The spectral localizer for probing the topology in nonlinear materials and beyond

Due to the local nature of the interactions in nonlinear photonic topological insulators, attempts to describe their topology using traditional band structure approaches are not suitable. Here, we develop a general framework for classifying the topology of nonlinear materials. Using the so-called spectral localizer to define local topological markers, we provide 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 spectral localizer can be used to probe the time-evolution of nonlinearly induced topological domains within a system.