Tomáš BZDUŠEK : Topology of multigap insulators and multifold band degeneracies

Energy spectra of crystalline matter are characterized by band structures. When the energy bands are separated by a spectral gap at the Fermi level, the system constitutes an insulator. In contrast, when the conduction and valence bands coalesce near the Fermi level, forming a band degeneracy, the system realizes a semimetal. Both insulators and semimetals can be characterized by topological invariants that enable quantized responses in the bulk and robust metallic states on the boundary. The program of classifying such features has been largely completed for systems with either global symmetry (Altland-Zirnbauer classes) or crystalline symmetry (space groups).

In this seminar, I will discuss recent insights into topological features of band structures that are revealed by partitioning the spectrum with multiple energy gaps. Initially developed in the context of non-Abelian braiding of band degeneracies in momentum space [1], this approach later led to the formulation of a broader class of multigap [2] and delicate [3] topological insulators. Over the past year, these ideas have been extended nontrivially in two further directions. On the one hand, it has been shown that, through higher gauge theory, some of these refined invariants correspond to integrals of a tensor Berry curvature, leading to a quantized nonlinear optical response [4]. On the other hand, multifold band degeneracies (formerly studied in space groups using little-group irreps) have been topologically characterized for Altland-Zirnbauer classes in terms of linking invariants [5]. These findings point to deeper topological principles underpinning multigap and multiband spectral features in crystalline matter.

[1] Q.S. Wu, A. A. Soluyanov, and T. Bzdušek, Non-Abelian band topology in noninteracting metals, Science 365, 1273–1277 (2019).

[2] A. Bouhon, T. Bzdušek, and R.-J. Slager, Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102, 115135 (2020).

[3] A. Nelson, T. Neupert, A. Alexandradinata, and T. Bzdušek, Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond, Phys. Rev. B 106, 075124 (2022).

[4] W. J. Jankowski, R.-J. Slager, and G. Palumbo, Probing Tensor Monopoles and Gerbe Invariants in Three-Dimensional Topological Matter, arXiv:2507.22116 (2025).

[5] A. A. Iliasov, Z. Guba, T. Yoshida, A. Tiwari, and T. Bzdušek, Topological characterization of multifold band degeneracies in Altland-Zirnbauer symmetry classes, arXiv:2607.13016 (2026).