Non-Abelian multi-gap topology of materials

We address the quantization of non-Abelian multi-gap topological charges in crystalline materials’ band structures. Recent experimental and theoretical advances have addressed these uncharted multi-gap topological phases that arise upon braiding nodal points formed by successive energy bands, which can in turn be quantified by 2D invariants such as Euler class. We here address subtleties of this physics in intrinsic 1 spatial dimension and in 1D Brillouin paths of higher dimensional lattices and show that a priori the non-Abelian multi-gap charges are not quantized. Rather, we find that only under strict conditions one retrieves meaningful 1D quantization, therefore putting a subset of recent results and claims in a critical context and providing for the necessary conditions to have 1D multi-gap topology. Our results finally pave the way to define generic models in both intrinsic 1D and sub-dimensional contexts that can aid pursuits in (meta-) material realizations. The outlook will be the systematic cataloging of the non-Abelian multi-gap topology of crystalline materials.