Kink-antikink asymmetry and impurity interactions in topological mechanical chains

TL;DR

This paper investigates topological kinks and antikinks in a diatomic rotor chain with broken inversion symmetry, revealing how topological boundary terms cause asymmetry in their energies and interactions, supported by theoretical analysis and physical prototypes.

Contribution

It introduces a nonlinear field theory for topological kinks in mechanical chains, highlighting the role of boundary terms in breaking kink-antikink symmetry and their impurity interactions.

Findings

Kinks propagate without Peierls-Nabarro barrier due to topological boundary term.

Kink-antikink energy degeneracy is broken by the boundary term.

Impurities affect kink interactions, confirmed by physical prototypes.

Abstract

We study the dynamical response of a diatomic periodic chain of rotors coupled by springs, whose unit cell breaks spatial inversion symmetry. In the continuum description, we derive a nonlinear field theory which admits topological kinks and antikinks as nonlinear excitations but where a topological boundary term breaks the symmetry between the two and energetically favors the kink configuration. Using a cobweb plot, we develop a fixed-point analysis for the kink motion and demonstrate that kinks propagate without the Peierls-Nabarro potential energy barrier typically associated with lattice models. Using continuum elasticity theory, we trace the absence of the Peierls-Nabarro barrier for the kink motion to the topological boundary term which ensures that only the kink configuration, and not the antikink, costs zero potential energy. Further, we study the eigenmodes around the kink and antikink configurations using a tangent stiffness matrix approach appropriate for pre-stressed structures to explicitly show how the usual energy degeneracy between the two no longer holds. We show how the kink-antikink asymmetry also manifests in the way these nonlinear excitations interact with impurities introduced in the chain as disorder in the spring stiffness. Finally, we discuss the effect of impurities in the (bond) spring length and build prototypes based on simple linkages that verify our predictions.