Fig. 9: Workflow for the identification of the topological boundary modes at finite temperatures.

Summary of the vibrational modes and phonon bands calculations performed. It is worth noting that the dynamical matrix approach is a one or two-dimensional model (see Supplementary Methods) and the normal mode analysis (NMA) and molecular dynamics (MD) methods involve three-dimensional force field calculations with the programmes CHARMM c45b1103 and Gromacs 2021.3104. All MD simulations are performed at finite temperature.