Fig. 2: Bulk-heavy boundary correspondence in phonon Su-Schrieffer-Heeger (SSH) analogues: point-mass polymer model vs. fully-atomistic polyyne density functional tight-binding simulation.

a An increase in the mass value at the boundary in a topological phonon SSH phonon analogue (pSSH) point-mass chain results in the expression of a vibration or phonon preferentially localised next to the heavy boundary. The example depicts the intensity of the longitudinal eigendisplacements (of eigenmode no. 26) for the topological (left hand side) and trivial (right hand side) cases before and after imposing a heavy-boundary condition for a pSSH system of 52 units. Mass ratio m_H/m_b (heavy mass divided by bulk mass) increased from 1 to 1.1. Spring constant ratio κ₁/κ₂ = 3 (1/3) for the topological (trivial) case. Eigenvalue spectra as a function of the heavy mass are also shown (up to five-fold increase). Supplementary Figs. 5–7 elaborates on the bulk-heavy boundary correspondence for additional phonon SSH analogues. b Atomistic polyyne modelled using density-functional tight binding (DFTB) methods. The phosphaalkyne C₅₀P₂, featuring triple P ≡ C bonds, realises a pSSH topological phase with a heavy-boundary mode at 1700 cm⁻¹ (left hand side), whereas the chloroalkyne C₅₀Cl₂, with Cl–C single bonds, is a trivial phase with no boundary mode. The longitudinal eigenmode displacements are depicted transversally for clarity.