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Homotopic insights into topological band theory

Homotopy theory is a branch of mathematics that classifies maps between a pair of spaces. In particular, two maps are marked as equivalent in the homotopic description if they can be continuously deformed into one another. Such a definition of equivalence resembles topological invariants of energy band structures, which characterize features unaffected by continuous deformations of the Bloch Hamiltonian in the momentum space. 

In this seminar, I will first review some historical milestones in the development of topological band theory, namely the Haldane model, the Kane-Mele model, and symmetry indicators. Afterwards, I will focus on my own work and discuss how homotopy theory provides complementary insights into the characterization of energy bands. Three concrete applications will be considered: (1) classification of band nodes in semimetals and superconductors, (2) multi-gapped topological obstructions which induce non-Abelian braiding of band nodes, and (3) delicate topological insulators which are overlooked by other classification methods. For each of these theoretical results, the relevance for concrete crystalline solids will be considered.