Topological phase transition in mechanical honeycomb lattice

Topological materials provide a new tool to direct wave energy with unprecedented precision and robustness. Three elastic topological phases, the valley Hall, Chern and spin Hall insulators, are currently studied, and they are achieved separately in rather distinct configurations. Here, we explore analytically various topological phase transitions for in-plane elastic wave in a unified mass-spring honeycomb lattice. It is demonstrated that the three elastic topological phases can be realized in this single lattice by designing mass, stiffness or introducing Coriolis' effect. In particular, the interface between valley Hall and Chern insulators is found to support topological interface mode for the first time. Perturbation method is used to derive the analytic effective continuum model in the neighbor of band degeneracy, and the physics in topological phase transitions are revealed through evaluation of topological invariants. The topologically protected interface states, their decaying profile as well as the pseudo-spin-indicating polarization specific for elastic wave are systematically analyzed, and these results are further confirmed numerically by Bloch wave analysis of domain wall strip and transient simulation of finite sized sample. This study offers a concise and unified analytical model to explore topology nature of elastic wave, and can provide intuitive guidance to design of continuum mechanical topological materials.