The acoustic Dirac equation as a model of topological insulators

The dynamical equations of motion of a discrete one-dimensional harmonic chain with side restoring forces is analogous to the relativistic Klein-Gordon equation. Dirac factorization of that 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 dynamical equations under P and T 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 one with a non-conventional topology (Berry phase of π). In this 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 to recover the original spinor. The interface between topologically conventional and non-conventional chains supports topological surface states. The Dirac-factored equations of motions of the one-dimensional harmonic chain with side springs can serve as a model for the investigation of the properties of acoustic topological insulators. [Work supported by NSF Award No. 2242925.]