The spectral localizer for probing the topology in nonlinear materials*

Over the past decade, photonic topological insulators have attracted significant interests due to their promise to achieve one-way propagations with their topological edge modes that are robust against fabrication defects or disorders. Recently, nonlinear topological insulators have garnered substantial attention as they have both enabled the discovery of new physics due to inter-particle interactions and may have applications in photonic devices such as topological lasers and frequency combs. However, there is currently no theoretical framework to rigorously classify topology in such systems due to the local nature of nonlinearities. Previous attempts to classify the topology of nonlinear systems have required significant approximations, and made use of the traditional topological band theory, which linear at its core. 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.