​​Measuring non-Abelian topological charges in the band structure of an optical kagome lattice​

Band degeneracies in real Bloch Hamiltonians carry topological charges associated with rotations of the eigenstate frame. In lattices with three or more bands, such as the kagome lattice, these charges are non-Abelian so that their ability to annihilate is path-dependent. 

Here we present experimental measurements of frame charges using ultracold bosons in an optical kagome lattice. Our approach employs an interferometric protocol in which a single wave packet sequentially probes two band nodes along a chosen path, allowing the relative frame charges to be extracted from the measured band populations. Due to their non-Abelian nature, the relative frame charges that we measure depend on this choice of path. By tuning the lattice potential, we also demonstrate that the relative charges can be changed by braiding nodes in momentum space. For comparison, we also perform measurements in an optical honeycomb lattice, for which the frame charges are Abelian.