Classifying material topology in real space using matrix homotopy

Abstract: The classification of topological phases of matter has traditionally relied on assumptions about the underlying material, requiring that the material be an insulator with a well-defined band structure. However, many experimentally relevant systems violate one or more of these assumptions, raising fundamental questions about how topology should be defined and diagnosed in realistic settings. In this talk, I will present an overview of these outstanding challenges and describe how a real-space, operator-based approach called the spectral localizer framework provides a platform-agnostic unifying theory capable of addressing these challenges. In particular, the spectral localizer enables the formulation of local, energy-resolved topological markers that remain well-defined in systems lacking a global spectral gap, translational symmetry, or sharp interfaces. This perspective allows one to meaningfully classify gapless heterostructures, such as photonic systems embedded in air, and can be applied directly to continuum models without first finding a low-energy approximation. Beyond stable topological phases, the spectral localizer also offers new insights into recently identified classes of topology, including fragile topological phases that induce Wannier obstructions resulting in novel forms of quantum materials. I will discuss applications of these ideas to nonlinear polariton systems to achieve reconfigurable topological interfaces and to electronic platforms such as two-dimensional electron gases in semiconductor heterostructures to show the emergence of Hofstadter’s butterfly for intermediate scales of the periodic potential’s strength, demonstrating the versatility of the spectral localizer as a general tool for topological classification in modern materials physics.