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Recently, the spectral localizer framework has emerged as an efficient approach for classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous definitions for concepts such as topological solitons and topological dynamics, where a system’s occupation induces a local change in its topology due to 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*-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 for 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. Finally, we provide some mathematically oriented comments on the C*-algebraic origins of this framework, including a discussion of real C*-algebras and graded C*-algebras that are necessary for incorporating physical symmetries. Looking forward, we hope that this tutorial will serve as an approachable starting point for learning the foundations of the spectral localizer framework.
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Probing topology in nonlinear topological materials using numerical K-theory
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
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- Award ID(s):
- 2110398
- PAR ID:
- 10494704
- Publisher / Repository:
- Arxiv
- Date Published:
- Journal Name:
- Physical Review B
- Volume:
- 108
- Issue:
- 19
- ISSN:
- 2469-9950
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1103/PhysRevB.108.195142
