Interferometry of Non-Abelian Charges of Band Nodes in the Euler Class

In systems with a real Bloch Hamiltonian, such as those symmetric under a composition of a π-rotation and time-reversal, band nodes can be characterised by a non-Abelian frame-rotation charge. The ability of these band nodes to annihilate pairwise is path dependent, since by braiding nodes in adjacent gaps the sign of the charges can be changed. Here, we theoretically present and numerically confirm two methods to experimentally demonstrate this non-Abelian braiding process in ultracold atomic systems. While these non-Abelian charges cannot be captured by conventional techniques, we present specific interferometry techniques capable of accessing the relative sign of the band nodes. Our results open up a promising avenue for measuring non-Abelian charges of band nodes and the experimental verification of braiding procedures directly, which can be applied in a variety of settings including the recently discovered anomalous non-Abelian phases arising under periodic driving.