Robust separation of topological in-plane and out-of-plane waves in a phononic crystal

Valley degree of freedom, associated with the valley topological phase, has propelled the advancement of the elastic waveguide by offering immunity to backscattering against bending and weak perturbations. Despite many attempts to manipulate the wave path and working frequency of the waveguide, internal characteristic of an elastic wave such as rich polarization has not yet been utilized with valley topological phases. Here, we introduce the rich polarization into the valley degree of freedom, to achieve topologically protected in-plane and out-of-plane mode separation of an elastic wave. Accidental degeneracy proves its real worth of decoupling the in-plane and out-of-plane polarized valley Hall phases. We further demonstrate independent and simultaneous control of in-plane and out-of-plane waves, with intact topological protection. The presenting procedure for designing the topologically protected wave separation based on accidental degeneracy will widen the valley topological physics in view of both generation mechanism and application areas. Judicious design of acoustic metamaterials has so far permitted control of topologically-protected waves in one plane of a two-dimensional system. Here, the symmetry of a phononic crystal is tuned geometrically to realise independent tuning of both in-plane and out-of-plane topological modes in the same frequency range.

Elastic wave, phononic fluctuation in a solid medium, has some unique characteristics exhibiting distinguished applications related to the waveguide. Because elastic wave speed is significantly slower than the photonic wave, elastic wave devices can be used to manipulate the photonic information compactly with phonon-photon coupling 36 . The pervasiveness of elastic vibration emphasizes the elastic wave routing and focusing for the energy harvesting 37 . One other peculiar characteristic of the elastic wave is that it has both longitudinal and transverse wave components, namely, rich polarization, while acoustic wave has only longitudinal one and photonic wave has only transverse one. These two components of elastic wave have been utilized independently in both engineering and science owing to their distinct physics, although hybridization and conversion between variously polarized components of elastic wave are quite common. Accordingly, filtering out a singly polarized wave or splitting the elastic waves according to their polarization has been one of the crucial questions in controlling the elastic wave, and various phononic crystals (PCs) have been designed to solve this problem based on the polarization bandgap 38,39 , wave phase difference 40 , and tensor engineering 41 .
Elastic PCs possessing valley Hall phase also have been limited to deal with only one polarized component of elastic wave that the lifted Dirac degeneracy belongs to, in the formation of the valley Hall phase. Mostly in two-dimensional PCs, the valley topological phase acts only on either in-plane polarized (IPP) wave [26][27][28][29] or out-of-plane polarized (OPP) wave 24,25,30,31 leaving the other one as redundancy. However, because IPP and OPP waves have different physics, Dirac degeneracy of IPP and OPP waves can be set to overlap so that IPP and OPP valley Hall phases are generated simultaneously in the same frequency. This gives the valley Hall phase the potential to be utilized for the mode separation of elastic waves and may serve the unparalleled robustness in the elastic wave mode separation, named topologically protected separation.
The challenge on the separation exists obviously, regarding the strategy generating valley Hall phase. A common strategy utilizing the deterministic Dirac degeneracy cannot break the link between IPP and OPP valley Hall phases since the degeneracy depends on the point group symmetry without distinguishing IPP and OPP waves. Accidental Dirac degeneracy, a complementary set of the deterministic Dirac degeneracy, only has the possibility to differentiate IPP and OPP valley Hall phases. Though only a few topological PCs were reported utilizing accidental degeneracy, and they are also limited to deal only one polarized component 27 . Moreover, sufficient condition for constructing accidental degeneracy has not been explored so far.
Here, we demonstrate the topologically protected elastic wave separation into IPP and OPP waves with two-dimensional PCs possessing superposed valley Hall phases, i.e., superposition of IPP and OPP valley Hall phases, based on accidental Dirac degeneracy. Sufficient conditions for constructing accidental degeneracy and strategies exploiting the degeneracy for multiply polarized waves in the identical medium are found and utilized for the separation of IPP and OPP topological edge states. First, we design the PCs with three tunable geometric parameters, so that the set of parameters establishing the deterministic Dirac degeneracy is connected in the parameter space. Then, the continuity of Bloch frequency with respect to the geometric parameters guarantees the existence of accidental degeneracy. This accidental degeneracy together with the following topological phase diagram reveals the separation of IPP and OPP phases, so all possible superposed topological phases. We find a pair of PCs possessing the same bandgap by Jaccard index in this setup for the topologically protected separation. A topologically protected edge state with any polarization can be created utilizing the pair of PCs and their valley pairs. Using these edge states, topologically protected independent routing of IPP and OPP waves is demonstrated as well as the topologically protected separation into IPP and OPP waves.

Results
Design of phononic crystals with three tunable parameters. The designed hexagonal-lattice PC is a truss-like structure with periodic hexagonal-like holes as shown in Fig. 1a, b. The substrate is a 2 mm thick linear elastic plate with the properties of conventional acrylic plastic (method), and the size of the unit cell is l = 10 mm with a thickness of the ligand wall t = 1.5 mm. Figure 1c, d shows the schematic representation of the unit cell without and with the perturbation, respectively. Two differentsized hemicylinders are attached to the hexagonal cell walls alternatively so that C 3 symmetry is preserved throughout the PC. Three tunable geometric parameters exist given as c, b 1 /B 1 , and b 2 /B 2 . c represents the nondimensionalized radius difference with the radiuses of the hemicylinders r 1 = (1 + c)r 0 and r 2 = (1 − c)r 0 . b 1 and b 2 represent the respective displacement of the hemicylinders displaced from the center of the hexagonal cell walls, while B 1 and B 2 are the maximum magnitudes that b 1 and b 2 can attain, respectively. For full restriction of the parameters b 1 and b 2 , see Supplementary Note 1 with Supplementary Fig. 1. The value of c is restricted to reside in [0,1], and the values of (b 1 /B 1 , b 2 /B 2 ) is restricted to range in |b 1 /B 1 | + |b 2 /B 2 | ≤ 1 for convenience. Among three parameters, only the locations of two hemicylinders (b 1 /B 1 , b 2 /B 2 ) are regarded as variables for perturbation, and the radius difference c is usually fixed.
Since the structure preserves the horizontal mirror plane (σ h ) symmetry or is equivalently symmetric with respect to its midplane, symmetric and antisymmetric waves corresponding to IPP and OPP waves, respectively, are preserved and differentiated. In the band diagram, polarization factor p 42 , defined to be 0 for purely IPP wave and 1 for purely OPP wave (method), verifies this correspondence on the frequency of interest (around bandgap or Dirac degeneracy) by having a value lower than 0.01 (IPP, blue) or higher than 0.86 (OPP, red) only as shown in Fig. 1e. The Dirac degeneracy appears when (c, b 1 /B 1 , b 2 /B 2 ) = (0, 0, 0), (0, 0.3, 0.3), and (0.25, 0, 0) for both IPP and OPP waves, while the bandgap appears when (c, b 1 /B 1 , b 2 /B 2 ) = (0.25, 0.3, 0.3). Dirac frequencies for the IPP and OPP wave band are adjacent when the Dirac degeneracies appear owing to the appropriate choice of the thickness dimension of the two-dimensional PCs. As the thickness changes, only OPP band frequency varies, but IPP band frequency does not change as shown in Supplementary Fig. 2.
Dirac degeneracies that appeared in Fig. 1e reveals to be the deterministic Dirac degeneracy protected by some point group symmetry of wavevector k, as follows. When the PC is unperturbed [(b 1 /B 1 , b 2 /B 2 ) = (0,0)], the energy band features the deterministic Dirac dispersion at both valleys as a result of C 3v symmetry 35 (mirror symmetry as well as C 3 symmetry) regardless of the value of c. Contrastively, when the PC is perturbed [(b 1 /B 1 , b 2 /B 2 ) ≠ (0,0)], the mirror symmetry breaks and only C 3 symmetry is intact unless c is not 0 nor 1, eliminating the deterministic Dirac degeneracy. If c is 0 or 1, deterministic Dirac degeneracy can appear even if the PC has nonzero perturbations (b 1 /B 1 , b 2 /B 2 ). For the PC with c = 0, the radiuses of the hemicylinders are all the same, so the energy band features deterministic Dirac degeneracy at both valleys if b 1 /B 1 = b 2 /B 2 as a result of C 6 symmetry 35 . When c = 1, the Dirac dispersion appears when b 1 /B 1 = 0 regardless of the value b 2 /B 2 , since the hemicylinder with radius r 2 vanishes (r 2 = 0). Supplementary  Fig. 3, which is the extended version of Fig. 1e, shows the band diagrams for all these deterministic Dirac degeneracies, together with the bandgaps or accidental degeneracies in the case when there is no point group symmetry generating deterministic Dirac degeneracy.
Accidental Dirac degeneracy from deterministic Dirac degeneracy. We had found all PCs featuring the deterministic Dirac degeneracy. At least one among three conditions must be satisfied for the degeneracy: (I) 0 < c < 1 and b 1 /B 1 = b 2 /B 2 = 0; (II) c = 0 and b 1 /B 1 = b 2 /B 2 ; and (III) c = 1 and b 1 /B 1 = 0 (Fig. 2a). By merging these three conditions in parametric space (c, b 1 /B 1 , b 2 /B 2 ), a set of all parameters enabling the deterministic Dirac degeneracy is revealed to be a connected set, P DD (Deterministic Dirac degeneracy) , as shown in Fig. 2b. This connectedness property is important because it ensures that the components of all deterministic Dirac degeneracies are identical regardless of the parameter value (c, b 1 /B 1 , b 2 /B 2 ) for each IPP wave and OPP wave.
To reveal the components of deterministic Dirac degeneracy, two degenerated states at valley K of the unperturbed PC [(c, b 1 /B 1 , b 2 /B 2 ) = (0, 0, 0)] are depicted for each IPP and OPP waves in Fig. 2c. In every state, one sublattice between p and q moves while the other is still, as observed in kinetic energy distribution. The four sequential displacement profiles taken at one-third of a period intervals reveal the time-varying motion of the states. For IPP degenerated states, each moving sublattice rotates circularly 16 , while for OPP degenerated states, moving sublattice shows gyro motion 25 . The direction of the gyro or circular rotation are opposite between two degenerated states for both IPP and OPP waves, proving that two degenerated states have opposite chirality 25 . For the notation (symbol), moving sublattice p/q (triangle/inverted triangle) and the sign indicating the chirality ± (filled/unfilled) of the motion are used 16 , and colored to indicate whether it belongs to IPP wave (blue) or OPP wave (red). Following this notation (symbol) rule, states p− (triangle unfilled) and q+ (inverted triangle filled) reveals as the components of deterministic Dirac degeneracy, for both IPP and OPP waves. The degenerated states at valley K′ also can be deduced from that of valley K owing to the preserved time-reversal symmetry.
The states p− and q+ for both IPP and OPP waves are preserved even when the deterministic Dirac degeneracy breaks by a slight change of b 1 /B 1 or b 2 /B 2 , owing to the intact C 3 symmetry 11 . Accordingly, as long as the parameter (c, b 1 /B 1 , b 2 /B 2 ) of the PC is in the neighborhood of P DD , the PC must have states p− and q+ on band-edge frequencies at valley K, for both IPP and OPP waves. We consider the difference of the frequencies between states p− and q+, in the neighborhood of P DD to verify the existence of accidental degeneracy whether for IPP wave or OPP wave. Δω must be continuous with respect to the geometric parameters (c, b 1 /B 1 , b 2 /B 2 ) with its image covering both some positive and negative values, so there must be a surface including P DD in the parameter space which satisfies Δω = 0 (Supplementary Note 2). Thus, nonzero perturbations (b 1 /B 1 , b 2 /B 2 ) in the vicinity of (0,0) which allow the PC to have the Dirac degeneracy at valleys always exist for any c between 0 and 1, verifying the existence of accidental Dirac degeneracy for all c between 0 and 1.
Topological phase diagram with superposed valley hall phase.
The PC with lifted degeneracy or bandgap, generated by a slight change of (b 1 /B 1 , b 2 /B 2 ) from the PC possessing Dirac degeneracy, turns out to have a nontrivial valley topological phase owing to its valley pseudospins named p− and q+ states. The frequency order of the split degenerated-states (a sign of Δω) directly discloses the valley Hall phase 16 , whereas, in the strict sense, it is nontrivial valley Chern number C V K revealing the valley Hall phase. Two valley Hall phases may arise for each IPP and OPP wave as shown in Fig. 3a: IPP phase i 0 for Δω < 0 (or C V K = 0.5) and i 1 for 0 < Δω (or C V K = −0.5); and OPP phase o 0 for 0 < Δω (or C V K = 0.5) and o 1 for The topological phase of the PC for general perturbations (b 1 /B 1 , b 2 /B 2 ) with fixed c is visualized by the topological phase diagram. As shown in Fig. 3c, when c is fixed to be 0, topological phase transition takes place only on the deterministic degeneracy line b 1 /B 1 = b 2 /B 2 , so the topological phase boundary for IPP wave and OPP wave matches perfectly. Consequently, IPP phase i 0 (i 1 ) and OPP phase o 0 (o 1 ) are inherently tied in the PC with c = 0, so only i 0 o 0 and i 1 o 1 are allowed to appear among four possible superposed topological phases. Contrastively, in the PC with c = 1 in which deterministic degeneracy line exists as well, the topological phase diagram shows that the IPP phase i 0 (i 1 ) and OPP phase o 1 (o 0 ) are inherently tied, so only i 0 o 1 and i 1 o 0 are allowed to appear (Fig. 3d). In any case, the deterministic Dirac degeneracy indeed turns out to be inadequate for our purpose of decoupling the IPP and OPP edge states.
We already proved the existence of accidental Dirac degeneracy for all of c between 0 and 1. To figure out the formation nature of the accidental Dirac degeneracy line, the topological phase diagram for IPP wave and OPP wave independently presents for the recognized cases c = 0 and 1 in Fig. 3d. In each IPP(OPP) topological phase diagram, we define a continuous function of c, θ IPP = θ IPP (c) (θ OPP = θ OPP (c)), as the tangent angle to degeneracy line at (b 1 /B 1 , b 2 /B 2 ) = (0,0) having i 1 (o 1 ) phase on its right side. Because the difference between θ IPP and θ OPP , is 0 (mod 2π) for c = 0, and π (mod 2π) for c = 1, there must be some c 0 between 0 and 1 satisfying so that all possible superposed topological phases, i 0 o 0 , i 1 o 1 , i 0 o 1 , and i 1 o 0 , emerge on the neighborhood of (b 1 /B 1 , b 2 /B 2 ) = (0,0) with the equal portions. Figure 3e shows the topological phase diagram for c = 0.25, in which all superposed topological phase emerges with nearly the same proportion. θ IPP and θ OPP with respect to c and topological phase diagram for various c is drawn in Supplementary Fig. 7.
Topologically protected edge state from accidental Dirac degeneracy. For two topological PCs to form the topologically protected edge state, they need to have not only different topological phases, but also the common bandgap frequencies. Specially, since we deal with IPP and OPP waves simultaneously, four bandgaps should share the same frequency range: each IPP and OPP wave bandgap for both the two PCs. Jaccard index J, the index gauging the similarity between two sets, is utilized here to describe the correspondence between bandgaps, while it attains 0 if two bandgaps have no intersection, and 1 if two bandgaps coincide (Supplementary Note 3). Figure 4b shows the Jaccard index between IPP and OPP bandgaps with respect to perturbation variables b 1 /B 1 and b 2 /B 2 for the PCs with c = 0.25, whose topological phase diagram is shown in Fig. 4a. Since the intersection of bandgaps vanishes on the degeneracy lines, the Jaccard index is 0 on both IPP and OPP degeneracy lines and gets larger as the perturbation goes farther from the degeneracy lines. Consequently, the Jaccard index has a significantly large value following two crossing lines L 1 and L 2 approximately, and the eigenfrequencies following these lines clearly show the significant correspondence between IPP and OPP bandgaps ( Supplementary  Fig. 8). Next, to find two PCs with different topological phases and common bandgap, the Jaccard index is calculated again between intersected bandgaps along L 1 and L 2 lines (Fig. 4c). In Fig. 4c, every point with a significantly large Jaccard index represents two topological PCs, candidates for generating the topologically protected edge states with superposed topological phases.
On the final selection of the PCs for the valley topologically protected edge state, the intervalley mixing should be carefully considered, because large intervalley mixing may directly disturb the topological protection. The reduction of the numerically calculated valley Chern number compared to the ideal magnitude of 0.5 reveals the intervalley mixing level, as the magnitude represents the localization level of Berry curvature near the valleys: the larger the magnitude of the valley Chern number, the smaller the intervalley mixing 30 . (In this work, deviation of the transmission from 0 dB is shown in transmission spectra in Supplementary Figs. 5c and 12b, d, f, and they reveal the relation between intervalley mixing and valley Chern number together with the numerically calculated valley Chern number in Supplementary Figs. 4 and 9.) The magnitude of the intervalley mixing is strongly and positively correlated with perturbation intensity, so limited perturbation intensity is required for small intervalley mixing 43 . However, limited perturbation intensity also leads to the limited width of the bandgap, which means a narrow working frequency for the topological protection. Thus, a moderate level of perturbation intensity is required for the valley PC to have both negligible intervalley mixing and significant bandgap 19 .
Among all candidates sorted by Jaccard index (Fig. 4c) Supplementary Fig. 9a), and the common bandgap wider than 5% (Fig. 4d). Their mirror-symmetry pairs with perturbations (b 1 /B 1 , b 2 /B 2 ) = (−0.25, −0.15) and (−0.15, 0.35), named S 01 and S 00 , respectively in the same way, also satisfy the desired conditions with the same common bandgap undoubtedly. Now, four PCs (S 00 , S 01 , S 10 , and S 11 ) having all distinct superposed topological phases with common bandgap are successfully designed. For clarity, Bloch wave functions and Berry curvatures for the four PCs were drawn in Supplementary Fig. 9. Topological phase transition among S 11 , S 10 , and S 01 , revealed by Bloch frequency with Bloch states at valley K, clearly shows that the formation of accidental Dirac degeneracy can differ according to the polarization of elastic wave, and indicate how to design polarized and topologically protected edge states (Fig. 4e).
The number of topologically protected edge states emerging at the boundary between two PCs is the same as the difference of topological index between the two PCs, as the bulk-boundary correspondence states. Figure 5a shows the domain wall S 01 S 10 , in which S 01 is located on the upper domain and S 10 is located on the lower domain, together with its band diagram. Because the difference of ideal valley Chern number between S 01 and S 10 is 1 for both IPP and OPP waves, one IPP edge state and one OPP edge state must emerge on the domain wall S 01 S 10 if the intervalley mixing is negligible according to the bulk-boundary correspondence. Supplementary Fig. 10a proves that the blue and red solid lines in the band diagram correspond to these IPP and OPP edge states, respectively, validating that the intervalley mixing is negligible in this domain wall. In the same manner, domain wall S 01 S 11 possesses one IPP topological edge state only ( Fig. 5b and Supplementary Fig. 10b), and domain wall S 11 S 10 possesses one OPP topological edge state only ( Fig. 5c and Supplementary Fig. 10c) according to the bulk-boundary correspondence. The IPP edge state in the domain wall S 01 S 11 and the OPP edge state in the domain wall S 11 S 10 resemble IPP and OPP edge states of the domain wall S 01 S 10 respectively, because S 11 has the same topological phase with S 10 for IPP wave and, with S 01 for OPP wave (Supplementary Fig. 10). The band diagrams for the other domain walls are shown in supplementary Fig. 11. Topological protection against sharp angles at the proposed domain walls are also demonstrated in Supplementary  Fig. 12.
Waveguiding dependent on elastic wave polarization. Separation of the elastic wave into IPP and OPP waves can be achieved by PC constructed with the three domains S 10 , S 01 , and S 11 shown in Fig. 6a. The separation mechanism is depicted in Fig. 6b. The hybridized vibration, a mixture of IPP wave and OPP wave, is injected into the PC through port 1. Both IPP wave and OPP wave can be guided to the junction from port 1 through the S 10 S 01 domain wall based on the nontrivial edge states. The wave is separated into IPP wave and OPP wave following the bulkboundary correspondence at the junction. The IPP wave transmits toward port 2 from the junction because S 11 has IPP topological phase (i 1 ) same as S 10 (i 1 ) but different from S 01 (i 0 ). In contrast, the OPP wave transmits toward port 3 because the OPP topological phase of S 11 (o 1 ) is different from S 10 (o 0 ) but the same as S 01 (o 1 ). As a result, the hybridized vibration injected through port 1 gets split into IPP and OPP waves through ports 2 and 3, respectively. Moreover, since every wave path for the separation is the boundary of two domains with different topological phases, the wave separation process, as well as the wave propagation, is topologically protected. IPP displacement and OPP displacement fields are presented in Fig. 6c, d, respectively, for harmonic hybridized vibration input to port 1, showing perfect separation of the IPP and OPP waves. Here, no standing wave is observed despite the wave path bend, showing the robustness of topologically protected waves. The separation performance for more generalized input to port 1 is demonstrated in Supplementary Fig. 13c, d. More evidences demonstrating the topologically protected separation including broadband characteristics within the topological bandgap of the separation and robustness against various disorders are presented in Supplementary  Figs. 13, 14, respectively. Moreover, we tested if the separation performance is maintained even with the nonlinear effect in Supplementary Fig. 16. We found that the separation performance is preserved even when the maximum displacement is several times larger than the plate thickness with a nonlinear effect.
In addition to the separation, IPP and OPP waves can be guided simultaneously and independently in the same platform.
They can be merged, separated, or even cross each other, since we have all superposed topological phases. This independent routing ability of IPP and OPP waves has great potential for information transfer because IPP and OPP guided waves can be utilized as two different non-crosstalk channels. Figure 7a shows the PC design with three decided domains S 10 , S 11 , and S 01 , and one undecided domain X for independent routing of IPP and OPP waves. Irrelevantly to X, the information can be transferred through IPP wave via port 2 or through OPP wave via port 1 (Fig. 7b). By choosing the appropriate domain for X, the IPP wave and OPP wave can be transferred to either port 3 or port 4, independent of each other. For instance, when the domain X is S 00 , because the IPP phase is the same between S 10 and S 11 , and between S 00 and S 01 , and because the OPP phase is the same between S 11 and S 01 , and between S 10 and S 00 , the IPP wave is routed toward port 4, and the OPP wave is routed toward port 3, crossing each other. The rest cases are presented in Fig. 7c, verifying the potential of elastic waves in the information transfer. Supplementary Fig. 17 also demonstrates the crossing of IPP and OPP guided waves, but together with their topological protection against the bent path.

Discussion
In this work, we have utilized the rich polarization of elastic wave together with valley degree of freedom via the accidental Dirac degeneracy, to achieve the topologically protected elastic wave separation according to the polarization of the wave. IPP and OPP valley Hall phases, which are usually dealt separately, are designed to generate the superposed valley phase which can be used to deal with all components of elastic wave in the frequency of interest. In this process, all the valley Hall phase is provoked via the accidental Dirac degeneracy instead of deterministic degeneracy, so that IPP and OPP valley Hall phases can be coupled arbitrarily without sticking to each other. The generation mechanism of accidental degeneracy is for the first time proposed with the deductive reasoning based on deterministic Dirac degeneracy, as no sufficient mechanism generating the accidental degeneracy have been revealed to the best of our knowledge. Strategy to achieve the topologically protected separation with the accidental degeneracy is also suggested with the aid of the Jaccard index.  6 Topologically protected elastic wave separation into in-plane polarized (IPP) and out-of-plane polarized (OPP) wave. a Configuration of phononic crystals (PCs) with Y-junction made of S 10 , S 01 , and S 11 for the separation of elastic waves. All edges between two PCs are zigzag edges, in which intervalley scattering is negligible. b Separation mechanism of an elastic wave at the Y-junction. The Violet arrow represents the hybridized vibration or wave injected to port 1. IPP and OPP wave trajectory are denoted as arrows, blue-colored for IPP and red-colored for OPP wave respectively, where ideal valley Chern numbers at valley K of each PCs are present to support the existence of the corresponding edge states. c, d In-plane displacement field (c) and out-of-plane displacement field (d) of the configuration for the hybridized vibration of 30 kHz injected to port 1. Recently, Wen, X. et al. suggested the topologically protected edge state based on accidental degeneracy using rotating scatterers 27 . However, the rotating scatterers do not necessarily guarantee the existence of accidental Dirac degeneracy, and accordingly, such accidental degeneracy cannot be tuned easily. In this work, we have tuned the accidental degeneracy line to effectively separate the IPP and OPP topological phase, owing to the definite existence of accidental degeneracy offered by mechanisms adopted. Regarding the elastic wave separation using the topological property, separation according to the valleypolarization has been demonstrated numerically and experimentally based on helical-valley modes 32 . Elastically polarized IPP and OPP waves had been hybridized to generate helical mode, thus separated wave was linear-mode-hybridized as well. In this work, we have differentiated IPP and OPP waves from the beginning, and the superposed valley Hall phase is used to separate them, instead of helical-valley mode. Thus, this work is distinguished by both the standard of separation, and physics underlying the separation. Lastly, separation of IPP and OPP waves has been demonstrated based on the polarization bandgap 38,39 , wave phase difference 40 , and tensor engineering 41 . What differentiates this work from them is the topological protection during the separation process. In this work, thanks to the topological protection, the separation can be maintained even when the structure is a little bit perturbed, and energy or information can be preserved with transmission close to unity during the separation process.
While we have only concentrated on the frequency around 30 kHz in this work, the working frequency for the separation can be tuned according to the application area owing to the scalability of elastic PCs. In the ultrasonic or higher frequency range, independent controllability for IPP and OPP waves can be utilized in signal processing or information transfer, by treating the elastic PCs as the multiple channels, which have no crosstalk with each other. In audible or lower frequency range, elastic wave manipulation can be used in energy harvesting or vibration isolation, while independent controllability for IPP and OPP waves enables each IPP or OPP wave to be applied independently as needed. This work can also be adopted in sensor or actuator applications, since 2D motion can be converted into two 1D motion by the separation of IPP and OPP waves (or vice-versa by merging IPP and OPP waves). We note that the same procedure can also be applied to steer the transverse electric (TE) mode and transverse magnetic (TM) mode independently and simultaneously in photonic devices.

Methods
Numerical simulations. The commercial software COMSOL Multiphysics Structural Mechanics (Solid Mechanics) module, which is based on the finite element method (FEM), is used for numerical simulations. The computer models used for the simulations are three-dimensional solid, not the two-dimensional plate except for Supplementary Figs. 5c and 12b, d, f: In these figures, 2D FEM based on Mindlin-Reissner plate theory is used. The substrate is assumed to have the properties of acrylic plastic with a density of 1190 kg m −3 , Young's modulus of 3.2 GPa, and Poisson's ratio of 0.35. We note that acrylic plastic was chosen for the ease of fabrication to conduct the experiments later on, and the same phenomenon in this work can be realized with the properties of other substrates. The maximum element size of the mesh is set to be less than one-thirtieth of the wavelength of 30 kHz shear wave in acrylic plastic. Eigenfrequency study is implemented with Bloch periodicity boundary conditions for calculating the energy band diagram for both bulk and edge, while single primitive cell is used for the bulk and 24 × 1 supercell composed of two different primitive cells are used for the edge 23 . Polarization factor 42 , is calculated for each state to differentiate IPP and OPP waves in the band diagram, where (u, v, w) is the displacement field. After the differentiation of the state, IPP and OPP band is drawn independently with 48 (51) points for the bulk (edge), equispaced in k-space. On the other hand, for the full-field simulation, frequency-domain perturbation study is implemented with a low-reflecting boundary (for 3D FEM) or spring foundation (for 2D FEM) on the outermost boundary of configuration. The in-plane and out-of-plane displacement field is calculated by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u j j 2 þ v j j 2 p and |w | , respectively, and the polarization factor for each port is calculated by the expression given in Eq. (4) with the volume comprised of 10 × 3 primitive cells at each port, where (u, v, w) is complex-valued displacement field with respect to the harmonic perturbations.
Computation for topological phase diagram and Dirac degeneracy angle.
To classify the topological phase of PCs without calculating the valley Chern number, and are calculated for each state in addition to the polarization factor p where (u X , v X , w X ) is displacement field at sublattice X. Since every valley state at valley K is either p− or q+, the sign of η IPP and η OPP and value of p determines the valley Hall phase of the PC. A topological phase diagram is drawn by determined valley Hall phases, with 20,201 points when c = 0.25, with 5101 points in Supplementary  Fig. 6, and with 841 points for the rest. For the Dirac degeneracy angle θ IPP and θ OPP , valley Hall phases are determined for PCs with perturbations (b 1 /B 1 , b 2 /B 2 ) = 0.1(cos γ, sin γ) where γ varies from 0°to 180°with 181 steps. Then, θ IPP and θ OPP are determined with error less than 0.5°by x-intercept of the graph Δω = ω p− − ω q+ with respect to γ for IPP and OPP waves.
Calculation of valley Chern numbers. From the Bloch wave function or displacement field of the primitive cell, integration of Berry curvature over an infinitesimal area or line integration of Berry connection around the area can be calculated by the phase evolution of Bloch wave function along with the counterclockwise rotation. The infinitesimal area used here is square 44

Data availability
All data needed to reach the conclusions of this study are included in this published article and/or the Supplementary Information. Additional data related to this study are available from the corresponding author on reasonable request.