Simultaneous pseudospin and valley topological edge states of elastic waves in phononic crystals made of distorted Kekulé lattices.

Topological metamaterials protected by the spatial inversion symmetry mainly support single type edge state, interpreted by either the quantum valley Hall effect or the quantum spin Hall effect. However, owing to the existence of the complicated couplings and waveform conversions during elastic wave propagation, realizing topologically protected edge states that support both pseudospin and valley degrees of freedom in elastic system remains a great challenge. Here, we propose a two-dimensional Kekulé phononic crystal (PC) that can simultaneously possess pseudospin- and valley-Hall edge states in different frequency bands. By inhomogeneously changing the elliptical direction in a Kekulé lattice of elliptical cylinders, three complete phononic bandgaps exhibiting distinct topological phase transitions can be obtained, one of which supports a pair of pseudospin-Hall edge states and the other hosts valley-Hall edge states in the low and high frequency regime. Furthermore, a sandwiched PC heterostructure and a four-channel cross-waveguide splitter are constructed to achieve selective excitation and topological robust propagation of pseudospin- and valley-momentum locking edge states in a single configuration. These results provide new possibilities for manipulating in-plane bulk elastic waves with both pseudospin and valley degrees of freedom in a single configuration, which has potential applications for multiband and multifunctional waveguiding.