Nonlinear breathers with crystalline symmetries

TL;DR

This paper explores the relationship between nonlinear localized excitations called breathers and linear Wannier states in lattice models with crystalline symmetries, providing analytical tools and applying them to specific models.

Contribution

It develops a formalism to predict nonlinear breather properties and links them to Wannier states in models with crystalline symmetries, extending understanding of nonlinear lattice dynamics.

Findings

Analytical predictions of breather spectrum and symmetry data

Application to nonlinear Su-Schrieffer-Heeger and kagome models

Establishment of correspondence between breathers and Wannier states

Abstract

Nonlinear lattice models can support "discrete breather" excitations that stay localized in space for all time. By contrast, the localized Wannier states of linear lattice models are dynamically unstable. Nevertheless, symmetric and exponentially localized Wannier states are a central tool in the classification of band structures with crystalline symmetries. Moreover, the quantized transport observed in nonlinear Thouless pumps relies on the fact that -- at least in a specific model -- discrete breathers recover Wannier states in the limit of vanishing nonlinearity. Motivated by these observations, we investigate the correspondence between nonlinear breathers and exponentially localised Wannier states for a family of discrete nonlinear Schr\"odinger equations with crystalline symmetries. We develop a formalism to analytically predict the breathers' spectrum, center of mass and symmetry data, and apply this to nonlinear generalizations of the Su-Schrieffer-Heeger chain and the breathing kagome lattice.