We illustrate a procedure that defines and converts non-Abelian charges of Weyl nodes via braid phase factors, which arise upon exchange inside the reciprocal momentum space. This phenomenon derives from intrinsic symmetry properties of topological materials, which are increasingly becoming available due to recent cataloguing insights. Specifically, we demonstrate that band nodes in systems with C2T symmetry exhibit such braiding properties, requiring no particular fine-tuning. We further present observables in the form of generalized Berry phases, calculated via a mathematical object known as Euler form. We demonstrate our findings with explicit models and a protocol involving three bands, for which the braid factors mimic quaternion charges. This protocol is implementable in cold atoms setups and in photonic systems, where observing the proposed braid factors relates to readily available experimental techniques. The required C2T symmetry is also omnipresent in graphene van-der-Waals heterostructures, which might provide an alternative route towards realizing the non-Abelian conversion of band nodes.

We present a general methodology towards the systematic characterization of crystalline topological insulating phases with time reversal symmetry (TRS). In particular, taking the two-dimensional spinful hexagonal lattice as a proof of principle we study windings of Wilson loop spectra over cuts in the Brillouin zone that are dictated by the underlying lattice symmetries. Our approach finds a prominent use in elucidating and quantifying the recently proposed "topological quantum chemistry" (TQC) concept. Namely, we prove that the split of an elementary band representation (EBR) by a band gap must lead to a topological phase. For this we first show that in addition to the Fu-Kane-Mele Z2 classification, there is C2T -symmetry protected Z classification of two-band subspaces that is obstructed by the other crystalline symmetries, i.e. forbidding the trivial phase. This accounts for all nontrivial Wilson loop windings of split EBRs that are independent of the parameterization of the flow of Wilson loops. Then, by systematically embedding all combinatorial four-band phases into six-band phases, we find a refined topological feature of split EBRs. Namely, we show that while Wilson loop winding of split EBRs can unwind when embedded in higher-dimensional band space, two-band subspaces that remain separated by a band gap from the other bands conserve their Wilson loop winding, hence revealing that split EBRs are at least "stably trivial", i.e. necessarily non-trivial in the non-stable (few-band) limit but possibly trivial in the stable (many-band) limit. This clarifies the nature of fragile topology that has appeared very recently. We then argue that in the many-band limit the stable Wilson loop winding is only determined by the Fu-Kane-Mele Z2 invariant implying that further stable topological phases must belong to the class of higher-order topological insulators. arXiv:1804.09719v4 [cond-mat.mes-hall]

We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on C 2 T-symmetric systems that have gained recent attention, for example, in the context of layered van-der-Waals graphene heterostructures, we relate these insights to homotopy groups of Grassmannians and flag varieties, which in turn correspond to cohomology classes and Wilson-flow approaches. We furthermore make use of a geometric construction, the so-called Plücker embedding, to induce windings in the band structure necessary to facilitate nontrivial topology. Specifically, this directly relates to the parametrization of the Grassmannian, which describes partitioning of an arbitrary band structure and is embedded in a better manageable exterior product space. From a physical perspective, our construction encapsulates and elucidates the concepts of fragile topological phases beyond symmetry indicators as well as non-Abelian reciprocal braiding of band nodes that arises when the multiple gaps are taken into account. The adopted geometric viewpoint most importantly culminates in a direct and easily implementable method to construct model Hamiltonians to study such phases, constituting a versatile theoretical tool.