that may be attributed to the CCW magnon
with a linewidthG 2 = 10 μeV, and (iii) an ex-
citation atE 3 = 80 μeV that may be attrib-
uted to HM2 and HM3 with a linewidthG 3 =
6 μeV. Thus, the predicted energies of the most
intense excitations are in excellent agreement
with experiment. The GM is thereby topolog-
ically trivial (C= 0), whereas the CCW mode,
n= 3, is topologically nontrivial withC= +1.
In contrast to the skyrmion lattice plane, the
spectra for momentum transfers along the
skyrmion tubes,q||, are not sensitive toBem,
where typical polarized TAS intensities using
setup 2 are shown in Fig. 4 (see figs. S10 to S13
and S16 for further data and the associated
NSF intensities, representing purely incoherent
nuclear scattering). In excellent agreement
with the calculated spectra shown in Fig. 4,
C1 and C2, the TAS exhibits pronounced max-
ima with substantial weight. A detailed inspec-
tion of the calculated spectra ( 17 ) allows these
maxima to be attributed to the CCW mode,
previously identified at the zone center using
microwaves ( 32 ). These are conventional, topo-
logically trivial, strongly dispersive spin wave
branches.
Moreover, using setup 2, polarized TAS al-
lowed us to discern pronounced individual
modes that are clearly nonreciprocal [i.e.,
E(–q||)≠E(q||)]. Namely for wave vectorsparallel and antiparallel toH, the structure
factor exhibited a pronounced nonreciprocity
[i.e.,Sij(q,E)≠Sij(–q,E)] already evident in
Fig. 4, C1 and C2. Here Fig. 4, A1 and B1, dis-
play the same scans under inversion of
momentum transfer for the same field orien-
tation (the same applies to the scans shown in
Fig. 4, A2 and B2). Related data under field
inversion are presented in figs. S13 and S16.
This well-known nonreciprocity originates in
the combination of Dzyaloshinskii-Moriya and
dipolar interactions as discussed in ( 41 ); it is
characteristic for chiral materials ( 42 ). In MnSi
it was previously observed in the field-polarized
( 22 , 24 ) and conical phases ( 34 ), as well as in
the paramagnetic regime ( 26 ).
The topological magnon bands in the skyrmion
lattice of MnSi we report here are highly un-
usual in several ways. First, the connection
between the nontrivial topological winding in
real space and the topological magnon bands
in reciprocal space may be quantitatively cap-
tured within a universal framework based on
three material-specific parameters: the energy
separating the conical and field-polarized state,
Ec2; the helical wave vector,kh; and the sus-
ceptibility in the conical state,cintcon( 17 ). Second,
the Landau levels and topological band struc-
ture are observed starting at the lowest-lying
states up to high energies. Third, we expect
contributions to the thermal Hall effect, which,
however, will not be quantized given the bo-
sonic character of the magnons. Fourth, an
intimate analogy exists between quantum Hall
phenomena and the emergent Lorentz force,
Landau levels, and the topological magnon
band structure caused by the nontrivial real-
space topology of skyrmion lattices in chiral
magnets. On the basis of the bulk/boundary
correspondence, our observations in the bulk
imply the existence of chiral edge states that
may be technologically useful—for instance, as
a directional coupler for quantum technologies.REFERENCESANDNOTES- G. E. Volovik,J. Phys. C 20 , L83–L87 (1987).
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SCIENCEscience.org 4 MARCH 2022•VOL 375 ISSUE 6584 1029
Fig. 4.Polarized neutron scattering intensity and calculated magnon spectra of MnSi for momentum
transfers parallel to the skyrmion lattice tubes. Left: Experimental data. Right: Calculations of the magnon
spectra. Thin gray lines represent the magnon spectraE(q); red and blue shaded lines denote the magnetic
response tensor,c′′ijðÞq;E, for the two spin-flip scattering processes. The line thickness of the magnetic
response tensor reflects the spectral weight. Energy and momentum transfers are provided in two
corresponding scales. (A1,A2,B1, andB2) Polarized TAS intensities for selected momentum transfers and
field values chosen to highlight well-defined, dispersive, nonreciprocal magnons. 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. (C1andC2) Calculated magnon spectra. The location of the experimental data
shown in (A) and (B) is marked by gray boxes (see figs. S10 to S13 and S16 for further data).
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