### Nonzero acoustic spin in waveguide with symmetry breaking

According to the physical meanings possessed in angular momenta of acoustic waves, the SAM density can be described as (refs. ^{5,6,7,8}):

$${bf{s}}=frac{rho }{2omega }{rm{Im}}[{{bf{v}}}^{* }times {bf{v}}],$$

(1)

where *ρ* is the mass density, *ω* is the frequency, *c* is acoustic velocity, and **v** is acoustic velocity field. According to the definition in Eq. (1), the acoustic SAM density will be nonzero for the local circularly (or in general, elliptically) polarized profile of acoustic velocity fields. To satisfy the circularly polarized velocity field conditions, we introduce nonsymmetric boundary settings to break the *x*-mirror symmetry of waveguide, shown in Fig. 1b, which will led to significant phase delay for *v*_{x} compared with *v*_{z}. By general symmetry argument, setting ({{mathcal{M}}}_{y})(*y* → −*y*) as the *y*-mirror operation, we have ({{mathcal{M}}}_{y}{s}_{x}(y){{mathcal{M}}}_{y}^{-1}=-{s}_{x}(-y)), ({{mathcal{M}}}_{y}{s}_{y}(y){{mathcal{M}}}_{y}^{-1}={s}_{y}(-y)) and ({{mathcal{M}}}_{y}{s}_{z}(y){{mathcal{M}}}_{y}^{-1}=-{s}_{z}(-y)). So for the nondegenerate waveguide mode, when the cross-section has the *y*-mirror symmetry, the cross-section SAM *S*_{x} = ∫*s*_{x}*d**x**d**y* and *S*_{z} = ∫*s*_{z}*d**x**d**y* will definitely vanish due to the cancellation of odd-symmetric SAM densities on the waveguide section. Similarly, the *x*-mirror symmetry of the system will lead to the vanishing *S*_{y} = 0 and *S*_{z} = 0 due to the odd-function *s*_{y,z}(*x*) = −*s*_{y,z}(−*x*). Therefore, to induce the nonzero SAM, e.g., *S*_{y} ≠ 0 for nondegenerate modes, one could break the ({{mathcal{M}}}_{x}) symmetry.

Considering the combination of the sound hard boundary (*ϕ* = 0) and the metasurface-induced soft boundary (*ϕ* = *π*) in Fig. 1b, we can analytically obtain that the waveguide mode indeed processes the nonzero total SAM as (see “Method”):

$${bf{S}}=frac{pi | {p}_{0}{| }^{2}}{rho {omega }^{3}}kR{J}_{0}^{2}(kappa R){{bf{e}}}_{y},$$

(2)

where *J*_{0}(*κ**r*) is the zero-order Bessel function of the first kind, **e**_{y} is the unit vector in *y*-direction, *k* is the longitudinal momentum component with **k** = *k***e**_{z} the wave vector along *z*-direction, (kappa =sqrt{{omega }^{2}/{c}^{2}-{k}^{2}}) is the transverse wave vector, *c* is the sound speed in the air, *R* is the radius of the waveguide, and *p*_{0} is the pressure field amplitude. Based on these nonsymmetric boundary conditions, the meta-waveguide eigenmodes will naturally carry nonzero SAM *S*_{y} ≠ 0 in the *y*-direction, depicted in Fig. 1c. Specially, this SAM is tightly locked to the momentum direction that reversing the momentum *k* will flip the spin, which resembles the spin-momentum locking in QSHE^{2,11}. By using the spin sources (circularly polarized acoustic dipoles)^{5,6} corresponding to different SAM, we can excite the waveguide modes selectively, as shown in Fig. 1d.

It should be mentioned that this spin-momentum locking feature of meta-waveguide modes is different from the cases in the surface evanescent modes^{1,30} due to nontrivial bulk topology and bulk-edge correspondence^{25,31}, where the field strength is strongly localized at (meta-)surfaces. For our waveguide modes, the SAM-distinguished dispersion is attributed to the structure-dependent wave interference and the resulting chiral velocity field wrapped by the phase-delayed metasurface. Specifically, the acoustic wave is spatially confined well in the area near the normal surface but severely attenuated in the area near the metasurface (see Fig. 1c, d), the latter of which is caused by the destructive interference due to the phase-delayed reflective metasurfaces. This transverse attenuation induces effective chiral fields perpendicular to the waveguide propagation direction, similar to the case in Gaussian longitudinal wave beams^{5}.

### Acoustic metasurface structures

We exploit the side bar structures to realize arbitrary phase-delayed reflective acoustic wave metasurfaces, as shown in Fig. 2a, b. The side bar arrays on the waveguide boundary can be regarded as acoustic meta-atoms^{32}. These meta-atoms can reflect the acoustic wave with arbitrary phase delay when they become resonant^{33}. As such, we can find that the waveguide mode can carry nontrivial acoustic SAM *S*_{y} ≠ 0, and the SAM is strongly associated with the linear momentum responsible to the propagation shown in Fig. 2c. This momentum-dependent SAM will led to the opposite SAM texture of modes for different momentum excitations in Fig. 2d. Specially, the SAM densities is associated with the near-circularly (elliptically) polarized velocity fields. As demonstrated in Fig. 2e, metasurface waveguide modes with *k* > 0 at different frequencies will result in the top-viewed anticlockwise elliptically polarized velocity fields corresponding to *s*_{y} > 0, which make opposite propagating modes of opposite polarizations become approximately orthogonal to each other. This momentum-locked SAM profile of bulk modes within the waveguide will apparently reduce couplings between forward propagating and reflected scattering modes. To experimentally verify the SAM densities inside the metasurface waveguide, we perform experimental measurements (see “Method”) and compare them with analytic theory predictions Eq. (2) and numerical simulations, as shown in Fig. 3. One can see that the experimental measurements are in good agreements with theoretical and numerical results. Indeed, the *y*-mirror symmetry makes the odd-symmetric *s*_{x} with ({s}_{x}(y)={{mathcal{M}}}_{y}{s}_{x}(y){{mathcal{M}}}_{y}^{-1}=-{s}_{x}(-y)), resulting in the vanishing *S*_{x} = ∫*s*_{x}*d**x**d**y* = 0 due to the cancellation in the integral. However, the *x*-mirror symmetry breaking leads to the non-perfect cancellation *s*_{y}(*x*) ≠ −*s*_{y}(−*x*), facilitating the nonzero *S*_{y} ≠ 0.

### Experimental observations of acoustic spin transport

With this special-boundary-defined spin texture and the inherent spin-momentum locking shown in Eq. (2) and Fig. 3, the transport in the metasurface waveguide would be robust compared with conventional spinless waveguide modes. As demonstrated in Fig. 4a, we bend the metasurface waveguide with the bending angle *θ*. In the Fig. 4b, the simulated transmission *T* of the bended metasurface waveguide will become better than the conventional circular duct waveguide, especially for *θ* ∈ [0, *π*/2], which will be nearly 100% transmission. This is assisted by the tight acoustic spin-momentum locking of the acoustic spinful mode in the meta-waveguide (see Eq. (2) and Fig. 2), which for opposite transport directions (momentum) has opposite SAMs. In particular, the nonsymmetric transmission *T*(*θ*) ≠ *T*(−*θ*) can be found for the metasurface waveguides with opposite bending angles, which indicates that the spinful waveguide mode will pass more easily through corners decorated by metasurfaces. This might be attributed to the asymmetric distribution of the waveguide field, i.e., the pressure fields mainly localize on one half of the waveguide cross-section without metasurface decorations. In the following, we are going to experimentally explore the acoustic spin transport in several typical examples, including robust transport, spin-selective routing and rotating spin under effective magnetic field.

*Robust transport with corner-scattering suppression due to tight spin-momentum coupling*: We first simulate the acoustic field of a U-shape normal acoustic waveguide without metasurface when incident wave imposed at one port. Results show that the strong backscattering will happen at bending corners, as depicted in Fig. 5a. As a contrast, we simulate the acoustic field when incident acoustic wave into the similar U-shape waveguide but with metasurface. From the simulation results shown in Fig. 5a, we can see the acoustic waves successfully pass through the U-shape meta-waveguide with low scattering loss at corners.

To verify these spin-related backscattering-suppressed effects, we perform experiments to measure the transport of U-shape acoustic waveguide with/without metasurfaces as shown in Fig. 5b. From the experimental results in Fig. 5c, we can see that the U-shape has little effects on the transmission of the metasurface waveguide at frequency region 2.85–2.95 kHz, which is exactly around the resonant frequency of the side bar structures. But in contrast, the waveguide without metasurface does not hold a clear acoustic SAM texture and it will suffer strong scatterings leading to severely attenuated transmission. Compared with conventional waveguides, this robustness transport of acoustic spinful mode in metasurface waveguides is assisted by the nonzero SAM and its tight spin-momentum locking relation, resulting in that the back-propagation requires spin flips to opposite one. Acoustic SAM textures represent polarization profiles of velocity fields. To ensure the spin (polarization) matching between opposite propagating acoustic modes of opposite SAMs, backscattering requires strong scatters that reverse spin. Indeed, scattering behaviors highly depend on the type of scatters and the interaction with wave modes. For example, certain types of impurities and defects will break topological transport protections in topological insulators without preserving time-reversal symmetry so that backscattering occurs with spin flip^{34,35}. In fact, many scatters in metasurfaces will possess spin-related scattered/reflected properties^{14,36,37}. From experimental results, one can see that here the metasurface waveguide modes with nonzero SAM is insensitive to the present corner defects.

*Spin-selective wave router based on opposite acoustic-spin-dependences of different directions*: As discussed before, the momenta of metasurface waveguide modes are tightly coupled to the SAM of modes, which means that the wave will propagate along the direction selectively based on the SAM match, when facing multiple transport channels with opposite SAMs. To observe the selective phenomena, we design a T-shape structure and confirm the spin-selective router by simulation results as shown in the Fig. 5d. From Fig. 1, we know that the acoustic mode to the left export is the same up-spin state (*S*_{y} > 0) as the incident acoustic wave, while the right export has opposite down-spin state, so that the acoustic wave will choose the left direction with the same up-spin. If we rotate the incident waveguide around incident axis for 180°, the incident acoustic wave will possess a flipped SAM. The right export will be the same down-spin state (*S*_{y} < 0) as the incident acoustic wave, while the left export has opposite up-spin state. As such, the acoustic wave now will choose the right direction with the same down-spin as expected. This is reminiscent of the spin-Hall-like effect for electrons and optics^{36,37}.

Experimentally, we input the acoustic wave from the port and measure the transmission at sides A/B shown in Fig. 5e. The experiment results in Fig. 5f show that the acoustic wave chooses one side (A) to propagate and leave no output for the other one (B). At the crossing point, the SAM of modes will make the acoustic wave propagate along the direction with the matched SAM. Moreover, it shows that this spin-selective transport will also be insensitive to the backscattering at T corner, due to the asymmetric spatial distribution of acoustic spinful waveguide modes and the assistance of spin-momentum locking.

*Gradually rotated metasurface waveguide for rotating acoustic spin and phase modulator*: As is known from Fig. 1, the acoustic wave polarization confined within the meta-waveguide forms complicated SAM texture, which highly depends on the boundary geometry settings of metasurfaces. This indicates that following a gradually rotating meta-waveguide boundary, the evolving polarization of acoustic wave will result in rotating SAM textures, as shown in Fig. 6a. By rotating metasurface boundary with an angular velocity **Ω** along *z-*axis, we find that the variation of acoustic SAM texture in *x**y* plane will follow the rotating evolution of velocity polarized profile ((frac{partial }{partial z}{{bf{e}}}_{i}={mathbf{Omega}} ,times {{bf{e}}}_{i})) as: (frac{partial }{partial z}{bf{S}}={mathbf{Omega}} , times {bf{S}}). This is a reminiscent of “acoustic spin Bloch equation” that resembles the equation of motion of electron spin procession under an effective “magnetic field” **Ω**^{38}. The *z-*axis plays the role of a pseudo-time so that the wave propagation along this pseudo-time will experience effective Larmor precession for acoustic spin with the Larmor frequency Ω.

As the consequence of this effective Larmor precession of acoustic spin, the total acoustic wave phase accumulated during the propagation in rotating metasurface waveguide will have contributions from two individual phases Θ = Θ_{0} + *γ*: the unperturbed phase Θ_{0} = *k**L* related with waveguide length *L* and wave vector *k*; and the additional phase (gamma =frac{k{R}_{0}^{2}}{2L}{theta }^{2}) induced by adiabatic variation of meta-boundary conditions, where *θ* = Ω*L* is the final rotation angle (See Supplementary Note 3). To experimentally observe the phase *γ* induced by rotating SAM texture, we rotate the boundary settings of metasurface waveguide slowly along *z* direction and measure the corresponding transmitted phase, as shown in Fig. 6b at frequency *f* = 2.85 and 2.9 kHz, respectively. The measured phases *γ* with different rotating rate Ω after the pseudo-time *L* are in good agreement with theoretical results. Results indicate that the gradually rotating metasurface waveguide is also a phase modulator that has implications in acoustic interferometers.