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Recently, it was shown that surface electromagnetic waves at interfacesbetween continuous homogeneous media (e.g., surface plasmon-polaritons atmetal-dielectric interfaces) have a topological origin [K. Y. Bliokh et al.,Nat. Commun. 10, 580 (2019)]. This is explained by the nontrivial topology ofthe non-Hermitian photon helicity operator in the Weyl-like representation ofMaxwell equations. Here we analyze another type of classical waves:longitudinal acoustic waves corresponding to spinless phonons. We show thatsurface acoustic waves, which appear at interfaces between media withopposite-sign densities, can be explained by similar topological features andthe bulk-boundary correspondence. However, in contrast to photons, thetopological properties of sound waves originate from the non-Hermitianfour-momentum operator in the Klein-Gordon representation of acoustic fields.

Klein-Gordon representation of acoustic waves and topological origin of surface acoustic modes