Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory

Abstract

Recently, the spectral localizer framework has emerged as an efficient approach to classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous definitions for concepts like topological solitons and topological dynamics, where a system's occupation induces a local change in its topology due to the nonlinearity. For systems embedded in radiative environments that do not possess a shared bulk spectral gap, this framework enables the identification of local topology and shows that local topological protection is preserved despite the lack of a common gap. However, as the spectral localizer framework is rooted in the mathematics of $C^{\ast }$-algebras, and not vector bundles, understanding and using this framework requires developing intuition for a somewhat different set of underlying concepts than those that appear in traditional approaches to classifying material topology. In this tutorial, we introduce the spectral localizer framework from a ground-up perspective, and provide physically motivated arguments for understanding its local topological markers and associated local measure of topological protection. In doing so, we provide numerous examples of the framework's application to a variety of topological classes, including crystalline and higher-order topology. We then show how Maxwell's equations can be reformulated to be compatible with the spectral localizer framework, including the possibility of radiative boundary conditions. To aid in this introduction, we also provide a physics-oriented introduction to multi-operator pseudospectral methods and numerical $K$-theory, two mathematical concepts that form the foundation for the spectral localizer framework.

29 pages, 11 figures

References in corpus (28)

• Topological Photonics
• Classification of topological insulators and superconductors in three spatial dimensions
• Photonic Analogue of Two-dimensional Topological Insulators and Helical One-Way Edge Transport in Bi-Anisotropic Metamaterials
• Chiral Quantum Optics
• Scheme to Achieve Silicon Topological Photonics
• Photovoltaic Hall effect in graphene
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Analogs of quantum Hall effect edge states in photonic crystals
• Topological characterization of periodically-driven quantum systems
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals
• Direct observation of corner states in second-order topological photonic crystal slabs
• Wannier representation of Z_2 topological insulators
• Exponential localization of Wannier functions in insulators
• Photonic crystal nanocavity based on a topological corner state
• Entanglement Spectrum of a Disordered Topological Chern Insulator
• Active topological photonics
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Topological Photonic Quasicrystals: Fractal Topological Spectrum and Protected Transport
• Quantized Nonlinear Thouless Pumping
• Topological Hofstadter Insulators in a Two-Dimensional Quasicrystal
• Optical isolation with nonlinear topological photonics
• Equality of the bulk and edge Hall conductances in a mobility gap
• Nonlinear imaging of nanoscale topological corner states
• Self-induced topological transition in phononic crystals by nonlinearity management
• Spectral truncations in noncommutative geometry and operator systems
• Quantised Bulk Conductivity as a Local Chern Marker
• Multivariable pseudospectrum in $C^{\ast }$-algebras