[Paper Review] Topological Invariant and Anomalous Edge States of Strongly Nonlinear Systems

Di Zhou, D. Zeb Rocklin+2 more|arXiv (Cornell University)|Dec 28, 2020

Mechanical and Optical Resonators1 references4 citations

TL;DR

This paper establishes a nonlinear Berry phase formalism for strongly nonlinear classical systems, demonstrating quantization via reflection symmetry and linking mode amplitude to topological phase transitions. It identifies anomalous edge modes that decay to plateaus governed by nonlinear fixed points, enabling bulk-boundary correspondence and proposing experimental platforms in photonics and electronics.

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 Schrodinger 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.

Motivation & Objective

• To define a proper Berry phase for nonlinear bulk modes in strongly nonlinear classical systems.
• To characterize topological phases in 1D generalized nonlinear Schrödinger equations under strong nonlinearity.
• To establish bulk-boundary correspondence for nonlinear topological edge modes.
• To identify the role of mode amplitude in driving topological phase transitions.
• To propose experimentally realizable systems—passive photonic and active electrical—for observing these nonlinear topological states.

Proposed method

• Develops an analytic strategy to demonstrate quantization of the nonlinear Berry phase under reflection symmetry.
• Identifies mode amplitude as a control parameter for topological phase transitions.
• Applies bulk-boundary correspondence to link nonlinear bulk modes with edge states.
• Analyzes decay behavior of edge modes, showing they relax to plateaus determined by nonlinear fixed points.
• Constructs a theoretical framework for nonlinear topological invariants in strongly nonlinear regimes.
• Proposes passive photonic and active electrical systems as viable experimental platforms.

Experimental results

Research questions

• RQ1 How can the Berry phase be consistently defined in strongly nonlinear classical systems?
• RQ2 What role does mode amplitude play in inducing topological phase transitions in nonlinear systems?
• RQ3 How do nonlinear topological edge modes behave spatially, particularly their decay dynamics?
• RQ4 To what extent does bulk-boundary correspondence hold in nonlinear topological systems?
• RQ5 What experimental systems can realize these nonlinear topological edge states?

Key findings

• The nonlinear Berry phase is quantized due to reflection symmetry, establishing a topological invariant in strongly nonlinear regimes.
• Mode amplitude acts as a control parameter that drives transitions between distinct topological phases.
• Anomalous topological edge modes decay away from boundaries and stabilize at plateaus determined by the fixed points of the nonlinearity.
• Bulk-boundary correspondence is preserved, linking nonlinear bulk modes to localized edge states.
• The edge modes exhibit non-trivial spatial decay profiles governed by the nonlinearity's fixed points.
• Passive photonic and active electrical systems are proposed as feasible experimental realizations of the predicted nonlinear topological phenomena.