Topological characterization of classical waves: The topological origin ofmagnetostatic surface spin waves

K. Yamamoto1), G. C. Thiang2), P. Pirro3), K.-W. Kim4), K. Everschor-Sitte5) and E. Saitoh6), 1)1): R. G. for Spin-energy transformation science., JAEA 2): The Univ. of Adelaide3): TU Kaiserslautern 4): KIST 5): JGU Mainz 6): The Univ. of Tokyo

A ferromagnet consists of numerous microscopicmagnets called spins, and fluctuations of the individual
spins can propagate as spin waves. The spin waves are acurrent focus of research in spintronics since they are
considered a potential alternative for electrons as a carrierof information. There is a particular type of spin waves
called magnetostatic surface spin waves (MSSWs), which
was discovered in 1960 [1]. In contrast to the ordinary
spin waves that diffuse through the three-dimensionalbulk of the magnet, MSSWs propagate along the surface
and only in one particular direction (Fig. 1), and areanomalously robust against back-scatterings. Because of
the chiral propagation and the immunity to backscatterings, they are almost exclusively used infundamental studies of spin-wave transport in
microstructures.
In the quantum mechanical studies of elecrtrons insolids, the recent advent of topological materials sciencehas shown that chiral surface states robust against
scatterings are a hallmark of topologically non-trivial
bulk band structures. The above mentioned features ofMSSWs are particularly reminescent of topological Chern
insulators so that there has been peculation over their
topological origin for a while, though without a proof. In
this work, we have theoretically validated this claim, andrevealed the precise topological nature of the spin wave
band structure and its relation to the chirality androbustness of MSSWs [2]. Along the way, we haveidentified a mathematical structure that is absent inquantum mechanical electrons but occurs only forclassical waves, whereby extending the realm of
topological materials science into an unexplored newterritory.
MSSWs arise from the dipole-dipole interactions
between magnetic moments. We have carefully studied
the band structure of spin waves that is determined by thelinearised classical Heisenberg model including thedipole-dipole terms. The key observation was that theresulting spin-wave Hamiltonian can be written in acanonical form that allows for calculating winding
numbers around any point in the Birllouin zone. Thewinding number around the origin of the Brillouin zone
was computed to be equal to -2. This implies the presenceof topologically protected vortex lines in the threedimensional Brillouin zone of the spin waves as indicatedby red and blue cylinders in Fig. 2. By appropriately
generalising the bulk-edge correspondence established forelectrons in solids, we successfully demonstrated that thechiral propagtion and the back-scattering immunity ofMSSWs are direct consequences of the topologicalFig.2 Schematic Brillouin zone structure for spin waves driven bydipole-dipole interactions. There exists a vortex line carrying thewinding number charge -2 passing through the origin (red cylinder)
and another at the Brillouin zone corner (blue cylinder) that maintainthe charge neutrality.

winding structure in the Brillouin zone.

This work was supported by Transregional Collaborative
Research Center (SFB/TRR) 173 SPIN+X of the DFG, JSPS
KAKENHI Grant No. JP18H05855, Australian ResearchCouncil Grant No. DE170100149, the German Research
Foundation (DFG) Grants No. EV 196/2-1 and No. SI 1720/2-1,
the KIST Institutional Program, JST-ERATO “Spin QuantumRectification”, and AIMR Tohoku University.

References[1] J. R. Eshbach and R. W. Damon, Phys. Rev. 118, 1208
(1960).
[2] K. Yamamoto et al., Phys. Rev. Lett. 122, 217201 (2019).
Fig.1 Unidirectional propagation of MSSWs along the surface. B is
the magnetic field and M is the magnetic moment.