Hybrid Topological Euler and Stiefel-Whitney Phases in Elastic Metamaterials

Recent advances in multigap topological phases—characterized by invariants like Euler and second Stiefel-Whitney classes across multiband subspaces—highlight their dependence on non-Abelian braiding of momentum-space band degeneracies in adjacent gaps. Here, we theoretically predict and experimentally demonstrate that two of such topological phases can coexist within a single system using vectorial elastic waves. The inherent coupling between different polarization modes enables non-Abelian braiding of nodal points of multiple energy band gaps and results in hybrid Euler and Stiefel-Whitney topological insulator phases. We furthermore unveil the central role played by the topologically stable Goldstone modes’ degeneracy. Our findings represent the first realization of hybrid phases in vectorial fields exhibiting topologically nontrivial Goldstone modes, paving the way for bifunctional applications that leverage the hybrid topological edge and corner states.