Predicting Degeneracy and Topological Properties in 2D Phononic Band Diagrams*

Phononic crystals are macroscopic elastic structures that have periodic modulations in material properties like shear modulus and density. We can draw analogies between band diagrams in phononic crystals that describe wave propagation and electronic band diagrams using the well-established tools in hard condensed matter theory. By applying representation theory to the symmetry properties of 2D phononic crystals, we can predict the existence or absence of degenerate wavemodes that correspond to intersections in the phononic band diagrams. Numerical calculations of these phononic band diagrams confirmed theoretical predictions for all seventeen 2D crystallographic groups. As in electronic systems, we also investigate the topological nature of phononic bands. Using band representations, which can be expressed as linear combinations of Wannier functions, we infer whether a phononic band has trivial or non-trivial topology. Using the above approach, we can design phononic crystals with target band structure properties.