Band modulations and topological transitions in a one-dimensional periodic bead-on-string chain

Abstract

We study band modulations and topological transitions in a one-dimensional periodic bead-on-string chain. Using an exact transfer-matrix formulation of the wave equation with periodically modulated mass density, combined with numerical spectral searches and tabletop experiments, we characterize band gaps and localized midgap states. We interpret these states by mapping the system to the Su-Schrieffer-Heeger (SSH) model and its low-energy (1+1)-dimensional Dirac theory. This framework reveals that the robust states are topological solitons bound to boundaries or engineered domain walls in the Dirac mass. Through this mapping, we provide an intuitive account of how band structure controls topological phase changes in mechanically realizable lattices.

Figures (20)

• Figure 1: Schematic of the experimental setup in (a) manual and (b) automated configurations. Both share the same mechanical components: beads threaded on a tensioned string, sliding clamps for boundary conditions, and PASCO WA-9613 driver/detector coils. The manual configuration employs a function generator and lock-in amplifier for detailed mode mapping, while the automated configuration uses an SR785 dynamic signal analyzer for frequency sweeps.
• Figure 2: Kronig-Penney band diagram for the uniform chain with $a=20cm$, $T=8.963374N$, $\rho_0=0.549g/m$, and $m=0.114g$. Shaded regions indicate allowed frequency bands.
• Figure 3: Evolution of the first three bands with bead mass $m$. (a) First band, (b) second band, and (c) third band. In each panel, the lower edge of the band remains fixed, while the upper edge shifts downward as $m$ increases. This illustrates that modes near the bottom of a band are insensitive to bead mass, whereas those near the top are strongly affected.
• Figure 4: Band-bottom modes for termination offset $\tau=0$. Circles mark the bead positions. (a) Mode at the bottom of the second band; (b) mode at the bottom of the third band. In both cases the beads lie exactly at nodes of the displacement, so the eigenfrequencies are insensitive to changes in bead mass.
• Figure 5: Band-top modes for termination offset $\tau=a/2$. Circles mark the bead positions. (a) Mode at the top of the first band; (b) mode at the top of the third band. Here the beads are located at points of nonzero amplitude, causing the eigenfrequencies to shift with bead mass.