On non-trivial braiding of nodal points in “momentum space” (i.e. in the Brillouin torus):
QuanSheng Wu, Alexey A. Soluyanov, Tomáš Bzdušek, Non-Abelian band topology in noninteracting metals, Science 365 (2019) 1273-1277 (arXiv:1808.07469 ,doi:10.1126/science.aau8740)
Apoorv Tiwari, Tomáš Bzdušek, Non-Abelian topology of nodal-line rings in PT-symmetric systems, Phys. Rev. B 101 (2020) 195130 (doi:10.1103/PhysRevB.101.195130)
Arguments that some effects in topological phases of matter are “unstable” or “fragile” in that the relevant deformation class of their valence bundles over the Brillouin torus is not their class in topological K-theory (as assumed by the K-theory classification of topological phases of matter) but an unstable homotopy class (what may be called a class in generalized nonabelian cohomology) such as of maps to a Grassmannian space (or more general flag variety) classifying (systems of) sub-bundles of a trivial vector bundle of fixed finite rank:
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