Combinatorics

Physics

Topology (electrical circuits)

Pure mathematics

Mathematics

Topological quantum number

Abelian group

Quantum mechanics

Nature

In the last few decades, topological phase1–11 has emerged as a new classification of matter states beyond the Ginzburg–Landau symmetry-breaking paradigm. The underlying global invariant is usually well characterized by integers, such as Chern numbers or winding numbers—the Abelian charges12–15. Very recently, researchers proposed the notion of non-Abelian topological charges16–19, which possess non-commutative and fruitful braiding structures with multiple (more than one) bandgaps tangled together. Here we experimentally observe the non-Abelian topological charges in a time-reversal and inversion-symmetric transmission line network. The quaternion-valued non-Abelian topological charges are clearly mapped onto an eigenstate-frame sphere. Moreover, we find a non-Abelian quotient relation that provides a global perspective on the distribution of edge/domain-wall states. Our work opens the door towards characterization and manipulation of non-Abelian topological charges, which may lead to interesting observables such as trajectory-dependent Dirac/Weyl node collisions in two-dimensional systems16,17,20, admissible nodal line configurations in three&nbsp;dimensions16,19,20, and may provide insight into certain strongly correlated phases of twisted bilayer graphene21. Non-Abelian topological charges and edge states in a PT-symmetric transmission line network are experimentally observed, and a non-Abelian quotient relation for the bulk–edge correspondence is found.

Experimental observation of non-Abelian topological charges and edge states