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Topologically protected elastic waves in phononic metamaterials

Topological states of quantum matter exhibit unique disorder-immune surface states protected by underlying nontrivial topological invariants of the bulk. Such immunity from backscattering makes topological surface or edge states ideal carriers for both classical and quantum information. So far, topological matters have been explored only in the realms of electronics and photonics, with limited range of bulk properties and largely immutable materials. These constraints thus impose severe performance trade-offs in experimentally realizable topologically ordered states. In contrast, phononic metamaterials not only provide access to a much wider range of material properties, but also allow temporal modulation in the non-adiabatic regime. Here, from the first-principles we demonstrate numerically the first phononic topological metamaterial in an elastic-wave analogue of the quantum spin Hall effect. A dual-scale phononic crystal slab is used to support two effective spins of phonon over a broad bandwidth, and strong spin-orbit coupling is realized by breaking spatial mirror symmetry. By preserving the spin polarization with an external load or spatial symmetry, phononic edge states are shown to be robust against scattering from discrete defects as well as disorders in the continuum. Our system opens up the possibility of realizing topological materials for phonons in both static and time-dependent regimes.