The exploration of synthetic degrees of freedom (SDOFs) has revolutionized classical physics, enabling the emulation of relativistic and topological phenomena in artificial materials. This includes the realization of Dirac and Weyl physics and various topological phases of matter in photonic and mechanical systems. Metamaterials, with their designer properties and synthetic spin degrees of freedom, have significantly impacted materials science and engineering by offering new ways to manipulate electromagnetic and acoustic fields. Beyond technological applications, these metamaterials provide experimental platforms for exploring fundamental scientific concepts. Previous research has focused on phenomena like Klein tunneling, Weyl points, Fermi arcs, and topological edge and surface states in systems engineered to emulate specific Hamiltonians. Synthetic gauge potentials acting on SDOFs are crucial for uncovering this new physics. By introducing symmetries or inducing controlled symmetry breaking, pseudo-magnetic fields can be emulated, acting on pseudo-spins. In acoustics, this has led to demonstrations of the Zak phase, valley-Hall effect, Dirac cones, Weyl points, and higher-order topological phases. This paper focuses on using pseudo-spins, arising naturally from evanescent fields or engineered via symmetries, to control acoustic wave radiation, guiding, and steering. The experimental demonstration of the pseudo-Spin-Hall effect is achieved in two metamaterial systems exhibiting coupled angular and linear momenta.

The literature extensively covers the use of synthetic degrees of freedom to emulate relativistic and topological phenomena in various wave systems. The realization of Dirac and Weyl physics, topological insulators, and higher-order topological phases has been reported in photonic and phononic crystals. Studies have demonstrated phenomena such as Klein tunneling, Weyl points, Fermi arcs, and topological edge states in carefully designed systems. The concept of synthetic gauge potentials and the manipulation of symmetries to generate pseudo-magnetic fields has been a central theme in this research. In acoustics, previous work has successfully demonstrated phenomena analogous to those observed in other wave systems, including Zak phase, valley-Hall effect, Dirac cones, Weyl points, and higher-order topological states in specifically designed acoustic lattices. These studies laid the groundwork for the current research by establishing the feasibility of manipulating acoustic waves using synthetic degrees of freedom and demonstrating the potential for novel functionalities.

The researchers experimentally demonstrated the pseudo-Spin-Hall effect in two types of acoustic metamaterials. The first system consisted of a perforated film (high-density polyethylene with an array of square holes) designed to support evanescent acoustic surface waves. First-principles calculations using COMSOL Multiphysics were employed to determine the dispersion relations and characterize the chirality of the velocity fields. The chirality, representing the degree of circular polarization of the velocity field, was shown to be non-uniform, exhibiting hotspots above the holes. Directional excitation was achieved by placing a circularly polarized source at these hotspots. The source was implemented using two orthogonal linear acoustic transducers with a 90° phase shift to create right- and left-handed excitations. The second system involved a two-dimensional acoustic kagome lattice composed of an array of acoustic resonator trimers coupled via narrow rectangular channels. The coupling strength between resonators was controlled by adjusting the channel positions. The kagome lattice was analyzed using the tight-binding model and first-principles finite-element method calculations in COMSOL. The chirality of the modes was quantified based on the C3 symmetry of the lattice. Directional excitation was achieved by placing circularly polarized sources in the bulk of the crystal. Finally, the presence and characteristics of topological edge states were investigated in the kagome lattice. A supercell calculation was used to determine the band structure and chirality of edge states. Experiments were conducted on a 3D-printed kagome lattice with appropriate boundary conditions to support edge states. Transmission measurements along the edges were conducted with sources of varying angular momenta to test the pseudo-spin locking of the edge states.

The study yielded several key findings: 1. **Directional Excitation of Evanescent Waves:** In the perforated film metamaterial, the researchers successfully demonstrated directional excitation of evanescent acoustic surface waves by exploiting the non-uniform chirality of the velocity field. Placement of a circularly polarized source at a chirality hotspot resulted in highly directional wave propagation. 2. **Synthetic Pseudo-Spin in Kagome Lattice:** The acoustic kagome lattice exhibited synthetic transverse pseudo-spin locked to linear momentum. First-principles calculations and experimental measurements confirmed the control of mode propagation both in the bulk and along the edges. 3. **Directional Excitation in Kagome Lattice:** Circularly polarized sources of opposite handedness were shown to excite modes propagating along specific directions in the Brillouin zone (Γ-K or Γ-K'), demonstrating directional excitation due to the coupling of angular and Bloch momenta. 4. **Pseudo-Spin Locking in Topological Edge States:** Topological edge states in the kagome lattice were found to exhibit pseudo-spin polarization, which reversed for opposite wavenumbers. This pseudo-spin locking enabled robust directional excitation of the edge states. Experiments confirmed that the handedness of the source determined the direction of edge state propagation. 5. **Experimental Validation:** All key findings were supported by both numerical simulations and experimental measurements, providing strong evidence for the observed phenomena.

This research demonstrates significant advancements in the control and manipulation of acoustic waves using the concept of synthetic pseudo-spin. The successful experimental demonstration of the pseudo-Spin-Hall effect in two different metamaterial systems highlights the versatility and potential of this approach. The findings offer new possibilities for designing acoustic devices with enhanced functionalities, such as highly directional emitters and waveguides. The observation of pseudo-spin locking in topological edge states suggests potential applications in robust and unidirectional acoustic signal transmission. The work extends the capabilities of acoustic metamaterials beyond the limitations of their inherent scalar nature, paving the way for advanced acoustic devices and the exploration of new topological phenomena in acoustic systems. The results have broad implications for developing new acoustic devices and technologies, such as highly directional sound sources, waveguides, and acoustic sensors. Future research could explore more complex metamaterial designs and explore the potential for integration with other technologies.

This study successfully demonstrated the control of acoustic waves using synthetic pseudo-spin in two distinct metamaterial systems: a perforated film and a kagome lattice. The observed directional excitation of evanescent waves and pseudo-spin-locked bulk and edge modes opens new possibilities for designing highly efficient and directional acoustic devices. The observed pseudo-spin locking mechanism, even in the absence of true spin, expands the possibilities for topological acoustic devices. Further research could explore the integration of this approach with other functionalities and the development of more complex metamaterial designs.

The experimental setup involved specific material choices and geometric parameters, limiting the immediate generalization of the findings to other materials or designs. While the observed effects are significant, the efficiency of directional excitation might be improved by optimizing the source design and metamaterial parameters. The range of frequencies over which the effects are observed may also be system-specific and require careful design considerations for practical applications. Future work could investigate the influence of various factors such as material properties and structural imperfections on the robustness and efficiency of the observed phenomena.