Fig. 1

Elastic pseudospins and helical edge states. a Topologically protected interface (yellow lines) formed by two different elastic insulators (perforated phononic crystal on the same plates). These two elastic insulators are identical in lattice constant a (3a₀), plate thickness (\(\sqrt 3\,\times\, \)0.4a₀), and radius of perforated holes r (\(\sqrt 3\,\times\, \)0.18a₀) but different hole-center distance characterized by b. When b is reduced from (right) b = 1.0a₀ to (left) b = 1.12a₀, the resulted band diagram evolution indicates the occurrence of a band inversion process for the p_x/p_y and d_x2−y2/d_xy modes (similar to p/d orbitals of electrons), corresponding to a topological transition from an ordinary insulator [OI, zero spin Chern number (Cs)] to a topological insulator [TI, none-zero Cs] with an overlapped bulk bandgap. b Projected band diagram of the TI–OI interface and a zoomed-in view illustrating two elastic helical edge states (characterized by two elastic pseudospins). The grey dispersions correspond to shear-horizontal waves and are not excited in our study. c Out-of-plane displacements of the degenerate modes indicating their evolutionary relationship: bulk p and d modes hybridize to form a pair of normal modes, i.e., one symmetric mode \(S = \left( {p_x + d_{x2 - y2}} \right)/\sqrt 2\) and one anti-symmetric mode \(A = \left( {p_y + d_{xy}} \right)/\sqrt 2\), used as the basis to construct the two pseudospins, S + iA and S − iA, protected by the pseudo (fermi-like) time reversal symmetry (T_p² = −1), as \(+ {\it{S}}\mathop{\longrightarrow}\limits^{{{\it{T}}_{\it{p}}}} + {\it{A}}\mathop{\longrightarrow}\limits^{{{\it{T}}_{\it{p}}}} - {\it{S}}\mathop{\longrightarrow}\limits^{{{\it{T}}_{\it{p}}}} \cdot \cdot \cdot\). d, e Experimentally recorded temporal evolution of elastic field distribution of the out-of-plane displacement in the vicinity of the TI–OI interface when the elastic wave is excited from the bottom of the interface (upward-traveling) and from the top (downward-traveling), respectively. These time-domain results in a half period (from t = 0 to t = π/ω) strongly confirm the existence of two spin-momentum locked elastic pseudospins represented in the S/A basis: time-dependent anti-clockwise elastic pseudospin + (i.e., S + iA) and clockwise elastic pseudospin− (i.e., S − iA), respectively