Topological non-Hermitian phononic metamaterials

Topological phases in phononic systems enable robust and unconventional control of wave propagation. However, realistic acoustic, elastic, and mechanical platforms are open, while loss or gain, active feedback, and spatiotemporal modulation provide tunable routes to non-hermiticity and nonreciprocity. Non-hermiticity reshapes complex-frequency spectra, eigenstates, and bulk-boundary correspondence beyond conventional Hermitian band theory, thereby giving rise to a wide range of unconventional wave phenomena. In this review, we provide a unified overview of topological non-Hermitian phononic crystals (PCs) and metamaterials from one to three dimensions, emphasizing physical mechanisms. In one-dimensional (1D) systems, non-Hermitian extensions of the Su–Schrieffer–Heeger (SSH) model reveal the emergence of the non-Hermitian skin effect (NHSE) and the breakdown of conventional Bloch band theory, necessitating the generalized Brillouin zone (GBZ) formalism and non-Bloch topological invariants. In two-dimensional (2D) systems, the interplay of non-hermiticity with lattice symmetry gives rise to exceptional degeneracies, anisotropic and higher order skin localization, and valley- and pseudospin-dependent transport. In three-dimensional (3D) systems, non-Hermitian Weyl phases support exceptional rings, complex Fermi arcs, and surface states governed by biorthogonal topology. Finally, we discuss non-hermiticity in space–time-modulated systems, non-Abelian exceptional-point braiding, and synthetic Landau-level physics as emerging directions. This review aims to offer a unified perspective on non-Hermitian topology in PCs and to stimulate further developments toward next-generation intelligent elastic metamaterials and topological wave devices.