Elastic gapless helical edge states on the translation-dislocation interface between identical 2D phononic patterns with Kekule distortion

We present a theoretical and experimental study of the elastic-wave topological edge states that propagate along the dislocation interface of two pieces of material with identical pattern only up to a rigid-body translation. This seems to violate our knowledge in topological materials about phase transition. It is due to that the elastic-wave analogue to the quantum spin Hall material is based on a mechanism distinct from the quantum mechanical counterpart. Quantum electronic topological phases of materials are classified by associated topological invariants. Examples include Chern numbers for quantum (anomalous) Hall insulators, valley Chern numbers for quantum valley Hall insulators, and Z2 invariants for quantum spin Hall topological insulators. Specifically, the Z2 invariant is connected to the unique property of fermionic systems whose wavefunction acquires a negative sign upon two consecutive applications of the time-reversal operator. This property leads to Kramers degeneracy and results in fully decoupled counter propagating edge states. However, this property is not acquired by the classical-wave systems. Instead, in our proposed system, a combination of spatial and temporal symmetry is utilized in replacing the time-reversal operator to synthesize the Kramers degeneracy. Correspondingly, a local topological charge is defined based on the difference of integrated pseudospin-resolved Berry curvature as an alternative to the Z2 invariant. Its value depends on the position of the reference frame, hence the same phononic structure can be in different topological phases upon shifting and allows edge states to exist on the dislocation interface. Our experiment result confirms the gapless spectrum and the robustness against sharp bends along the path, of the edge states.