Acoustic frequency filter based on anisotropic topological phononic crystals

We present a design of acoustic frequency filter based on a two-dimensional anisotropic phononic crystal. The anisotropic band structure exhibits either a directional or a combined (global + directional) bandgap at certain frequency regions, depending on the geometry. When the time-reversal symmetry is broken, it may introduce a topologically nontrivial bandgap. The induced nontrivial bandgap and the original directional bandgap result in various interesting wave propagation behaviors, such as frequency filter. We develop a tight-binding model to characterize the effective Hamiltonian of the system, from which the contribution of anisotropy is explicitly shown. Different from the isotropic cases, the Zeeman-type splitting is not linear and the anisotropic bandgap makes it possible to achieve anisotropic propagation characteristics along different directions and at different frequencies.

Most of the previous research focuses on the topological property of isotropic systems with global (or complete) bandgaps. Limited efforts have been devoted to anisotropic systems with directional (or partial) bandgaps 29 . However, anisotropy grants more degrees of freedom in manipulating wave propagation and adds more complexity in the corresponding mathematical modeling, it would be interesting to investigate the consequences of breaking certain symmetries in an anisotropic system. For example, the topology evolution of a directional bandgap when the TR symmetry is broken and the subsequent wave propagation behaviors may bring rich physics and render more applications.
Here, we explicitly study the topological properties of anisotropic systems. We find that a two-dimensional (2D) anisotropic phononic crystal, under broken TR symmetry, possessing a combined topologically nontrivial global and a directional bandgap. The global bandgap is attributed to the broken TR symmetry, while the directional bandgap is a signature of the anisotropy. The combined bandgap enables the frequency filter functionality of the phononic crystal: a particular boundary either supports a topologically protected edge state or prohibits wave propagation, depending on the working frequency. Such phononic crystal offers a platform to engineer the topology through multiple parameters including TRS broken perturbation, geometric parameter, direction and frequency. The TRS broken perturbation is contributed by the external applied air flow, and without that, the system exhibit various topologically trivial bandgaps, global, directional, or combined, depending on geometric parameters. By applying the gradually increased external air flow, the system may experience topological transitions from a conductor or a normal insulator to a Chern insulator. We further consider the contribution of the anisotropy and find that along a certain direction, the bandgap topology is associated with the frequency.
To characterize the phase transitions and capture the physical essence of anisotropy, we develop an effective Hamiltonian and classify the topological properties. Potential applications are discussed as well.

Results
Theoretic Model. The two-dimensional anisotropic phononic crystal considered here is composed of a square array of acoustic waveguides. As illustrated in Fig. 1(a), the unit cell with lattice constant = a m 2 contains a hollow ring with inner and outer radii = .
r m 0 5 1 , respectively, connected by four rectangular waveguides. While the lengths of these waveguides are identical, the widths of them are different, giving rise to anisotropic coupling along different directions between neighboring units. We set the width as indicates the widths of the horizontal (vertical) waveguides and is tunable. For simplicity but without loss of generality, the ratio of anisotropy κ is fixed to be 2.5. Inside the ring, the air flows counterclockwise with a velocity field distribution V ve  = θ , where  θ e denotes the azimuthal unit vector. The acoustic wave propagation obeys the irrotational aero-acoustics equation 30 .
The band structure of the phononic crystal with = . d m 0 03 x without airflow is shown in Fig. 2(a), which exhibits three states at the Brillouin zone center. Their field patterns possess symmetries denoted as d, px, py, a convention used in classifying electron orbitals 31 . The eigenfrequencies of these states depend on the size of the rectangular waveguides. Such band structure may be modeled by the tight-binding approximation and the corresponding onsite energy of states φ d , φ px , py φ (in free space) are d ε , ε px , py ε , respectively. When the airflow is introduced, the effective Hamiltonian, under the basis of ( , , ) , is written as:  where t ij l (l = x, y represents the x or y directions, i, j indicates the orbital d, px, py) is the coupling coefficient of two states φ i and φ j between two neighboring rings and E t k t k 2 cos( ) 2 cos( ) . ∆z represents a perturbation, induced by the airflow, that breaks TR symmetry and is proportional to the strength of the airflow ν. At the Γ point x y , the eigenvalues of the effective Hamiltonian are E d (0), , respectively, where the function f t t t ( ) ( )(1 ) pxpx x pypy y κ = + − vanishes when the system is isotropic, i.e., κ = 1.

Topological phase transition.
In the frequency region around 140 Hz, only fundamental mode is propagating in each narrow rectangular waveguide, making the coupling coefficients and the eigenfrequencies almost proportional to the width of the waveguides when there is no airflow, as shown in Fig. 1(b). When the airflow is applied, which can be viewed as a Zeeman-type perturbation 27 characterized by the ∆z term in Eq. (1), the eigenfrequencies of ϕ + , ϕ − and ϕ d as functions of the flow strength are plotted in Fig. 1(c). Here, ϕ + , ϕ − and ϕ d denote the Bloch states at the Brillouin zone center, which are different from the free space states mentioned earlier.. The intersecting point A in Fig. 1(c) indicates the band inversion between branches associated with ϕ + and d ϕ states. Such inversion, according to the Haldane model 32 , reveals the occurrence of a topological transition. For instance, when = .
d m 0 04 x (the case shown in Fig. 1(c)), the system is a trivial insulator for v m s 7 7 / < .
, and a Chern insulator for larger v. Because eigenfrequencies of states ϕ + and ϕ d depend on v, d x , and κ, the transition points at a fixed ratio of anisotropy κ form a curve in the phase diagram displayed in Fig. 1(d).
The phase diagram suggests that the band topology can be tuned by changing widths of the rectangular waveguides and/or the velocity of the airflow. When there is no airflow, the system exhibits a global bandgap when d x is small, as shown in Fig. 2(a). The global bandgap gradually closes as d x increases towards a critical value = . d m 0 041 s , where the accidental degeneracy of state d ϕ and py ϕ occurs, and induces a semi-Dirac point 33 . Further increasing d x opens a directional bandgap along the ΓX direction, highlighted in gray in Fig. 2(b).

Frequency filter in anisotropic system. Such a directional bandgap means that the wave propagation is
forbidden along the ΓX direction while it is allowed along the other directions. This property is summarized schematically in Fig. 3(a). It is not clear about the changes to this directional bandgap when the TR symmetry is broken. In the following, we consider an example of an anisotropic phononic crystal with d m 0 06 and v m s 20 / = . Its band structure is plotted in Fig. 2(b) in blue curves, which shows the induced air flow opens a global bandgap and the original directional bandgap almost remains unchanged. A schematic of the band diagram of this system is illustrated in Fig. 3(b), where the global bandgap with a bandwidth f g ∆ is marked in blue. From the Eq. (1), we can prove that this global bandgap is topologically nontrivial with a nonzero Chern number C = 1. Figure 3(b) also shows that the gap size along the ΓX direction is f d ∆ , which is larger than ∆f g , meaning that the gap along the ΓX direction would exhibit a mixed behavior that cannot be simply defined as trivial or nontrivial. To examine the topological property of this anisotropic phononic crystal, we study the edge states. It is well known that for a system possessing a topologically nontrivial bandgap, there exists a gapless edge state at its interface with a trivial insulator. We calculate the band structures of two different supercells consisting of 1 × 16 unit cells. The first one is infinite along the ΓY direction and terminated by rigid boundaries along the ΓX direction. The gapless edge state is clearly shown in Fig. 3(c). The second one is infinite along the ΓX direction. In this case, the edge state only exists within the frequency region f g ∆ . − .

Hz
Hz (140 7 145 2 ), and below the edge state there is a bandgap covering a frequency range . − .

Hz
Hz 136 1 140 7 with gap width ∆ − ∆ f f d g , as shown in Fig. 3(d). Therefore, for a sample with a boundary along the ΓX direction, it will "select" the type of propagating waves according to the frequency. As illustrated schematically in Fig. 3(b), the sample supports one-way propagation edge state along the ΓX direction at ω 2 , while a bulk state propagating along the ΓY direction is supported at 1 ω .

Discussions and Conclusions
To verify our predictions of the frequency-dependent propagating behavior, we perform finite-element simulations of a finite-sized sample. It contains 20 × 40 unit cells. We impose a source, with two different frequencies ω 1 and 2 ω , at the bottom of the sample. At 2 ω , the system behaves as a Chern insulator with a topological protected edge state propagating at the boundary as shown in Fig. 4(a), which resembles the property in an isotropic acoustic Chern insulator. However, the strikingly difference occurs at ω 1 . In the isotropic case, ω 1 corresponds to bulk state frequency, and the wave propagation is supported everywhere including the boundary 28 . While in our  anisotropic Chern insulator, the system possesses a directional bandgap that forbids wave propagation along the ΓX direction at ω 1 , the wave cannot propagate along the boundary as manifested in Fig. 4(b). Thus, the boundary can be viewed as a frequency filter.
In conclusion, we report our design of an anisotropic topological phononic crystal which can work as a frequency filter. It exhibits a tunable topological transition point as well as a tunable directional bandgap. The combined topological nontrivial global bandgap and the directional bandgap is systematically studied by using a tight-binding model and numerical simulations. We find the wave propagation behavior at a particular boundary depends on the frequency, and we demonstrate the functionality of a frequency filter: at certain frequencies, the boundary allows one-way propagation edge state, while at other frequencies, it forbids wave propagation. The mechanism is universal and would not be limited to acoustics. We believe our findings can inspire more designs and applications based on topological insulators.

Methods
Throughout the paper, the Finite Element Method (FEM) based on commercial software COMSOL Multiphysics is employed for the band structure computations and the simulations. Plane wave radiation boundary conditions are set on the outer boundaries of simulation domain. The largest mesh element size is set smaller than 1/20 of the shortest wavelength. The topological invariant is calculated based on Eq. (1).