Eigenmodes and winding arguments in mechanical spaces. (a,b) When exchanging the strength of alternating springs kappa_1 and kappa_2, two eigenmodes are inverted, as classified by winding arguments, such that seemingly identical Case i and Case ii springs chains, belong to vibrationally discontinuous spaces (c,d) When exchanging the strength of non-alternating springs chains which are seemingly different, eigenmodes are not inverted.

Atomistic molecular lattice showing thermally-robust boundary eigenmodes. (a) Eigenmode mapping at 72cm-1 localized at the edges of the Ribbon. Red/blue represents the highest/lowest amplitude. (b) Eigenmode analysis of two atomistic materials, one with (Ribbon) and without (Crystal) physical boundaries with vacuum. A new eigenmode at 72cm-1 appears when the boundary is opened, thereby defining the Ribbon. (c) The 72 cm-1 edge eigenmode appears shift to 69 cm-1 in molecular dynamic simulations at 10 K, as shown by the velocity autocorrelation function in (c). (d) The dynamic band structure along the highest symmetry direction (vector a_m; in panel a) for the corresponding edge mode shows a localized mode (green arrow) between dispersive bulk nodes (yellow arrows).