Kei Yamamoto,^{1, 2, ∗} Guo Chuan Thiang,³ Philipp Pirro,⁴

Kyoung-Whan Kim,5, 6 
Karin Everschor-Sitte,5 and 
Eiji Saitoh7, 8, 1
 
1 Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
2 Institut f¨ur Physik, Johannes Gutenberg-Universit¨at Mainz, 55128 Mainz, Germany
3 School of Mathematical Sciences, University of Adelaide, SA 5000, Australia
4 Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨at Kaiserslautern, 67663 Kaiserslautern, Germany
5 Institut f¨ur Physik, Johannes Gutenberg Universit¨at Mainz, 55128 Mainz, Germany
6 Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Korea
7 Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
8 Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

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.