Topological Invariants in Nonlinear Thouless Pumping of Solitons

TL;DR

This paper introduces a unified topological invariant for nonlinear soliton pumping in optical waveguides, bridging weak and strong nonlinearity regimes, and explaining quantized and fractional charge transport.

Contribution

It proposes a novel topological invariant applicable across different nonlinear regimes, linking Abelian and non-Abelian Chern numbers to soliton transport.

Findings

In weak nonlinearity, the invariant reduces to the Abelian Chern number.

In strong nonlinearity, the invariant involves the non-Abelian Chern number divided by the number of bands.

The approach explains both integer and fractional quantization of pumped charge.

Abstract

Recent explorations of quantized solitons transport in optical waveguides have thrust nonlinear topological pumping into the spotlight. In this work, we introduce a unified topological invariant applicable across both weakly and strongly nonlinear regimes. In the weak nonlinearity regime, where the nonlinear bands are wellseparated, the invariant reduces to the Abelian Chern number of the occupied nonlinear band. Consequently, the pumped charge is quantized to an integer value. As the nonlinearity increases, the nonlinear bands start to intertwine, leading to a situation where the invariant is expressed as the non-Abelian Chern number divided by the number of interacting bands. This could result in a fractional quantization of the pumped charge. Our unified topological invariant approach not only advances the understanding of the soliton dynamics, but also provides implications for the future design of nonlinear topological systems.