Revealing topological attributes of stiff plates by Dirac factorization of their 2D elastic wave equation

Dirac factorization of the elastic wave equation of two-dimension stiff plates coupled to a rigid substrate reveals the possible topological properties of elastic waves in this system. These waves may possess spin-like degrees of freedom associated with a gapped band structure reminiscent of the spin Hall effect. In semi-infinite plates or strips with zero displacement edges, the Dirac-factored elastic wave equation shows the possibility of edge modes moving in opposite directions. The finite size of strips leads to overlap between edge modes consequently opening a gap in their spectrum eliminating the spin Hall-like effects. This Dirac factorization tells us what solutions of the elastic wave equation would be if we could break some symmetry. Dirac factorization does not break symmetry but simply exposes what topological properties of elastic waves may result from symmetry breaking structural or external perturbations.