Synthetic Pseudo-Spin-Hall effect in acoustic metamaterials

In recent years, synthetic degrees of freedom (SDOFs) have significantly expanded the landscape of classical physics by enabling the emulation of relativistic and topological phenomena. Dirac and Weyl physics and a broad range of topological phases of matter have been successfully realized in artificial photonic and mechanical materials, facilitating unprecedented control over propagation and scattering of waves. Artificial electromagnetic and acoustic metamaterials endowed with synthetic spin degrees of freedom have enabled new approaches to scatter, trap, and radiate fields, and to explore advanced concepts such as Klein tunneling, Weyl points and Fermi arcs, and higher-order boundary states. A key ingredient to unveil this new physics is the use of synthetic gauge potentials acting on SDOFs. By engineering additional symmetries (e.g., sublattice, duality, crystalline) and using deliberate symmetry reductions, one can emulate pseudo-magnetic fields acting on pseudo-spins. In acoustics, this approach has demonstrated Zak phase in SSH acoustic lattices, valley-Hall effects, Dirac cones, Weyl points, and higher-order topological phases. Here the authors show that pseudo-spins, which can naturally arise for evanescent fields or be engineered via lattice symmetries, can be used to control radiation by guiding and steering in acoustic lattices with coupled angular and linear momenta. The work experimentally demonstrates a pseudo-spin-Hall effect in two types of acoustic metamaterial systems exhibiting such coupling.

The paper situates itself within a rapidly growing body of work on synthetic degrees of freedom enabling emulation of relativistic and topological physics in classical platforms. Prior studies in photonics and acoustics have realized Dirac and Weyl physics, including observations of Weyl points and Fermi arc surface states, and demonstrated topological edge, surface, and higher-order states in engineered lattices. In acoustics, synthetic gauge potentials and symmetry-engineered lattices have enabled demonstrations of Zak phase (SSH chains), valley-Hall transport, and 3D topological phenomena, along with higher-order topological phases (quadrupole, octupole, and higher). Recent theoretical works established analogies between acoustics and electromagnetics for evanescent and surface waves, predicting transverse spin and associated radiation forces and torques in acoustic fields despite their scalar pressure nature. These developments motivate the present study, which leverages evanescent-field transverse angular momentum and C3-symmetry-enabled pseudo-spin in kagome lattices to control directional excitation and propagation of acoustic modes.

Two complementary acoustic metamaterial platforms were designed, simulated, fabricated, and measured to realize and probe pseudo-spin–momentum locking.

1. Evanescent-field metasurface (hole array):

• Structure: A metasurface formed by a periodic array of square holes drilled in a mechanically rigid high-density polyethylene (HDPE) slab. The holes are isolated by HDPE walls. The fundamental acoustic mode in each hole oscillates axially; coupling occurs via free space between holes, producing guided, spoof-plasmon-like acoustic surface modes below the sound line.
• Simulation: First-principles finite-element simulations (COMSOL Multiphysics, Acoustic Module) computed dispersion and near fields. Chirality of the acoustic velocity field was quantified by projecting the normalized velocity vector onto right-circularly polarized (RCP) basis vectors in the X–Z and Y–Z planes, yielding a two-component chirality vector S = (Sxz, Syz). The projection was rescaled so that LCP maps to −1, in-phase to 0, and RCP to +1. Simulations revealed highly nonuniform chirality with hotspots above holes where polarization is nearly purely circular (RCP or LCP), and verified evanescent decay away from the surface.
• Experiment: Directional excitation was implemented by placing a circularly polarized source at a chirality hotspot. The source comprised two orthogonal speakers driven with ±90° phase shifts to realize RCP/LCP in the x–z plane. Fields were measured along a high-symmetry direction (Γ–X). Complex amplitudes from two speaker orientations (normal parallel vs perpendicular to the HDPE surface) were combined (addition for CCW, subtraction for CW) to emulate handed sources and reconstruct field maps. Measurements were compared to COMSOL simulations.

2. Kagome lattice with synthetic pseudo-spin:

• Unit cell and fabrication: A two-dimensional kagome lattice was realized using 3D-printed acoustic resonator trimers (cylindrical cavities) coupled by narrow rectangular channels. Lattice constant a0 = 2d = 42.63 mm; cylinder height h = 40.00 mm; radius R = 6.20 mm. Connector dimensions: gx f = 3.01 mm, gy = fy = 8.00 mm, gz = 4.47 mm. Intra-/inter-cell coupling asymmetry was engineered by placing outer connectors at top/bottom of cylinders and shifting inner connectors toward the center by dim = 6.00 mm to realize expanded (γ > ε) and shrunken (γ < ε) configurations. Trimers and boundary cells were printed in acrylic-based UV-curable resin, with narrow probe channels (port diameter D0 = 3.73 mm; upper port height Ha = 3.97 mm; lower port height 2Hd) for excitation and measurement; unused ports were capped. Cells interlock mechanically.
• Theoretical description and simulations: Due to nearest-neighbor connectivity, the system maps to a tight-binding model with inter-cell γ and intra-cell ε couplings. Synthetic pseudo-spin arises from C3 rotational symmetry of the trimer unit cell. Mode chirality was quantified using Φc = i log(⟨u_n(k,r)|R3|u_n(k,r)⟩), with R3 the 3-fold rotation operator and u_n the eigenstate at Bloch vector k; Φc ≈ 0 indicates monopole (in-phase), ±1 indicates circularly polarized (opposite chiralities), and 1/2 indicates dipole-like modes. First-principles COMSOL simulations generated band structures and mode profiles showing chirality evolution across the Brillouin zone, including handed circular polarization at K/K′ points for the low-frequency band and nearly circular polarization across much of the flat band.
• Bulk directional excitation experiment: To emulate an in-plane circularly polarized source in the trimer, measurements exploited C3 symmetry. A speaker was placed at the center site; amplitude and phase responses were recorded along a hexagonal path. Using three equivalent source positions (blue/orange/magenta) with relative phase shifts 0, +2π/3, and +4π/3, the full complex response for a circularly polarized source at each measurement site n was synthesized as An = Ablue + Aorange e^{i 2π/3} + Amagenta e^{i 4π/3}. Numerical simulations (large lattice with losses to suppress reflections) and experiments produced angular directionality diagrams, revealing handedness-dependent propagation along Γ–K (LCP) and Γ–K′ (RCP) directions over a broad k-range near ~5–6 kHz.
• Edge-state calculations and measurements: A strip supercell (10 cells) periodic in x and terminated in y was simulated to obtain edge bands; two edge-localized bands were identified with chirality φc varying with projected momentum ky. The lower-frequency edge band exhibited near-circular polarization as kx approached projected valley points (ka/2π = ±1/3). A triangular-shaped topological kagome lattice (expanded phase) supporting lower-edge-band states on all three boundaries was fabricated. Transmission along edges was measured by placing sources at a corner and detectors at the opposite edge center for both expanded (topological, γ > κ) and shrunken (trivial, γ < κ) lattices. Frequency sweeps over the lower-edge-band range were performed using sources with different handedness (LCP, RCP, linear). Instrumentation included an arbitrary waveform generator (Rigol DG822), directional microphones (EMM-6), and an external DAQ (AUBIO BOX USB 96); a LabVIEW-based FFT analyzer extracted amplitudes and phase differences. Spectra showed a bandgap in both lattices and a midgap transmission peak in the topological case corresponding to edge-state transport. Handedness-selective, unidirectional edge excitation was observed consistent with pseudo-spin–momentum locking.

Frequencies of interest centered around the fundamental trimer mode (~5200 Hz) for bulk studies and ~5600 Hz for the hole-array directional excitation example.

• Evanescent acoustic metasurface (hole array): First-principles simulations show spoof-plasmon-like guided modes below the sound line with strongly nonuniform transverse angular momentum (chirality) concentrated in hotspots above holes. The velocity field’s handedness is locked to propagation direction. Experiments using two orthogonal speakers with ±90° phase shifts, tuned to v ≈ 5600 Hz, demonstrated highly directional surface-wave excitation that reverses with source handedness.
• Kagome lattice pseudo-spin in the bulk: The trimer-based kagome lattice exhibits synthetic pseudo-spin from C3 symmetry. Band-structure calculations (expanded case, γ > ε) show chirality Φc varying across the Brillouin zone: the lowest band is LCP at K, RCP at K′, monopolar at Γ, and dipolar near M; the flat band is nearly circularly polarized except near Γ and M. Time-reversal pairs (±k) flip handedness, enabling handedness-dependent, valley-selective excitation. Numerical and experimental directionality diagrams confirm that LCP sources excite propagation along Γ–K, while RCP sources excite Γ–K′, over a broad k-range around 5–6 kHz. This behavior does not rely on dimerization (present also for γ = ε and γ < ε).
• Edge states and pseudo-spin locking: In a topological kagome lattice (expanded vs shrunken domains), strip supercell simulations reveal two edge bands; the lower edge band becomes circularly polarized near projected valley momenta (ka/2π = ±1/3). Experiments on a triangular topological sample show bandgaps in both topological and trivial structures but an additional midgap transmission peak only in the topological lattice, indicating edge-state transport. Handedness-controlled, unidirectional edge excitation is observed: LCP and RCP sources launch edge modes in opposite directions, evidencing pseudo-spin–momentum locking of edge states.
• Overall: The work demonstrates a synthetic pseudo-spin-Hall effect in acoustics via two mechanisms—evanescence-induced transverse spin and lattice-symmetry-induced pseudo-spin—enabling robust control of directionality in both bulk-like and edge transport regimes.

The study addresses how to endow scalar acoustic fields with effective spin degrees of freedom to achieve spin–momentum locking analogous to photonic spin-Hall effects. By exploiting (i) the intrinsic transverse angular momentum of evanescent acoustic fields in perforated films and (ii) C3-symmetry-enabled pseudo-spin in a kagome lattice, the authors show that the handedness of a local source determines the direction of energy flow both along metasurface-guided modes and along specific lattice directions (K vs K′) in the bulk, as well as along topological edges. These findings validate that synthetic pseudo-spins can be engineered and coupled to linear momentum in acoustic systems, providing a mechanism for directional excitation without magnetic biasing or moving media. The results are significant for acoustic wave control, as they enable selective launching, routing, and steering based on source polarization state, and extend pseudo-spin concepts beyond valley-Hall systems to lattices characterized by bulk polarization, where edge states are strongly gapped yet still exhibit pseudo-spin locking.

This work experimentally demonstrates a synthetic pseudo-spin-Hall effect in acoustic metamaterials using two complementary platforms: evanescent guided modes in perforated films with transverse spin locked to propagation, and a kagome lattice whose C3 symmetry endows bulk and edge modes with pseudo-spin that locks to momentum. Directional launching and transport are controlled by source handedness, enabling selective excitation along K/K′ directions and unidirectional edge propagation in topological configurations. These results introduce powerful design strategies for spin-controlled acoustic radiation and routing. Potential future directions include optimizing metasurface and lattice geometries to maximize chirality hotspots and coupling efficiency, extending the concepts to broadband and 3D architectures, integrating active or tunable elements for reconfigurable pseudo-spin control, and exploring interactions with scatterers or nonlinearities to realize spin-selective acoustic devices and robust signal-processing functionalities.