Stability of topological edge states under strong nonlinear effects

TL;DR

This paper investigates how strong nonlinear effects influence the stability and frequency of topological edge states in a modified Su-Schrieffer-Heeger model, revealing conditions for stable, localized high-amplitude edge states despite disorder.

Contribution

It introduces a nonlinear analysis of topological edge states with cubic onsite nonlinearity, identifying regimes of stability and robustness against disorder, which was not previously explored.

Findings

Nonlinear edge states become unstable due to various instabilities.

Stable high-amplitude edge states exist in certain nonlinear regimes.

Robustness of edge states persists under small disorder, especially with chiral symmetry.

Abstract

We examine the role of strong nonlinearity on the topologically-robust edge state in a one-dimensional system. We consider a chain inspired from the Su-Schrieffer-Heeger model, but with a finite-frequency edge state and the dynamics governed by second-order differential equations. We introduce a cubic onsite-nonlinearity and study this nonlinear effect on the edge state's frequency and linear stability. Nonlinear continuation reveals that the edge state loses its typical shape enforced by the chiral symmetry and becomes generally unstable due to various types of instabilities that we analyze using a combination of spectral stability and Krein signature analysis. This results in an initially-excited nonlinear-edge state shedding its energy into the bulk over a long time. However, the stability trends differ both qualitatively and quantitatively when softening and stiffening types of nonlinearity are considered. In the latter, we find a frequency regime where nonlinear edge states can be linearly stable. This enables high-amplitude edge states to remain spatially localized without shedding their energy, a feature that we have confirmed via long-time dynamical simulations. Finally, we examine the robustness of frequency and stability of nonlinear edge states against disorder, and find that those are more robust under a chiral disorder compared to a non-chiral disorder. Moreover, the frequency-regime where high-amplitude edge states were found to be linearly stable remains intact in the presence of small amount of disorder of both types.