Fig. 4: A point-mass double-chain atomistic ladder Su-Schrieffer-Heeger (daSSH) model and atomistic equivalent under thermal fluctuations.

The model is realised by additional springs connecting next-nearer neighbour masses, leading to a dynamical matrix depending on the masses m_C, m_M and angles θ and α, and bonds κ₁, κ₂, κ₄, κ₅, with each mass bonded to a semi-unmovable substrate (blue dot on the blue surface) via a spring κ₃. a Following the dynamical matrix approach for the daSSH model in the case of θ = α = 60°, two decoupled types of modes are recognized: longitudinal, including two bands showing band inversion; and a transversal flat mode. b Band dispersion from molecular dynamics (MD) simulations of an infinite-chain of 52 masses at finite temperature considers more values of θ and α around the equilibrium value of 60°. c A finite-chain with localised heavy-boundary masses induces a topological boundary mode (TBM marked with magenta arrow) at 13 THz. d The boundary mode is mainly localised at the second-to-last mass (blue). The inset shows the maximum of the power spectral density (PSD) around the frequency of the TBM. Dashed lines serve as view guides when moving from the boundary to the bulk of the chain. A snapshot of the MD simulations shows the first seven masses of a finite chain. Source data are provided as a Source Data file.