Non-Hermiticity has greatly broadened the scope of research on topological materials, especially on topological metamaterial because of the flexibility in the introduction of designed loss/gain or active control wherein. This creates a new frontier—the non-Hermitian topological metamaterials. In this work, we report the realization of non-Hermitian Möbius insulators for acoustic waves by incorporating the designed loss into acoustic crystals. Associated with the Möbius insulators, features like wave localization, Möbius twist in momentum space with $4\pi$ periodicity, and projective translation eigenvalues in phase space, are demonstrated. More specifically, the skin effect of Möbius edge states is identified. We also extend our study to the three-dimensional case, and the Möbius hinge states and the associated skin effect are observed. With designed loss to achieve Möbius insulators and reveal the skin effect of Möbius edge states, our work provides an avenue toward the study of non-Hermitian topological physics.

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• Acoustics
• Phononic crystals
• Acoustic metamaterials
• Topological materials
• Finite-element method