# Topological characterization of classical waves: the topological origin of magnetostatic surface spin waves

## TL;DR

This paper introduces a topological framework for classical waves, specifically magnetostatic surface spin waves, revealing their topological origin through vortex lines in the Brillouin zone and predicting surface modes without a bulk gap.

## Contribution

It develops a topological characterization for classical wave Hamiltonians, applying it to spin waves and identifying vortex lines as topological invariants, expanding topological matter beyond gapped systems.

## Key findings

- Surface modes appear without a bulk gap.
- Vortex lines in the Brillouin zone characterize topology.
- Bulk-edge correspondence applies to classical waves.

## Abstract

We propose a topological characterization of Hamiltonians describing classical waves. Applying it to the magnetostatic surface spin waves that are important in spintronics applications, we settle the speculation over their topological origin. For a class of classical systems that includes spin waves driven by dipole-dipole interactions, we show that the topology is characterized by vortex lines in the Brillouin zone in such a way that the symplectic structure of Hamiltonian mechanics plays an essential role. We define winding numbers around these vortex lines and identify them to be the bulk topological invariants for a class of semimetals. Exploiting the bulk-edge correspondence appropriately reformulated for these classical waves, we predict that surface modes appear but not in a gap of the bulk frequency spectrum. This feature, consistent with the magnetostatic surface spin waves, indicates a broader realm of topological phases of matter beyond spectrally gapped ones.