Topology in non-linear mechanical systems

TL;DR

This paper introduces a general method to define topological indices in non-linear mechanical systems using differential geometry, revealing robust solitons and expanding topological mechanics beyond linear theories.

Contribution

It presents a novel, approximation-free approach to topological invariants in non-linear systems, utilizing the Poincare-Hopf index to identify protected zero modes.

Findings

Defined a Z-valued topological invariant for non-linear zero modes

Identified a type of topologically protected solitons robust to disorder

Proposed a new direction for discovering non-linear topological states

Abstract

Many advancements have been made in the field of topological mechanics. The majority of the works, however, concerns the topological invariant in a linear theory. We, in this work, present a generic prescription of defining topological indices which accommodates non-linear effects in mechanical systems without taking any approximation. Invoking the tools of differential geometry, a Z-valued quantity in terms of the Poincare-Hopf index, that features the topological invariant of non-linear zero modes (ZMs), is predicted. We further identify one type of topologically protected solitons that are robust to disorders. Our prescription constitutes a new direction of searching for novel topologically protected non-linear ZMs in the future.