Berry Phases in the Bosonization of Nonlinear Edge Modes

TL;DR

This paper explores how nonlinear chiral density waves in topological insulator edge modes accumulate Berry phases, providing a new diagnostic tool for nonlinearity via a Lie-Poisson framework and explicit examples like the KdV equation.

Contribution

It introduces a novel connection between bosonization, Lie-Poisson dynamics, and Berry phases in nonlinear edge modes of topological insulators.

Findings

Berry phases can be computed explicitly for nonlinear wave profiles.

Wave periodicity in time leads to measurable Berry phases.

The KdV equation models nonlinear quantum Hall edge modes effectively.

Abstract

We consider chiral, generally nonlinear density waves in one dimension, modelling the bosonized edge modes of a two-dimensional fermionic topological insulator. Using the coincidence between bosonization and Lie-Poisson dynamics on an affine U(1) group, we show that wave profiles which are periodic in time produce Berry phases accumulated by the underlying fermionic field. These phases can be evaluated in closed form for any Hamiltonian, and they serve as a diagnostic of nonlinearity. As an explicit example, we discuss the Korteweg-de Vries equation, viewed as a model of nonlinear quantum Hall edge modes.