The acoustic Dirac equation as a model of topological insulators

The dynamical equations of motion for a discrete, one-dimensional harmonic chain with side restoring forces are analogous to the relativistic Klein–Gordon equation. Dirac factorization of the discrete Klein–Gordon equation introduces two equations with time reversal (T) and parity (P) symmetry-breaking conditions. The Dirac-factored equations enable the exploration of the properties of the solutions of the dynamic equations under PT symmetry-breaking conditions. The spinor solutions of the Dirac factored equations describe two types of acoustic waves: one with a conventional topology (Berry phase equal to 0) and the other with a non-conventional topology (Berry phase of π). In the latter case, the acoustic wave is isomorphic to the quantum spin of an electron, also known as an “acoustic pseudospin,” which requires a closed path, corresponding to two Brillouin zones (BZs), to restore the original spinor. We also investigate the topology of evanescent waves supported by the Dirac-factored equations. The interface between topologically conventional and non-conventional chains exhibits topological surface states. The Dirac-factored equations of motion of the one-dimensional harmonic chain with side springs can serve as a model for the investigation of the properties of acoustic topological insulators.