Entwined Brillouin Klein bottles induced by a non-Abelian gauge field

The Brillouin Klein bottle induced by ${\mathbb{Z}}_{2}$ gauge field has been recently proposed and has sparked growing interest. The momentum-space nonsymmorphic symmetry that it respects fundamentally modifies the Bloch band theory and expands the topological classification. Previous works limited to systems under ${\mathbb{Z}}_{2}$ gauge field have been developed, leaving the interaction between non-Abelian gauge fields and symmetries untouched. Here, we report theoretical and experimental realization of entwined Klein bottles with two orthogonal sub-bottles induced by ${U}_{2}$ non-Abelian gauge field, which stems from the hopping between two inseparable pseudospins. The bilayer acoustic crystals with non-Abelian coupling are constructed here to explore different topologies from the traditional Klein bottles under the Abelian fields. This research sheds light on both first-order and higher-order topologies unexplored in non-Abelian gauge fields, and enriches the topological classification with gauge symmetry beyond the existing paradigms.