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

TL;DR

This paper investigates topological phase transitions and localized states in a one-dimensional bead-on-string chain, using transfer-matrix analysis, experiments, and mapping to the SSH model and Dirac theory.

Contribution

It introduces a novel mechanical system analysis by linking band modulations to topological states via an exact transfer-matrix approach and theoretical mappings.

Findings

Identification of band gaps and localized midgap states.

Mapping of the mechanical system to the SSH model and Dirac theory.

Demonstration of topological solitons bound to boundaries or domain walls.

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.