Unified approach to Floquet lattices, topological insulators, and their nonlinear dynamics

TL;DR

This paper introduces a unified analytical framework for Floquet topological insulators, combining linear and nonlinear dynamics, and applies it to various lattice structures, revealing complex wave behaviors including spirals, localization, and collapse.

Contribution

It develops a comprehensive method to analyze Floquet topological insulators and their nonlinear dynamics, providing new insights into wave behavior in different lattice geometries.

Findings

Analytic calculation of Berry connection, curvature, and Chern number.

Observation of spiral patterns and localized structures in nonlinear regimes.

Identification of wave collapse at higher nonlinearity levels.

Abstract

A unified method to analyze the dynamics and topological structure associated with a class of Floquet topological insulators is presented. The method is applied to a system that describes the propagation of electromagnetic waves through the bulk of a two-dimensional lattice that is helically-driven in the direction of propagation. Tight-binding approximations are employed to derive reduced dynamical systems. Further asymptotic approximations, valid in the high-frequency driving regime, yield a time-averaged system which governs the leading order behavior of the wave. From this follows an analytic calculation of the Berry connection, curvature and Chern number by analyzing the local behavior of the eigenfunctions near the critical points of the spectrum. Examples include honeycomb, Lieb and kagome lattices. In the nonlinear regime novel equations governing slowly varying wave envelopes are derived. For the honeycomb lattice, numerical simulations show that for relatively small nonlinear effects a striking spiral patterns occurs; as nonlinearity increases localized structures emerge and for somewhat higher nonlinearity the waves collapse.