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**Authors: **Olga
Petrova and Oleg Tchernyshyov

*Department of Physics and Astronomy,
Johns Hopkins University, 3400 N. Charles St.*

Ordered arrays of skyrmion
lines and baby skyrmions have been revealed in three-dimensional compounds and
thin films via neutron scattering experiments [1,2] and real-space observation
[3,4]. Their nonzero topological charge suggests that the low-frequency
dynamics of these systems is dominated by the spin Berry phase. Zang *et al.* approached this problem by
treating skyrmions as interacting particles that are subject to a Lorentz-like
force [5]. However, the point-particle approximation is hard to justify because
(a) the skyrmion size is comparable to the distance between skyrmions [3,4] and
(b) the spatial Fourier spectrum of a skyrmion crystal is dominated by Bragg
peaks with the lowest nonzero q, with much weaker higher harmonics [1]. Thus a
skyrmion crystal should be viewed not as a collection of particles, but as a
superposition of waves, namely three helices whose wavevectors form an
equilateral triangle. Such a spin-density wave would be favored by a cubic term
in the free energy, which arises in the presence of an applied magnetic field
[1]. We have reexamined the dynamics of a skyrmion crystal treating it as a
spin-density wave.

We derive the spin-wave spectrum first for a single helix and then for three phase-locked helices. The low-frequency spin waves are Goldstone modes associated with the broken translational symmetry of the crystal. Their dispersion is determined by the elastic moduli of the skyrmion crystal and by the kinetic terms of the effective Lagrangian, which include both a Berry-phase term reflecting the nontrivial topology of skyrmions and a kinetic energy inherited by the skyrmion crystal from single helices. In the absence of the Berry phase, one finds two branches of excitations with sound-like dispersions, transverse and longitudinal phonons [6]. The Berry-phase term acts like an effective magnetic field, mixing the longitudinal and transverse vibrations into a gapped cyclotron mode and a twist wave with a quadratic dispersion [7].

This work was supported in part by the US National Science Foundation under Award No. DMR-1104753.

[1] S. Muehlbauer, B.
Binz, F. Jonietz, C. Peiderer, *et al.*,
Science **323**, 915 (2009).

[2] W. Muenzer, A.
Neubauer, T. Adams, *et al*., Phys.
Rev. B **81**, 041203 (2010).

[3] X. Z. Yu, Y. Onose, N.
Kanazawa, J. H. Park, *et al*., Nature **465**, 901 (2010).

[4] X. Z. Yu, N. Kanazawa,
Y. Onose, K. Kimoto, *et al*., Nat.
Mater. **10**, 106 (2011).

[5] J. Zang, M. Mostovoy,
J. H. Han, and N. Nagaosa, Phys. Rev. Lett. **107**, 136804 (2011).

[6] T. R. Kirkpatrick and
D. Belitz, Phys. Rev. Lett. **104**,
256404 (2010).

[7] O. Petrova and O.
Tchernyshyov, Phys. Rev. B **84**,
214433 (2011).

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