Hermitian and non-hermitian higher-order topological states in mechanical metamaterials

Higher-order topological insulators exhibit clear hierarchical boundary states, providing an amazing platform for robust wave manipulations in mechanical or acoustic systems. Recently, this theory has been extended to nonHermiticity-induced higher-order topology, which enables actively controllable topological transport. Nevertheless, most related studies focus on the non-Hermitian quadrupole topology, associated with the coexistence of negative and positive couplings to constitute a pi-flux lattice, which hindered the realizations in classical wave systems. Here, we propose a novel tight-binding model without negative couplings, which supports nonHermiticity-induced higher-order topological states. In the Hermitian case, the middle band gaps host the vanishing bulk polarization and the associated topology is dependent only upon the associated edge polarization. By introducing the external loss as non-Hermitian components, we reveal that therein the trivial structures can also host topological corner states, featured by the quantized nonzero edge polarizations in the biorthogonal basis. Furthermore, such a topological phase transition induced by non-Hermiticity can be realized for almost all of our lattices, except for the isotropic case where the middle band gaps never open. Basing on the theoretical design, we map the lattice models into the mechanical metamaterials to access the analogous higher-order topology in the elastic wave systems for both Hermitian and non-Hermitian physics. The simulated band structures match well with theoretical solutions. And the robust higher-order topological states are also identified. Our work paves the way to implement the non-Hermitian higher-order topology and offers the possibility for wave manipulations in non-Hermitian systems.