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

## TL;DR

This paper demonstrates that surface acoustic waves at media interfaces have a topological origin linked to the Klein-Gordon representation of acoustic fields, revealing a new topological perspective for classical sound waves.

## Contribution

It introduces a topological explanation for surface acoustic waves based on the non-Hermitian four-momentum operator in the Klein-Gordon framework, extending topological concepts beyond electromagnetic waves.

## Key findings

- Surface acoustic waves are explained by topological features.
- Topological properties originate from the non-Hermitian four-momentum operator.
- Surface waves occur at media with opposite-sign densities.

## Abstract

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