plane in the vicinity of the nuclear [110] Bragg
reflection (i.e.,G[110]= [110]; see the schematic
depictions in Figs. 2, 3, and 4). Accordingly,
the associated theoretical predictions shown
inFigs.2,3,and4werecalculatedforapro-
jection onto the subspace perpendicular to the
nuclear Bragg vectorG[110].
We begin with the magnon spectra in the
skyrmion lattice plane, where the emergent
Landau levels are expected. For the unpolarized
ToF surveys, the magnetic fieldHwas parallel
to [001] and therefore perpendicular to both
G[110]and the scattering plane, as depicted
in Fig. 2A1. This is denoted as setup 1. Because
the magnetic satellites were located within the
scattering plane, the NSF and SF scattering
contained both nuclear and magnetic contrib-
utions, additionally motivating TAS. Typical
magnon spectra forq⊥perpendicular toH
recorded with ToF spectroscopy are shown in
Fig. 2A1 and fig. S3. The scattering intensity is
presented in yellow/blue shading. Black lines
represent the calculated magnetic response
tensor,c^00 ijðÞq;E,wherethelinethickness
reflects the spectral weight. The intensity
compares well with the calculated spectra again
showninFig.2A2asexplicitlydemonstratedin
fig. S6, which is a color-shaded depiction of the
calculated dynamic structure factor with a
convolution of the ToF resolution.
The calculated spectra for wave vectors per-
pendicular to the skyrmion tubes,q⊥, shown
in Fig. 2A2 as gray lines, display the Landau
levels stemming from the nontrivial topology
of the skyrmion lattice; the calculated mag-
netic response tensor is shown as black lines,
as in Fig. 2A1. The weak dispersion of the
Landau levels is caused by variations inBem
as well as the magnon potential. Because the
Landau level wave functions of magnons for
q⊥are distinctly different from the plane wave
function of the neutron, the spectral weight
at elevated energies gets distributed across a
wide range of magnon bands. Thus, a broad
distribution of spectral weight, as compared
to the well-defined magnon branches along
theskyrmiontubesshowninFig.4,represents
a key signature of the nontrivial topology.
For a quantitative comparison between
theory and experiment, and to keep track of
spin-flip versus non–spin-flip processes with
respect to incoherent and nuclear scattering,
we performed high-resolution polarized TAS.
Because the predicted spacing of the Landau
levels of ~10 μeV is well below the resolution
of state-of-the-art TAS, we used a setup in
which the magnetic fieldHwas parallel toG
(i.e.,H[110]along [110]) such that the skyrmion
lattice stabilized perpendicular toH[110]with
a pair of its satellites parallel to a 1½ 10. This
is referred to as setup 2. The polarizationP
is aligned adiabatically withH[110]and hence
G, and thus the SF and NSF scattering were
purely magnetic and nuclear, respectively.
SCIENCEscience.org 4 MARCH 2022•VOL 375 ISSUE 6584 1027
Fig. 2. Magnon spectra of MnSi for momentum transfers perpendicular to the skyrmion lattice tubes.
Left: Experimental data. Right: Quantitative calculations of the magnon spectra. Thin gray lines represent
the magnon spectraE(q); black, red, and blue lines represent the magnetic response tensor,c′′ijðÞq;E.
The line thickness of the latter reflects the spectral weight. Colors red/blue and green denote spin-flip and
non–spin-flip processes, SF and NSF, respectively. Energy and momentum transfers are provided in two
corresponding scales. (A1) ToF scattering intensity for momentum transfersq⊥,1|| [1,1,0]. The color shading
represents the experimental scattering intensity. Black lines represent the calculated magnetic response
tensor shown also in (A2) (see fig. S6B for the calculated dynamic structure factor as convoluted with
the instrumental resolution). (A2) Calculated excitation spectra (gray) and magnetic response tensor (black).
(B1,B2,C1, andC2) Polarized TAS intensities for selected momentum transfers and field values. Curves
shown in red and blue shading represent the sum of the calculated dynamic structure factor convoluted
with the instrumental resolution (light-red/blue shading) and a quasi-elastic (QE) contribution attributed to
longitudinal fluctuations (gray shading) ( 17 ). The same quantitative scaling factor and the same QE
contribution was used for all TAS data shown in the main text and ( 17 ) for momentum transfers
perpendicular and transverse to the skyrmion latticeq⊥andq||, respectively. (D1andD2) Calculated
magnon spectra and magnetic response tensors. The location of experimental scans shown in (B) and (C)
is marked by gray boxes [see figs. S10 and S17 for further data including those of scan (ii)].RESEARCH | REPORTS