Topological Invariant and Anomalous Edge Modes of Strongly Nonlinear Systems

TL;DR

This paper extends topological phase characterization to strongly nonlinear classical systems, defining a nonlinear Berry phase, demonstrating quantization, and identifying topological edge modes with potential experimental implementations.

Contribution

It introduces a proper nonlinear Berry phase definition, demonstrates its quantization, and establishes bulk-boundary correspondence in strongly nonlinear 1D systems.

Findings

Nonlinear Berry phase can be properly defined and is quantized due to reflection symmetry.

Topological edge modes exist in nonlinear systems and decay to fixed points.

Proposed experimental setups for observing these nonlinear topological phenomena.

Abstract

Despite the extensive studies of topological states, their characterization in strongly nonlinear classical systems has been lacking. In this work, we identify the proper definition of Berry phase for nonlinear bulk modes and characterize topological phases in one-dimensional (1D) generalized nonlinear Schr\"{o}dinger equations in the strongly nonlinear regime. We develop an analytic strategy to demonstrate the quantization of nonlinear Berry phase due to reflection symmetry. Mode amplitude itself plays a key role in nonlinear modes and controls topological phase transitions. We then show bulk-boundary correspondence by identifying the associated nonlinear topological edge modes. Interestingly, anomalous topological modes decay away from lattice boundaries to plateaus governed by fixed points of nonlinearities. We propose passive photonic and active electrical systems that can be experimentally implemented. Our work opens the door to the rich physics between topological phases of matter and nonlinear dynamics.