Probing topology in nonlinear topological materials using numerical $K$-theory

Abstract

Nonlinear topological insulators have garnered substantial recent attention as they have both enabled the discovery of new physics due to interparticle interactions, and may have applications in photonic devices such as topological lasers and frequency combs. However, due to the local nature of nonlinearities, previous attempts to classify the topology of nonlinear systems have required significant approximations that must be tailored to individual systems. Here, we develop a general framework for classifying the topology of nonlinear materials in any discrete symmetry class and any physical dimension. Our approach is rooted in a numerical $K$-theoretic method called the spectral localizer, which leverages a real-space perspective of a system to define local topological markers and a local measure of topological protection. Our nonlinear spectral localizer framework yields a quantitative definition of topologically non-trivial nonlinear modes that are distinguished by the appearance of a topological interface surrounding the mode. Moreover, we show how the nonlinear spectral localizer can be used to understand a system's topological dynamics, i.e., the time-evolution of nonlinearly induced topological domains within a system. We anticipate that this framework will enable the discovery and development of novel topological systems across a broad range of nonlinear materials.

13 pages, 4 figures

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Cited by in corpus (11)

• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Real-space topological invariant for time-quasiperiodic Majoranas
• Optical control of topological end states via soliton formation in a 1D lattice
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• Nonlinear Corner States in Topologically Nontrivial Kagome Lattice
• Versatile Control of Nonlinear Topological States in Non-Hermitian Systems
• Local and energy-resolved topological invariants for Floquet systems
• Emergent topological phases and coexistence of gapless and spectral-localized Floquet quantum spin Hall states via electron-phonon interaction
• Quantitative measure of topological protection in Floquet systems through the spectral localizer
• Chiral solitary waves in a nonlinear topological insulator model
• Dimensional crossover of class D real-space topological invariants