In the last few decades, topological phase^{1,2,3,4,5,6,7,8,9,10,11} 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 charges^12,13,14,15. Very recently, researchers proposed the notion of non-Abelian topological charges^16,17,18,19, 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 systems^16,17,20, admissible nodal line configurations in three dimensions^16,19,20, and may provide insight into certain strongly correlated phases of twisted bilayer graphene²¹.

在过去的几十年中，拓扑阶段^{1、2、3、4、5、6、7、8、9、10、11}已经成为超越金茨堡-朗道对称破缺范式的物质状态的新分类。潜在的全局不变量通常由整数很好地表征，例如陈数或绕组数——阿贝尔电荷^12,13,14,15。最近，研究人员提出了非阿贝尔拓扑电荷的概念^16,17,18,19，它们具有非交换且富有成效的编织结构，其中多个（多个）带隙纠缠在一起。在这里，我们通过实验观察时间反转和反转对称传输线网络中的非阿贝尔拓扑电荷。四元数值的非阿贝尔拓扑电荷清楚地映射到本征态框架球体上。此外，我们发现了一个非阿贝尔商关系，它提供了关于边缘/畴壁状态分布的全局视角。我们的工作为非阿贝尔拓扑电荷的表征和操纵打开了大门，这可能会导致有趣的观测结果，例如二维系统中与轨迹相关的 Dirac/Weyl 节点碰撞^16,17,20，三个维度中的允许节点线配置^{16,19,20，并且可以提供对扭曲双层石墨烯21}的某些强相关相的洞察。