Tunable Topological Surface States of Three-Dimensional Acoustic Crystals

Topological design for band structures of artificial materials such as acoustic crystals provides a powerful tool to manipulate wave propagating in a robust and symmetry-protected way. In this paper, based on the band folding and breaking mechanism by building blocks with acoustic atoms, we construct a three-dimensional topological acoustic crystal with a large complete bandgap. At a mirror-symmetry domain wall, two gapped symmetry and anti-symmetry surface states can be found in the bandgap, originated from two opposite Su-Schrieffer-Heeger chains. Remarkably, by enforcing a glide symmetry on the domain wall, we can tune the original gapped surface states in a gapless fashion at the boundaries of surface Brillouin zone, acting as omnidirectional acoustic quantum spin Hall effect. Our tunable yet straightforward acoustic crystals offer promising potentials in realizing future topological acoustic devices.

In general, the topology of band structure space originates from its nontrivial geometry phase, e.g., Zak phase in 1D [41] or Berry phase in 2D [2], which is ill-defined at the degenerated point.
In other words, the degenerated point of band structure might be the phase transition point between trivial phase and some topological phase. An excellent way to realize topological insulators is to lift band degeneracy by breaking symmetries. One can lift the linear or quartic band degenerated point by either breaking time-reversal symmetry to obtain 2D quantum Hall effect [42], or breaking spatial symmetry to 2D topological valley [43] and quantum spin Hall effect [44]. Regarding the construction of degeneracy by building symmetries, there are mainly three approaches. The first one is to resort to group theory, creating a high symmetric structure with 2D irreducible representation such as E in C 3v [43] or C 4v [45] lattices. This kind of band degeneracy requires careful lattice optimization to obtain favorite dispersion and eliminate the influence of other bands. The second one is to construct accidental band degeneracy relying on elaborately designed parameters [46]. Last but not least, the band folding mechanism increases symmetries by enlarging the primitive unit cell. For example, with in-plane 2D folding, four-fold degeneracy can be constructed using a triple unit cell three times larger than the primitive one [44,47]. Likewise, with out-of-plane 1D folding, two two-fold Weyl points can reshape into one fourfold Dirac degeneracy [40,48]. Thus, extending the band  folding and breaking mechanism to a 3D Brillouin zone (BZ) would bring us a convenient way to realize topological insulators in 3D ACs.
In this paper, we propose a 3D AC with tunable topological surface states based on the 3D band folding and breaking mechanism. We start from one simplest 3D AC structure with a simple cubic lattice by building blocks. By doubling the unit cell in real space, the first isolated band of AC is folded in a 3D BZ, forming two-fold band degeneracy with a nodal plane. Then, we break the band degeneracy with a NaCl-like AC structure to get a large 3D complete bandgap. We can also obtain tunable topological surface states in the bandgap upon different symmetries of domain walls, e.g., mirror or glide symmetry. Our model shows the controllable and reconfigurable abilities of topological sound transport on a 2D plane.

Band Folding and Breaking in a 3D AC
To begin with, we try to construct an AC by building blocks, using two acoustic cavity-tube structures as shown in Figure 1A. The structure parameters are h 0.4a, l h′ 0.2a, where a is the lattice constant. These two cavities having different resonant frequencies act as two different acoustic atoms, represented by b and o. The connecting tubes act as the hopping of neighboring atoms. Thus, as shown in Figures  1B-D, a simple cubic lattice AC can be built using identical b-type acoustic atoms, where the primitive unit cell only contains one atom. Using band folding and breaking mechanism, we can double the primitive unit cell in real space with two atoms to create band degeneracy. Then, we break such degeneracy by replacing the nearest neighbors to be o-type acoustic atoms with a NaCl-like structure. Accordingly, their band structures will experience a folding and breaking process to open a 3D complete bulk bandgap ( Figures 1E-G). Here, we only focus and operate on the first band of such AC. In Figure 1C, the unit cell is chosen to be rhombic dodecahedral geometry which is twice large as that of a simple cubic case, associating with a half volume of bulk BZ in Figure 1F. The band degeneracy originates from additional symmetries in this non-primitive unit cell. In other words, the cubic BZ is folded into a truncated octahedral BZ in 3D.
We resort to the tight-binding approximation (TBA) method to further illustrate the degeneracy created by band folding. The TBA model can well describe our cavity-tube structure with definite local resonance and hopping effect. Considering two acoustic atoms (b and o) in a rhombic dodecahedral unit cell, the Bloch Hamiltonian on the basis of sublattices is: where H AB 2t[cos(ak x /2) + cos(ak y /2) + cos(ak z /2)], Bloch wavevector k (k x , k y , k z ), a represents the lattice constant, t represents the nearest-neighbor hopping, and m 1 (m 2 ) represents the onsite energy of the sublattice b (o). The results are shown in Figures 2A,C. In identical atoms case (m 1 m 2 1), the band structure possesses a two-fold degenerated nodal plane in the first 3D BZ, which is similar to the Fermi surface of copper (Cu). Then, we can lift the nodal plane to open a 3D complete bulk bandgap by changing m 2 2, corresponding to the broken half-lattice symmetry with a NaCl-like AC in Figure 1D.
In our model, the moderate structure parameters already ensure a large complete bandgap with a relative bandwidth of 36%, which facilities the propagation and manipulation of acoustic surface states. Increasing the contrast ratio of two acoustic atoms will bring a broader bandgap. The nodal plane in Figure 2B resembles a sphere with a nearly identical amplitude of Bloch vectors. It implies that the broken half-lattice symmetry would have almost the same effect on the band degeneracy along all directions, which minimizes the directional bandgap overlap

Tunable Topological Surface States on Different Domain Walls
Then, we focus on the surface states on the (001) surface, the same as those on (100) or (010) surfaces. Figure 3A is a supercell configuration without a domain wall along the z-direction. A periodical segment is denoted by T. Figure 3B shows the surface BZ. A typical domain wall can be introduced by adding a mirror symmetry at the interface with a T-T' configuration, where segment T' is the mirror counterpart of T ( Figure 3C). Due to the equivalence of each principal axis in a cubic lattice, T and T' also possess a relative a/2 shift along x-or y-direction. Figure 3D is the projected band structure without the domain wall, where the 3D complete bandgap agrees with Figure 1G. For the case of the mirror-symmetry domain wall, two gapped surface states in the bulk bandgap can be found ( Figure 3E).
These two surface states on the mirror-symmetry domain wall can be taken as an extension of two bound states of two z-direction SSH chains in the k xy plane. Here, we can exchange the onsite and hopping terms and denote the original connecting tube as u. Then, the cuboid supercell is composed of a pair of SSH chains, as 2 /) along the z-axis. Each chain gives one bound state at the domain wall according to 1D SSH with different Zak phases on each side of the interface [41,49]. In a 3D case, these two 0D bound states extend to two 2D surface states with dispersion under the coupling of two chains in the k xy plane. Along the boundaries of surface BZ, the in-plane (k xy plane) coupling is missing. Therefore, two surface states are separated. As inplane coupling increases from BZ boundary to center, these two surface states will gradually merge into bulk bands in opposite directions, forming two inverted bowl shapes ( Figure  3F). Due to the mirror symmetry M z : (x, y, z) → (x, y, −z) relative to the interface, two surface states can also be classified into symmetry and antisymmetry modes. Moreover, the interface is an anti-phase boundary formed by shifting a portion of the crystal lattice as well, which can support pseudospin-momentum locking [50,51]. The acoustic pseudospin can be defined by the rotating direction of the energy flow [52][53][54][55]. The acoustic field distributions of anti-symmetry surface mode P 1 and symmetry mode P 2 at X point, with counter-rotation of energy flow (black arrows), are presented in Figure 3G.
Besides the mirror-symmetry domain wall, our model can possess a glide-symmetry domain wall as well. Here, we consider an additional air layer (O) with a thickness of t x a/2 added on both mirror-symmetry and glide-symmetry domain walls ( Figures  4A,B). As the thickness parameter (t x ) increases from 0.1 to 1, for the mirror-symmetry case (T-O-T' configuration), the antisymmetry mode gradually emerges into bulk bands, and the symmetry one has a blue shift connecting the upper bulk band ( Figure 4C). On the other hand, for the glidesymmetry case (T-O-T configuration), the surface states are always two-fold degenerated at the boundaries of surface BZ ( Figure 4D). Such band degeneracy is the consequence of glide symmetries [56][57][58], i.e., G x : (x, y, z) → G(−x, y + a/2, z) and G y : (x, y, z) → G(x + a/2, −y, z). The 3D views of surface states with t x 1 are plotted in Figures 4E,F.
These flexible and tunable surface states can be used to realize reconfigurable directional acoustic filters. For example, the surface states have directional surface bandgap along Γ X direction in the mirror-symmetry case, which is different from the glide-symmetry case with gapless surface states. Their transmission spectra are shown in Figures 5A,B. In a wide frequency window from 0.58 to 0.76 c/a, Γ M transmission spectra maintain a high valve for both two cases. However, Γ X transmission is forbidden for the mirror-symmetry case in the frequency window from 0.70 to 0.76 c/a, but allowed for the glide-symmetry case. The acoustic field distributions at a frequency of 0.74 c/a in Figures 5C-F match well with projected band structures and transmission spectra. It should be noticed that T and T' are the same structures, only possessing a relative a/2 shift along the x-or y-direction. That means we can efficiently turn on/off sound transport by simply tuning the relative displacement between left-side and right-side ACs in the experiment. Omnidirectional Acoustic Quantum Spin Hall Effect In addition to the directional surface bandgap, we can also construct a complete surface bandgap by modifying the interface. In Figure 6A, we elaborately tune the aforementioned air-interface layer to another modified interface layer C with a/2 thickness composed of uniform o-type acoustic atoms. The mirror symmetry and glide symmetries still sustain in T-C-T' and T-C-T configuration, respectively. The projected band structures are shown in Figure 6B. A complete surface bandgap is constructed for the mirror-symmetry case, forbidding surface sound transport in all directions. For the glide-symmetry case, the surface states at the boundaries of the surface BZ are still degenerated but with flat dispersion.
In the glide-symmetry case, two SSH chains can be described as modified The interface is chosen to be identical o-type acoustic atoms. Like two unmodified chains in the mirror-symmetry case (T-T' configuration), each modified chain also gives one bound state. Around the surface BZ center with strong in-plane (k xy plane) coupling, these two bound states lift in both cases. However, the situation is different at the surface BZ boundaries without inplane coupling. The additional glide symmetry (T-C-T configuration) guarantees two-fold degeneracy, forming two buckled bowl-shaped fashion ( Figure 6C). In each k y (or k x ) slice, these two degenerate surface states turn into a pair of helical edge states, which implies that such glide-symmetry domain wall can act as an omnidirectional acoustic quantum spin Hall layer on a 2D plane [46,59].
The simulated transmission spectra for both mirror-symmetry and glide-symmetry cases are shown in Figure 6D, consisting of their gapped and gapless surface states. The corresponding acoustic field distributions at a frequency of 0.74 c/a are shown in Figures  6E,F. Note that there is a slight dip at a frequency of 0.70 c/a due to the linear degeneracy, resembling the ballistic transport of sound at the surface Dirac point [60,61].

CONCLUSION
To conclude, we construct a sizeable topological bandgap in an AC by building blocks based on a 3D band folding and breaking mechanism. By adjusting the interface symmetries of domain walls, we realize various kinds of topological surface states to turn on/off sound transport efficiently. The omnidirectional acoustic quantum spin Hall layer can be achieved under a glide-symmetry domain wall as well. Compared to 1D edge states in 2D models, these tunable surface states show great potentials in manipulating acoustic signals in multiple directions. Our simple yet flexible 3D AC may provide a route of realizing tunable acoustic devices such as acoustic filters on a 2D plane.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.