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Many free-fermion topological phases of matter such as the Chern insulator are characterised by topological quantum numbers assigned to single isolated bands. While such single-band phases are now well understood, intriguing features remain to be explored within topological band theory. I will explain how nodes in real Bloch Hamiltonians carry non-Abelian topological charges which arise from the geometry of the classifying space. Moreover, by braiding these nodes around each other in reciprocal space, it is possible to induce a ‘multi-band’ topological phase, where the two band subspace supporting the nodes is labelled with an integer, the Euler class. Another example of a multi-band topological invariant is the Hopf invariant, which characterises three dimensional complex phases and provides a solid state realisation of the Hopf fibration. Such systems can also host Chern numbers on each coordinate plane within the Brillouin zone; I will describe how the presence of such subdimensional invariants influences the bulk Hopf invariant. Finally, I will discuss a real topological phase in 3D which possesses a bulk Hopf invariant and 2D Euler classes. These systems have nontrivial quantum geometry, and appear to host unusual nodal line structures.

[1] arXiv:2405.17305 (2024)
[2] Nat. Phys. 16, 1137–1143 (2020)
[3] Phys. Rev. Lett. 101, 186805 (2008)
[4] Phys. Rev. B 94, 035137 (2016)
[5] Phys. Rev. B 108, 125101 (2023)

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