Designing topological interface states in phononic crystals based on the full phase diagrams

The topological invariants of a periodic system can be used to define the topological phase of each band and determine the existence of topological interface states within a certain bandgap. Here, we propose a scheme based on the full phase diagrams, and design the topological interface states within any specified bandgaps. As an example, here we propose a kind of one-dimensional phononic crystals. By connecting two semi-infinite structures with different topological phases, the interface states within any specific bandgap or their combinations can be achieved in a rational manner. The existence of interface states in a single bandgap, in all odd bandgaps, in all even bandgaps, or in all bandgaps, are verified in simulations and experiments. The scheme of full phase diagrams we introduce here can be extended to other kinds of periodic systems, such as photonic crystals and designer plasmonic crystals.


Introduction
The topological physics is growing rapidly in condensed matter physics, from quantum Hall effect [1] to topological insulators [2,3] and Weyl semimetals [4].
For one-dimensional (1D) periodic systems, the simplest topologically nontrivial phase exists in polyacetylene [21]. It is found that the band topology of this kind of systems can be characterized by Zak phases, which are quantized topological invariants as long as the unit cell possesses inversion symmetry [22]. In recent years, the topological description has been introduced in various classical counterpart such as 1D photonic crystals and phononic crystals (PCs) [23,24], and been successfully applied to predict the existence of interface states from the Zak phases of bulk bands.
In the system of a 1D PC, for example, a cylindrical waveguide with periodically alternating structures, the macroscopic controllability enables it to be a capable platform to realize the advanced concepts such as band inversion and topological phase transition. Recently, interesting topological phenomena, such as topological interface and edge states [24][25][26], valley-Hall effect [27], have been observed in acoustic systems.
In this work, we focus on the topologically induced interface states within arbitrary bandgap (or bandgaps) by judiciously designing the Zak phases of each constituent PC.
The existence of interface states can be equivalently predicted by the Zak phases, surface impedances, and transmission spectra. As an example, all possible existence of the interface states in the lowest four bandgaps is observed numerically and experimentally, including the interface states in any single bandgap, all odd bandgaps, all even bandgaps, and all bandgaps. The transmission spectra and spatial distributions of the pressure field for the interface states are measured and exhibit excellent agreement with the simulated results.

Results
Topological properties of 1D PCs. The 1D periodic system under study is shown in  plotted by red dashed and black solid lines, respectively. The dimensions for this structure (denoted by S1) are rA=1.3 cm, rB=1.7 cm, dA=8 cm, dB=6 cm. The frequencies of the lowest four band-edge modes at k=0 are indicated by orange (2.321 kHz), green (2.541 kHz), blue (4.664 kHz) and pink (5.046 kHz) dots, respectively. d, Simulated eigen-fields of pressure in the unit cell of S1, corresponding to the four labeled bandedge modes in c. These four eigenmodes can be classified into the even modes (first and last) or odd modes (second and third) according to the field distributions with respect to the inversion center.
The band structure of the 1D PC with sound hard boundaries can be obtained by the transfer matrix method (TMM) [28,29], where k is the Bloch wave vector,  is the angular frequency, va is the sound speed in air (343 m/s), a is the lattice constant (a=dA+dB), and  The topological property of bulk bands for the 1D PC can be represented by their Zak phases, which are defined as [24]: where n Zak is the Zak phase of n th bulk band, and  is the density of air. In general, n Zak can be any value if the choice of the unit cell is arbitrary. However, when the unit cell is chosen to be inversion symmetric with the inversion center being the middle of tube-A or tube-B, it can be proved that n Zak should be quantized as either 0 or . The Zak phase can take different values (0 or for different choices of the unit cell [22,23]. The quantized n Zak characterizes the topology of the corresponding band. When two semi-infinite PCs are connected at an interface, the condition for the existence of an interface state in the n th bandgap is that the impedances on both sides of the interface satisfy the condition: ZL+ZR=0 [23]. Here, ZL and ZR are respectively the impedances of the left-hand and right-hand PCs, and can be expressed by the reflection coefficient from the left (right) side of the interface as where Z0 is the impedance of the free space. As is known, for the semi-infinite phononic crystal, the surface impedance Z is purely imaginary in the bandgap, i.e. Z/Z0=i. According to the bulk-interface correspondence for the 1D PC, the sign of  (n) within the n th gap can be related to the Zak phases of the bulk bands below this gap by the following relation, as long as there is no band crossing below this gap [23]: or cyan (negative) strips, respectively. The sign of  (n) can be identified not only by Eq. (4) but also by the parity of the band-edge states below and above the corresponding bandgap [23].
The band-edge states are marked by the dots at the Brillouin zone boundary (k=±/a) and center (k=0), and their parities are represented by their colors with blue for odd and orange for even respectively to the odd and even modes. If the eigenmode at the lower edge of the bandgap has odd parity while the one at upper edge is even, then sgn(<0; otherwise, if the eigenmode at lower edge is even and the one at the upper edge is odd, then sgn()>0 [23].
Topologically induced interface states in n th gap can be achieved by different combinations of structures Si and Sj (i, j=1, 2, 3, 4, and i ≠ j) as long as  (n) of the left-hand and right-hand PCs take opposite sign, i.e. the colors in the n th bandgap for Si and Sj are different. As shown in Fig. 2, for the configuration of S1+S2 (or S3+S4), there should exist interface states in all the odd bandgaps. Whereas for the configuration of S1+S3 (or S2+S4), there are interface states in the even bandgaps. For S1+S4 (or S2+S3), the two PCs fall into the opposite phases in all bandgaps, consequently, interface states appear in all the bandgaps. Note that when the system is impinged by a plane wave, the inference for these 3 cases is not restricted to the first four bandgaps, but is applicable for all higher-ordinal bandgaps as long as the weak dispersion limit is satisfied [30].
Achieving interface states from the full phase diagram. Analogous to the band inversion in electronic systems [31,32], when the bandgap closes and reopens implies that the topological phase transition occurs in the system of PC. The loci of the band crossing point will separate the different phases of a certain bandgap, which can be characterized by the sign of (n) within this bandgap. With the condition of band crossing [23,30], band inversion can be achieved when the geometric parameters in Eq. (1) satisfy: (i) rA=rB, then the structure degenerates to the trivial case, i.e. a non-structured tube with a constant cross section; or (ii) dA/dB=(dm+d )/(dm−d )=n1/n2, where n1 and n2 are integers, as such the band crossing occurs at the (n1+n2) th bandgap [23]. According to these two criterions, all the phase transition lines in the phase diagram of a bandgap can be obtained. With the full phase diagrams in the (r,d ) space, we can construct topologically induced interface states in arbitrary bandgap(s). As shown in Table 1   The simulated transmission spectra of two connected PCs (6 unit cells for both the left and right PCs), for S1+S2, S1+S3, and S1+S4, respectively. h-j, The corresponding measured transmission spectra, where the transmission peaks arising from the interface states in each bandgap are highlighted by colored dots.
Further, the existences of topological interface states are experimentally verified, and the experimental set-up is shown in Fig. 5a. The sample contains two connected PCs, the right-hand one in red is manufactured with the geometric parameters of S1 and the lefthand one in blue is manufactured with the geometric parameters of S2,3,4 (only the photo of S1+S2 is shown). The junction is marked by the green arrow, and the PCs on either sides are truncated at the boundaries of an intact unit cell. Meanwhile, a loudspeaker and four microphones are used to generate input signals and to measure the transmission spectra, respectively. The measured transmission spectra for the three configurations are respectively shown in the lowest panel of Fig. 5. The transmission peaks marked by colored dots in the corresponding bandgaps confirm the existence of the interface states.
The experimental results verify that interface states appear in all odd bandgaps for configurations of S1+S2, whereas in all even bandgaps for configurations of S1+S3, moreover, in all bandgaps for configurations of S1+S4. The frequencies of interface states agree well with our theoretical predictions and simulated results.

Discussion
In summary, we have proposed a scheme to design interface states at the junction of two 1D  Data availability. The data which support the figures and other findings within this paper are available from the corresponding authors upon request.

Additional Information
Supplementary information is available in the online version of the paper.