acoustic down-spin state and up-spin state can lead to the one-way acoustic propagation property of the acoustic topological boundary. As shown in the lower spin state in FIG. 6 , both the upper topological boundary 12 and the lower topological boundary 14 support outward transmission, that is, sound waves are transmitted from the “cross” center 15 to the outlets 17 and 19 . In contrast, both the left topological boundary 11 and the right topological boundary 13 support inward transmission, ie acoustic wave inlets 16 and 18 to the center 15 of the "cross". When the spin state is an upward spin state, the acoustic transmission direction of each topological boundary is just opposite. Therefore, when the edge state of the left entry 16 is activated, the lower spin state will be activated. In this case, the inward transport mode of the left topological boundary 11 and the outward transport modes (associated with the lower spin state) of the upper topological boundary 14, lower topological boundary 12 are activated so that sound can propagate from the left inlet 16 To "Cross" center 15, from "Cross" center 15 to exits 17 and 19. Since the upper spin state of the right topological boundary state cannot be activated by the lower spin state, and the activated lower spin state only supports inward one-way propagation, the sound cannot propagate from the "cross" center 15 to the outlet 18.

(8) The acoustic topological insulator proposed in this patent uses the quantum spin Hall effect to design the prototype device 1 of the acoustic splitter.

Finally, it should also be noted that what is listed above is only a specific embodiment of the present invention. Apparently, the present invention is not limited to the above embodiments, and many deformations are possible, such as triangular shape, equilateral hexagonal shape and so on. All deformations that can be directly derived or associated by those skilled in the art from the content disclosed in the present invention should be considered as the protection scope of the present invention.

Claims (7)

1. An acoustic topological insulator 1 is composed of 400 regular triangular prisms 3 periodically arranged on a regular hexagonal flat plate 2 according to a triangular lattice, wherein the regular triangular prism 3 is divided into four parts, and the upper left part 7 and the lower right part 8 are clockwise The direction is rotated 30 degrees, the lower left part 9 and the upper right part 10 are rotated 30 degrees counterclockwise. 2. An acoustic topological insulator according to claim 1, characterized in that: the square plate 2 of the acoustic topological insulator 1 is evenly arranged with 400 circular holes 4 in a triangular lattice. 3 . The acoustic topological insulator according to claim 1 , wherein the lower part of the regular triangular prism 3 of the acoustic topological insulator 1 is a bolt structure 6 . 4. a kind of acoustic topological insulator according to claim 1, is characterized in that: the regular triangular prism 3 of acoustic topological insulator 1 is periodically fixed on 400 round holes 4 of regular hexagonal plate 2 with nut 7 according to triangular lattice superior. 5. according to claim 1 and 4 described a kind of acoustic topological insulator, it is characterized in that: the regular triangular prism 3 of acoustic topological insulator is divided into four parts, left upper part 7 and right lower part 8 rotate 30 degrees clockwise, left lower part Part 9 and upper right part 10 are rotated 30 degrees in the counterclockwise direction. 6. An acoustic topological insulator according to claims 1 and 5, characterized in that: the domain walls of the four regular triangular prisms 7, 8, 9 and 10 of the acoustic topological insulator form cross acoustic topological boundaries. 7. An acoustic topological insulator according to claims 1 and 6, characterized in that: the acoustic topological boundary of the acoustic topological insulator cross is divided into four sections 11, 12, 13 and 14.