Fig. 1: One-way transport and spin-momentum locking of Rayleigh waves.

Simulated spatial energy distribution for 2D (a) and 3D (b) continua with different B_3D/μ values, using a prescribed particle displacement at boundary point r following ψ_k(r). Green arrows denote the spin direction (parallel to the surface) and blue arrows denote the strong propagation direction. The relative displacement magnitude of the red points marked in (a and b) is evaluated to measure \(\left\vert {C}_{-{{{\bf{k}}}}}/{C}_{{{{\bf{k}}}}}\right\vert\), shown as yellow points in (c and d). The shear modulus (μ) is kept constant at 0.01 GPa throughout all the simulations, and the bulk modulus (B_3D) varied. c, d Dependence of the one-way propagation metric on the elastic moduli ratio in 2D (c) and 3D (d). The blue curves are analytic results, and the red lines are bounds. e, f The dependence of polarization of the Rayleigh wave at various depths when B_3D/μ = 0 (normalized using the decay length κ_t of the transverse polarized component of the edge mode), depicted as the ratio of the short to long axes (b/a) of the ellipse of polarization (analytic). The inset illustrates the polarization (blue ellipses) and ψ_−k(r)^* ⋅ ψ_k(r) (color) of the Rayleigh waves at different depths for 2D (e) and 3D (f). Sizes of the ellipses are proportional to the amplitude of the Rayleigh waves at the ellipses' center. g, h Dependence of the longitudinal (c_l), transverse (c_t), and Rayleigh (c_R) wave speeds (normalized by c_t) on the Poisson’s ratio (ν) in 2D (g) and 3D (h) (analytic). i A summary of the one-way transport metric in 2D and 3D media. j Dependence of Rayleigh wave speeds (c_R) on Poisson’s ratio (ν) in 2D continua at B = 10⁸, 10⁹, 10¹⁰ Pa (analytic), indicating that c_R approaches a finite value (\(\sqrt{2B/\rho }\), and ρ = 1000 kg/m³) as ν → − 1 (equivalently B/μ → 0) when B is kept constant, and μ is increased (see Supplementary Materials for details).