Fig. 1. Designing a phononic topological metal.

a Simulated acoustic eigen-pressure field p for a single acoustic block resonator at the first elementary resonance mode with its geometry parameters, 20 mm by 20 mm by 40 mm. p is shown in normalized units. b 3D-printed acoustic dimer consisting of one resonator belonging to the SSH layer and one resonator belonging to the metal layer, with the pair connected by an acoustic 3D printed bridge (red arrow points to one entrance to this bridge). The coupling bridge, parameterized by t_c, has width 3 mm, height 3 mm, and length 6 mm. c, d, e, f, g, h Schematics (c, d, e) and full wave simulations of the band structure (f, g, h) of the acoustic SSH lattice (c, f), acoustic metal lattice (d, g), and acoustic metallized SSH lattice (e, h). Band structures are shown in units of frequency squared to emphasize the symmetric spectrum due to chiral symmetry. The wave vector k lies in the first Brillouin Zone, which ranges from −π/a to π/a, where a = 52mm is the lattice constant. The couplings in the SSH lattice t_in and t_out are defined by channels with widths 15 mm (t_in) and 5 mm (t_out), and the same height 3 mm, and length 6 mm. The metal layer’s coupling t_M stems from a channel with dimensions width 7 mm, height 3 mm, and length 6 mm.