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

TL;DR

This paper introduces a numerical $K$-theoretic framework using the spectral localizer to classify and analyze the topology of nonlinear materials across various symmetry classes and dimensions, enabling new insights into topological modes and dynamics.

Contribution

It develops a general, real-space $K$-theoretic method for classifying nonlinear topological insulators, overcoming previous limitations of local nonlinearities.

Findings

Defines local topological markers using the spectral localizer.

Quantitatively distinguishes topologically non-trivial nonlinear modes.

Analyzes the time-evolution of topological domains in nonlinear systems.

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.