A two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states

Abstract

The present invention proposes a two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states, which relates to the technical field of phononic crystal design. It is composed of cuboid-shaped acoustic resonators arranged in rows and columns. Each two acoustic resonators are connected by two upper and lower coupling tubes. Moreover, imaginary on-site energies γ1 with only background loss and imaginary on-site energies γ2 with additional loss are respectively introduced at different lattice points of the two-dimensional phononic crystal structure. By gradually increasing the additional loss of the imaginary on-site energy γ2, the dispersion relation of the two-dimensional phononic crystal gradually changes from closed to open, and high-order topological corner states appear in the band gap. The emergence of non-Hermitian induced high-order topological phases lays a foundation for lossless transmission in acoustic devices.

Description

Technical Field

[0001] The present invention relates to the technical field of phononic crystal design, and more specifically, to a two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states. Background Art

[0002] A phononic crystal is a medium composed of a periodic composite of materials with different elastic properties, and is a new type of acoustic functional material with phononic energy bands and band gaps.

[0003] Hermiticity is the basis of quantum systems, which effectively guarantees real eigenvalues and orthogonal eigenstates. These Hermitian properties are usually used to define various quantum topological structures and accurately classify the topological phases of matter. Generally speaking, the way phononic crystals manipulate electromagnetic waves can achieve no gain and loss through the real part parameters of phononic crystals, which belongs to the category of Hermiticity.

[0004] Most of the existing high-order topologies are studied based on Hermitian systems. Although these classical topological systems follow Hermitian properties, since the system cannot exist without energy exchange with the outside world, there must be losses and / or gains, so they are essentially non-Hermitian in nature. Therefore, the research under the Hermitian system is ideal but overly limited. When considering a non-conservative system interacting with the outside world, its energy conservation is violated, and the concept of non-Hermitian dynamics is naturally introduced. Non-Hermitian physics has flourished in non-conservative systems because in physical systems existing in nature, it is difficult to achieve no energy exchange with the outside world, and there is more or less coupling between the system and the outside world. Therefore, there is energy gain or loss. Since its proposal, non-Hermitian physics has been widely studied and has shown new principles, phenomena and applications in open quantum systems, interacting electron systems, and classical systems with added gain and loss. For example, by introducing gain and loss, under the condition of satisfying space-time symmetry, the system appears special singular points in the complex energy plane, supporting characteristics such as unidirectional propagation and anomalous Fermi arcs.

[0005] Currently, the introduction of traditional non-Hermitian is achieved by introducing an imaginary part loss in the Hamiltonian. Therefore, non-Hermitian only makes the topological boundary states decay, has no influence on the real part of the dispersion and the eigenstates, and cannot determine the topological properties of the bands, because the induced band gaps can be topological or trivial. Whether the introduction of non-Hermitian can induce high-order topological phases and play a more important role in high-order topologies is unknown and difficult to further apply in practice, such as in the fields of acoustics and optics. Therefore, it is very important to study the crystal design of induced topological corner states in the non-Hermitian state for applications in the fields of acoustics and optics to achieve lossless transmission. Summary of the Invention

[0006] To solve the problem of the large limitations in the crystal structure design of the current Hermitian state, the present invention proposes a two-dimensional phononic crystal structure based on non-Hermitian-induced topological corner states. The two-dimensional phononic crystal structure is designed in the non-Hermitian state and induces topological corner states, laying a foundation for lossless transmission in acoustic devices.

[0007] To achieve the above technical effects, the technical solution of the present invention is as follows:

[0008] A two-dimensional phononic crystal structure based on non-Hermitian-induced topological corner states, wherein the two-dimensional phononic crystal structure is composed of cuboid-shaped acoustic resonators arranged in rows and columns. Each row or column includes four acoustic resonators, and two upper and lower coupling tubes are connected between every two acoustic resonators; the acoustic resonators in the outermost row or column of the phononic crystal structure include two virtual first acoustic resonators with only background loss and an on-site energy of γ1 and two virtual second acoustic resonators with additional loss and an on-site energy of γ2. The arrangement order of the acoustic resonators in the outermost row or column of the phononic crystal structure is: the first acoustic resonator, the second acoustic resonator, the second acoustic resonator, the first acoustic resonator; the acoustic resonators within the outermost row or column of the phononic crystal structure are four virtual first acoustic resonators with only background loss and an on-site energy of γ1.

[0009] Preferably, the size of each acoustic resonator is: 100mm × 100mm × 300mm, and the wall thickness of each acoustic resonator is 6mm.

[0010] Preferably, the lengths of the coupling tubes connecting every two acoustic resonators are the same.

[0011] Preferably, assuming that the acoustic resonators are arranged in a right-angle coordinate system in the two-dimensional phononic crystal structure, the acoustic resonator at the bottom left corner of the two-dimensional phononic crystal structure is located at the origin of the right-angle coordinate system, and the width ratio of the coupling tubes in the horizontal coordinate direction to the coupling tubes in the vertical coordinate direction is 5:1.

[0012] Preferably, when γ1 = γ2 ≠ 0, the dispersion relation of the two-dimensional phononic crystal is closed; when γ1 ≠ γ2 ≠ 0, as the gradually increasing additional loss γ2 is added, the dispersion relation of the two-dimensional phononic crystal changes from closed to gradually open.

[0013] Preferably, as the added additional loss γ2 increases, the band gap is opened, and topological phases appear within the band gap, including bulk states, boundary states, and corner states.

[0014] Preferably, the higher-order topology is symmetry-protected, and this structure has reflection symmetry and chiral symmetry.

[0015] Preferably, when the imaginary on-site energies γ1 and γ2 are introduced and γ1≠γ2≠0, the band gap of the two-dimensional phonon crystal is opened, forming a quadrupole topological insulator.

[0016] Preferably, the topological properties of the quadrupole topological insulator are obtained by calculating the Wilson loop twice. First, the Wannier bands of all occupied bands below the band gap are calculated along the x / y direction. The Wannier bands have a band gap. At the same time, the Wannier bands are divided into three parts: "0", "+", and "−", and the two parts of the "±" Wannier bands are symmetric, that is, the bulk polarization disappears, and the topology appears within the band gap of the Wannier bands. Therefore, a second Wilson loop calculation is performed on all occupied bands below the band gap of the Wannier bands to confirm that the edge Hamiltonian is a topological insulator with a quantized dipole moment and is caused by the quantized quadrupole moment.

[0017] Compared with the prior art, the beneficial effects of the technical solution of the present invention are:

[0018] The present invention proposes a two-dimensional phonon crystal structure based on non-Hermitian-induced topological corner states, which is composed of cuboid-shaped acoustic resonators arranged in rows and columns. Each two acoustic resonators are connected by two upper and lower coupling tubes. Moreover, imaginary on-site energies γ1 with only background loss and imaginary on-site energy γ2 with additional loss are introduced at different lattice points of the two-dimensional phonon crystal structure. By gradually increasing the additional loss of the imaginary on-site energy γ2, the dispersion relation of the two-dimensional phonon crystal gradually changes from closed to open, and high-order topological corner states appear in the band gap. The emergence of the non-Hermitian-induced high-order topological phase lays a foundation for lossless transmission in acoustic devices. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 Schematic diagram of the two-dimensional phonon crystal structure proposed in Embodiment 1 of the present invention;

[0020] Figure 2 Schematic diagram of the unit cell ball-and-stick model of the two-dimensional phonon crystal structure proposed in Embodiment 1 of the present invention;

[0021] Figure 3 Schematic diagram of the dispersion relation of the two-dimensional phonon crystal structure with uniform background loss proposed in Embodiment 2 of the present invention;

[0022] Figure 4 Schematic diagram of the dispersion relation of the two-dimensional phonon crystal structure after the additional loss increases proposed in Embodiment 2 of the present invention;

[0023] Figure 5 Schematic diagram of the bulk topological phase that appears in the second band gap of the two-dimensional phonon crystal structure when γ1≠γ2≠0 and the bulk band of the two-dimensional phonon crystal is opened as the additional loss γ2 increases, proposed in Embodiment 2 of the present invention;

[0024] Figure 6 It shows the schematic diagram of the topological phase of the edge state appearing at the band gap of the second energy band of the two-dimensional phononic crystal when γ1≠γ2≠0 and the bulk energy band of the two-dimensional phononic crystal is opened with the increase of the additional loss γ2 in Embodiment 2 of the present invention;

[0025] Figure 7 It shows the schematic diagram of the topological phase of the corner state appearing at the band gap of the second energy band of the two-dimensional phononic crystal when γ1≠γ2≠0 and the bulk energy band of the two-dimensional phononic crystal is opened with the increase of the additional loss γ2 in Embodiment 2 of the present invention;

[0026] Figure 8 It shows the simulation diagram of the eigenfrequency of the finite structure of the 4×4 two-dimensional phononic crystal structure when γ1≠γ2≠0 in Embodiment 3 of the present invention;

[0027] Figure 9 It shows the schematic diagram of the first wave function of the corner state at 504.709 Hz within the band gap proposed in Embodiment 3 of the present invention;

[0028] Figure 10 It shows the schematic diagram of the first wave function of the corner state at 504.709 Hz within the band gap proposed in Embodiment 3 of the present invention;

[0029] Figure 11 It shows the schematic diagram of the first wave function of the corner state at 504.710 Hz within the band gap proposed in Embodiment 3 of the present invention;

[0030] Figure 12 It shows the schematic diagram of the second wave function of the corner state at 504.710 Hz within the band gap proposed in Embodiment 3 of the present invention. Detailed implementation manners

[0031] The accompanying drawings are only for illustrative purposes and should not be construed as a limitation to this patent;

[0032] For better illustration of this embodiment, some parts of the accompanying drawings are omitted, enlarged or reduced, and do not represent the actual size;

[0033] For those skilled in the art, it is understandable that some well-known content descriptions in the accompanying drawings may be omitted.

[0034] The technical solutions of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0035] The description of the positional relationship in the accompanying drawings is only for illustrative purposes and should not be construed as a limitation to this patent;

[0036] Embodiment 1

[0037] As Figure 1As shown, this embodiment proposes a two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states. Refer to Figure 1 , this two-dimensional phononic crystal structure is composed of cuboid-shaped acoustic resonators arranged in rows and columns. Each row or column includes four acoustic resonators, and two upper and lower coupling tubes are connected between every two acoustic resonators. The acoustic resonators in the outermost row or column of the phononic crystal structure include two virtual first acoustic resonators with only background loss and an on-site energy of γ1, and two virtual second acoustic resonators with additional loss and an on-site energy of γ2 (in Figure 1 , the color of the second acoustic resonator is darker). The arrangement order of the acoustic resonators in the outermost row or column of the phononic crystal structure is: the first acoustic resonator, the second acoustic resonator, the second acoustic resonator, the first acoustic resonator. The acoustic resonators within the outermost row or column of the phononic crystal structure are four virtual first acoustic resonators with only background loss and an on-site energy of γ1.

[0038] It can be seen from Figure 1 that the first acoustic resonator and the second acoustic resonator are represented by different colors. In the two-dimensional phononic crystal structure shown in Figure 1 , the virtual second acoustic resonator with additional loss and an on-site energy of γ2 rotates counterclockwise or clockwise by 90 degrees with the first acoustic resonator in a certain outermost row as the rotation center to obtain the arrangement of the acoustic resonators in the outermost column perpendicular to it. In this embodiment, the size of each acoustic resonator is: 100mm × 100mm × 300mm, the wall thickness of each acoustic resonator is 6mm, and the lengths of the coupling tubes connecting every two acoustic resonators are the same, with a length of 180mm.

[0039] Assume that the acoustic resonators are arranged in a right-angle coordinate system in the two-dimensional phononic crystal structure. The left-bottommost acoustic resonator of the two-dimensional phononic crystal structure is located at the origin of the right-angle coordinate system, and the width ratio of the coupling tubes in the horizontal coordinate direction to the coupling tubes in the vertical coordinate direction is 5:1.

[0040] Figure 2 shows a schematic diagram of a tightly bound model of a single cell composed of 16 microcells. Corresponding to Figure 1 , different colors represent the first acoustic resonator and the second acoustic resonator respectively (for distinction, diagonal lines are added to the sites corresponding to the second acoustic resonator). Figure 2 The solid lines in x represent the coupling in the y-axis direction of the ordinate, and the dashed lines represent the coupling in the x-axis direction of the abscissa. W y represents the intracellular coupling in the x-axis direction, W x represents the intercellular coupling in the x-axis direction, V y represents the intercellular coupling in the y-axis direction, Wx (y) / V x (y) = 1, Wx(V x ) / Wy(V y ) = 5:1, and the lattice constant a of the primitive cell is 1120 mm.

[0041] Example 2

[0042] Generally speaking, higher-order topology is protected by symmetry, and this structure has reflection symmetry and chiral symmetry.

[0043] In order to further study the physical properties of the above two-dimensional phononic crystal structure model, its Hamiltonian was studied, and the tight-binding model was simulated according to the Hamiltonian of the structure, and the dispersion relation diagram of the above structure was obtained. Figure 3 It represents a schematic diagram of the dispersion relation of a two-dimensional phononic crystal structure with uniform background loss, that is, when γ1 = γ2 ≠ 0, that is, when all the resonators only have background loss, the dispersion relation diagram is in a state of band closure. Figure 4 It represents a schematic diagram of the dispersion relation of a two-dimensional phononic crystal structure after the additional loss increases, that is, when γ1 ≠ γ2 ≠ 0, as the gradually increasing additional loss γ2 is added, the dispersion relation of the two-dimensional phononic crystal changes from closed to gradually open. See Figure 4 , as the additional loss added increases, the originally completely closed energy band completely opens into four four-fold degenerate energy bands. With the opening of the band gap, the emergence of topological properties is bound to follow.

[0044] Example 4

[0045] In this example, Figure 4 the topological properties of the second band gap shown in Figure 5 , Figure 6 and Figure 7 are deeply studied. In the case of γ1 ≠ γ2 ≠ 0, as the additional loss γ2 added increases, the band gap of the energy band is opened, and topological phases appear in the band gap, including bulk states, boundary states and corner states. As shown in

[0046] The eigenfrequencies of a 4×4 two-dimensional phononic crystal structure are simulated, and the volume quadrupole moment caused by the additional loss added can be characterized in a way similar to the Hermitian case. With the introduction of the imaginary on-site energies γ1, γ2, and in the case of γ1 ≠ γ2 ≠ 0, the band gap of the two-dimensional phononic crystal is opened, forming a quadrupole topological insulator.

[0047] The topological properties of the quadrupole are obtained by calculating the Wilson loop twice. First, the Wannier bands of all occupied bands below the band gap are calculated along the x / y directions. It is found that the Wannier bands have a band gap. At the same time, the Wannier bands are divided into three parts: "0", "+", and "−", and the two parts of the "±" Wannier are symmetric. That is, we call this phenomenon the disappearance of bulk polarization. Then, we judge that the topology appears within the band gap of the Wannier bands. Therefore, we immediately perform a second Wilson loop calculation on all occupied bands below the band gap of the Wannier bands, which is also called the nested Wilson loop. The polarization result obtained by the calculation is 0.5, indicating that the edge Hamiltonian is a topological insulator with a quantized dipole moment and is caused by the quantized quadrupole. Due to the non-trivial quadrupole, high-order topological corner states can be found in a finite sample. To see this, numerical calculations are performed on the finite phonon crystal structure, and the Figure 8 characteristic frequencies obtained are plotted. In Figure 8 , the four frequencies within the innermost circle correspond to the Figures 9 - 12 wave function graphs of the four corner states shown. As can be seen from Figures 9 - 12 , when the frequency is around 504.7, the wave function probabilities of the four corner states are significantly higher than those in other parts, which further confirms the existence of the topological corner states and lays a foundation for the lossless transmission in acoustic devices in the future.

[0048] It should be clear that the described embodiments are only a part of the embodiments of the present application, rather than all embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope protected by the embodiments of the present application.

[0049] The terms used in the embodiments of the present application are only for the purpose of describing specific embodiments and are not intended to limit the embodiments of the present application. The singular forms of "a", "the", and "said" used in the embodiments of the present application and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term "and / or" used herein refers to and includes any or all possible combinations of one or more of the associated listed items.

[0050] When the following description refers to the accompanying drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the present application. On the contrary, they are merely examples of devices and methods consistent with some aspects of the present application as detailed in the appended claims. In the description of the present application, it should be understood that the terms "first", "second", "third", etc. are only used to distinguish similar objects, and do not have to be used to describe a specific order or sequence, nor can they be understood as indicating or implying relative importance. For those of ordinary skill in the art, the specific meanings of the above terms in the present application can be understood according to specific circumstances.

[0051] In addition, in the description of the present application, unless otherwise specified, "a plurality of" means two or more. "And / or" describes the association relationship of associated objects and indicates that three relationships can exist. For example, A and / or B can represent: A exists alone, A and B exist simultaneously, and B exists alone. The character " / " generally represents an "or" relationship between the associated objects before and after.

[0052] Obviously, the above embodiments of the present invention are merely examples for clearly explaining the present invention, and are not limitations on the embodiments of the present invention. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to enumerate all embodiments here. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention shall be included in the protection scope of the claims of the present invention.

Claims

1. A two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states, characterized in that, The two-dimensional phononic crystal structure is composed of cuboid-shaped acoustic resonators arranged in rows and columns. Each row or column includes four acoustic resonators, and two upper and lower coupling tubes are connected between every two acoustic resonators. The acoustic resonators in the outermost row or column of the phononic crystal structure include two virtual first acoustic resonators with only background loss and an on-site energy of γ1, and two virtual second acoustic resonators with additional loss and an on-site energy of γ2. The arrangement order of the acoustic resonators in the outermost row or column of the phononic crystal structure is: the first acoustic resonator, the second acoustic resonator, the second acoustic resonator, the first acoustic resonator; The acoustic resonators within the outermost row or column of the phononic crystal structure are four virtual first acoustic resonators with only background loss and an on-site energy of γ1; When γ1 = γ2 ≠ 0, the dispersion relation of the two-dimensional phononic crystal is closed; When γ1 ≠ γ2 ≠ 0, as the gradually increasing additional loss γ2 is added, the dispersion relation of the two-dimensional phononic crystal changes from closed to gradually open; As the added additional loss γ2 increases, the band gap of the energy band is opened, and topological phases appear within the band gap, including bulk states, edge states, and corner states; with the introduction of the virtual on-site energies γ1 and γ2, and in the case of γ1 ≠ γ2 ≠ 0, the band gap of the energy band of the two-dimensional phononic crystal is opened, forming a quadrupole topological insulator.

2. The two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states according to claim 1, characterized in that, The size of each acoustic resonator is: 100mm × 100mm × 300mm, and the wall thickness of each acoustic resonator is 6mm.

3. The two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states according to claim 2, characterized in that, The lengths of the coupling tubes connecting every two acoustic resonators are the same.

4. The two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states according to claim 3, characterized in that, Suppose the acoustic resonators are arranged in a right-angle coordinate system within the two-dimensional phononic crystal structure. The left-bottommost acoustic resonator of the two-dimensional phononic crystal structure is located at the origin of the right-angle coordinate system, and the width ratio of the coupling tubes in the transverse coordinate direction to those in the longitudinal coordinate direction is 5:

1.

5. The two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states according to claim 1, characterized in that, Higher-order topology is protected by symmetry. This structure has reflection symmetry and chiral symmetry.

6. The two-dimensional phononic crystal structure based on non-Hermitian induced topological corner states according to claim 1, characterized in that, The topological properties of the quadrupole topological insulator are obtained by calculating the Wilson loop twice. First, calculate the Wannier bands of all occupied bands under the band gap along the x / y direction. The Wannier bands have a band gap. At the same time, the Wannier bands are divided into three parts: "0", "+", and "−", and the two parts of the "±" Wannier bands are symmetric, that is, the bulk polarization disappears, and the topology appears within the band gap of the Wannier bands. Therefore, perform the second Wilson loop calculation on all occupied bands under the band gap of the Wannier bands to confirm that the edge Hamiltonian is a topological insulator with a quantized dipole moment, and it is caused by the quantized quadrupole moment.

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