M. Janoschek et al- Helimagnon Bands as Universal Excitations of Chiral Magnets

8/3/2019 M. Janoschek et al- Helimagnon Bands as Universal Excitations of Chiral Magnets

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Helimagnon Bands as Universal Excitations of Chiral Magnets

M. JanoschekPhysik Department E21, Technische Universitat Munchen, D-85748 Garching, Germany and

Laboratory for Neutron Scattering, Paul Scherrer Institut & ETH Zurich, CH-5232, Villigen, PSI

F. Bernlochner, S. Dunsiger, C. Pfleiderer, and P. BoniPhysik Department E21, Technische Universitat Munchen, D-85748 Garching, Germany

B. RoessliLaboratory for Neutron Scattering, Paul Scherrer Institut & ETH Zurich, CH-5232, Villigen, Switzerland

P. LinkForschungsneutronenquelle Heinz Maier-Leibniz (FRM II),

Technische Universitat Munchen, D-85748 Garching, Germany

A. RoschInstitute for Theoretical Physics, Universitat zu Koln, Germany

Kavli Institute for Theoretical Physics, Santa Barbara, USA(Dated: September 6, 2010)

MnSi is a cubic compound with small magnetic anisotropy, which stabilizes a helimagnetic spinspiral that reduces to a ferromagnetic and antiferromagnetic state in the long- and short-wavelengthlimit, respectively. We report a comprehensive inelastic neutron scattering study of the collectivemagnetic excitations in the helimagnetic state of MnSi. In our study we observe a rich varietyof seemingly anomalous excitation spectra, as measured in well over twenty different locations inreciprocal space. Using a model based on only three parameters, namely the measured pitch of thehelix, the measured ferromagnetic spin wave stiffness and the amplitude of the signal, as the onlyfree variable, we can simultaneously account for all of the measured spectra in excellent quantitativeagreement with experiment. Our study identifies the formation of intense, strongly coupled bandsof helimagnons as a universal characteristic of systems with weak chiral interactions.

PACS numbers:

I. INTRODUCTION

The spontaneous breaking of a continuous symmetryin a magnetically ordered state implies the existence ofGoldstone modes. Thus, the low-energy spin wave dis-persions of ferromagnets and commensurate antiferro-magnets are universal they can be deduced from simplesymmetry arguments and the description does not de-pend on microscopic details15. In ferromagnets, wherethe order parameter is a conserved quantity, the resultingspin waves have a quadratic dispersion while in antifer-romagnets the superposition of the normal modes of thesublattices leads to the well-known linear dispersion atlow energies6,7.In recent years complex forms of magnetic order re-lated to weak chiral interactions of non-centrosymmetricsystems, also referred to as Dzyaloshinsky-Moriya(DM) interaction attract great interest. Systems inwhich DM interactions play an important role in-clude multiferroics8, the parent compounds of the high-temperature superconductors9, thin magnetic films (Mnon a W substrate, or Fe on a Ir substrate10,11), heavyfermion superconductors12 and even itinerant-electronmagnets, like MnSi, which displays a skyrmion latticeat ambient pressure13 and an extended non-Fermi liquid

resistivity at high pressures14,15. As DM interactions areof growing importance in non-centrosymmetric systems

this raises the question for any universal properties of thespin excitations in systems with weak chiral interactions.Helimagnetic order is very well established in systemswith magnetic frustration such as rare earth elements likeTb or Ho. An excellent experimental and theoretical ac-count has been given by Jensen and Mackintosh16,17, butexcitation spectra perpendicular to the magnetic propa-gation vector have not been studied before. In addition ithas long been appreciated that helimagnetic order due tofrustration differs fundamentally from helimagnetic or-der due to DM interactions.18 The work on frustratedmagnets has been contrasted by studies of the tetragonalsystem Ba2CuGe2O7

19, in which the influence of DM in-teractions on the excitation spectra has been addressed.However, Ba2CuGe2O7 orders antiferromagnetically onlocal length scales. Due to the much stiffer spectrum ofan antiferromagnet compared to a ferromagnet, the ef-fects of DM interactions in a helical antiferromagnet aretherefore substantially weaker and qualitatively differentas compared to a helical ferromagnet like MnSi. More-over, the helimagnetic order in Ba2CuGe2O7 is distortedand shows higher order harmonic contributions of at least20%.We have therefore decided to study the perhaps simplest

arXiv:0907.5576v2

[cond-mat.str-el]

27Apr2010


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FIG. 1: (a) Depiction of a low-energy spin wave excitation ofa ferromagnet with wavelength ex. The spins precess aroundthe uniform magnetization. (b+c) Sketch of a similar excita-tion in the case of a helimagnet with a momentum, q, parallelto the propagation vector of the helix, kh. The wavelengthex is smaller (b) or much larger (c) than the pitch of thehelix h = 2kh . Note that only for this parallel configuration

the excitations of the ferromagnet and the helimagnet arevery similar for short ex. (d) Reciprocal space map aroundthe nuclear (110) Bragg peak. The locations of the magneticBragg satellites k1 to k4 associated with the four domainconfigurations of the helimagnetic order are shown as colorcoded throughout the text. The locations at which the spinexcitations were measured are shown by the open symbols.They may be grouped along three trajectories, where the res-olution ellipsoids are shown in gray shading. Note that all ofthe spectra recorded may be simultaneously accounted for bythe model described in the text using just one parameter (theintensity).

example for the effects of DM interactions, notably cu-bic chiral helimagnets, where the uniform magnetizationrotates slowly around an axis with a characteristic wavevector kh. As illustrated in Fig. 1, a helical state is es-sentially a ferromagnet on short length scales and an an-tiferromagnet for long distances. In fact, from a moregeneral viewpoint, all forms of complex order can be in-terpreted as a superposition of helimagnetic order20. Incases where the pitch of the helix is much larger than the

lattice spacing, h a, a helix can become remarkablystable against crystalline anisotropies. The Goldstonetheorem ensures that the excitations of an incommen-surate helimagnet are gapless (in case the helimagnet iscommensurate the gap is exponentially small)21,22. Inthis paper we report, that the spin excitations for a he-lix with a long pitch display an universal property: theformation of strongly coupled bands of helimagnons.

The B20 compound MnSi is ideally suited to study thecollective spin excitations of helimagnets due to DM in-teractions experimentally, because the magnetic prop-erties result from a clear separation of energy scalesin a metallic host. MnSi crystallizes in the non-centrosymmetric cubic space group P213 (a = 4.558 A).Below Tc = 29.5 K and in zero magnetic field a long-wavelength spin spiral with the spins perpendicular tothe propagation direction stabilizes. The periodicityh 180 A of the helix results from the competitionof ferromagnetic exchange interactions, as the strongestscale, and Dzyaloshinskii-Moriya (DM) interactions asa manifestation of weak spin- orbit coupling in crystalstructures without inversion center, on an intermediatescale23,24. The propagation direction of the spin spi-ral is finally locked to the cubic space diagonal throughvery small crystal field interactions, providing the weak-est scale. In comparison with the helical modulation, the

Fermi wave-vector is large kF 0.7 A1.MnSi has recently attracted great interest as the per-haps best candidate displaying a genuine non-Fermi liq-uid metallic state in a three-dimensional metal at highpressure14,15,25. Moreover, based on neutron scatteringand -SR at high pressures26,27, it has been suggestedthat the NFL behavior may be related to spin textureswith non-trivial topology2831. In fact, a skyrmion lat-tice, was recently identified unambiguously at ambientpressure in a small phase pocket just below Tc, which isbelieved to be stabilized by thermal fluctuations13. Toresolve the origin of these exciting properties it has be-come of great importance in its own right to establish atfirst a full account of the normal helimagnetic state andits spectrum of excitations. This has also inspired theexperimental study presented here.The three different energy scales governing the physicsof the helimagnetic state of MnSi result in four differentregimes for spin excitations. As a function of increas-ing momentum of the excitations these regimes may besummarized as follows. For the smallest momenta oneexplores the antiferromagnetic limit, q

kp, where a

linear spectrum of spin waves is expected. Here q isthe wavevector measured from the ordering vector kh ofthe helix and kp measures the weak pinning of the ori-entation of the helix to the crystalline lattice, which isfourth order in spin orbit coupling (kp is expected to beof the order ofk2h/kF and therefore extremely tiny). Theantiferromagnetic limit is followed by the helimagneticregime for kp q kh, where the pinning of the he-lix can be neglected. Here the universal properties arepredicted to consist of a dispersion similar to an antifer-


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romagnet, |q|n with n = 1 for propagation parallelto kh, whereas in the transverse direction the spectrum isquadratic with n = 2 characteristic of a ferromagnet21,32.Further increasing the momenta leads to the cross-overbetween the helimagnetic regime and ferromagnetic limit,kh q kc with kc kF deep in the magnetic phase.The magnetic structure is now probed on a length scaleshort compared to its pitch and a spectrum reminiscent

of a ferromagnet may be expected, because the systemlocally appears to be a ferromagnetically aligned. How-ever, as shown below this expectation differs dramaticallyfrom the observed behavior. Instead of ferromagneticmagnons multiple helimagnon bands are excited, whichmay be regarded as a new universal property of magneticmaterials. Finally, for the largest momenta, q kc, theferromagnetic spin waves cross over into the Stoner con-tinuum as established experimentally in a large numberof systems. The properties depend now on details of theFermi surface, i.e., they are no longer universal.Several pioneering inelastic neutron scattering studieshave been carried out in MnSi. In the mid 1970iesIshikawa and coworkers established for the first time inany magnetic metal the existence of paramagnon fluctu-ations over major portions of the Brillouin zone33. Theyfurther observed ferromagnetic spin waves when sup-pressing the helimagnetic state in an applied magneticfield33. More recently polarized inelastic neutron scat-tering established that the spin fluctuations in the para-magnetic state are chiral34 and the existence of spin-flipexcitations in a small applied field35,36.The magnetic phase diagram of MnSi is remarkably rich.For small magnetic fields between of 0.05 T and 0.15 Ta reorientation transition takes place and the helimag-netic modulation aligns with the applied field to form theso-called conical phase. The helical modulation is sup-

pressed for fields above 0.6 T. Finally, a small phasepocket just below Tc has been identified recently as thefirst example of a skyrmion lattice, thereby demonstrat-ing the inherent instability of the helical order to stabilizenon-trivial spin textures in small magnetic fields.For a comprehensive understanding it is consequently im-portant to measure the spin excitations in the prestine(zero-field cooled) helical state. This has not been at-tempted before, since four equally populated domainsform, all of which contribute to the full spectrum of thespin excitations, as described below. In the following wedenote these domains as follows: domain 1 correspondsto k1 (1,1,1), domain 2 corresponds to k2 (1, 1, 1),domain 3 corresponds to k3 (1,1, 1), and domain 4corresponds to k4 (1, 1,1) (cf. Fig. 1). We confirmedthat the domains are equally populated in our measure-ments.The experimental study of the excitation spectra of thehelimagnetic state are also extremely challenging becauseof the long wavelength of the helical modulation. Thismakes high-resolution measurements in a relatively smallportion of reciprocal space around a magnetic satellite re-flection defined by q k = 0.035 A1 necessary. In the

study described here we focus on the cross-over betweenthe ferromagnetic limit and the helimagnetic regime de-scribed above. In fact, the excitations for even smallermomenta deep in the helimagnetic regime and the antifer-romagnetic limit described above are technically not ac-cessible in present-day inelastic neutron scattering mea-surements, even for the most advanced neutron scatteringtechniques.

II. EXPERIMENTAL METHODS

A large single crystal of 8 cm3 grown by the Bridg-man method was studied. The paramagnetic spin fluctu-ations of the same single crystal were reported previouslyin Ref. 34. The single crystal displayed a lattice mosaic ofthe order 0.5. The specific heat, susceptibility and resis-tivity of small pieces taken from this large single crystalwere in excellent agreement with the literature, wherethe residual resistivity ratio (RRR) was of the order 100.The latter indicates good, though not excellent sample

purity.The bulk of our studies was carried out on the coldtriple-axis spectrometer TASP at the Paul Scherrer Insti-tut (PSI)37, with a few supplementary measurements onPANDA at the Forschungsneutronenquelle Heinz-MaierLeibnitz (FRM II). For the experiment the single crystalwas cooled with an ILL-type orange He-cryostat. Thesample was oriented with the [1,1,0] and [0,0,1] crystal-lographic directions in the scattering plane. In order toavoid second order contamination of the neutron beam aberyllium filter was inserted between the sample and theanalyzer.The experiments were performed around the nuclear

Bragg reflection (1, 1, 0) taking advantage of the largemagnetic structure factor of the corresponding magneticBragg reflections (cf. Ref. 33 and Fig. 1(c)). Two dif-ferent setups were used. For setup I the triple-axis spec-trometer was operated with fixed final wave vector kf= 1.2 A1. Additionally 20 Soller collimators were in-stalled in the incident beam and in front of the analyserwhereas 40 collimators were used in front of the detec-tor. In setup II we used kf = 1.4 A

1 and a collimationsetting: open-40-MnSi-40-80. The energy resolution insetup I and II was approximately 40 eV and 90 eV,respectively. The corresponding Q-resolution is depictedin Fig. 1. It is important to emphasize that the resolu-tion achieved was about the best currently available.In order to compare the model with the experimentalresults the theoretical scattering function S(Q, ) wasconvoluted with the resolution function of the triple-axisspectrometers we used38,39. For the fits presented in thisarticle a program called TASRESFIT40 has been devel-oped in PYTHON to perform the convolution of the spec-trometer resolution with the theoretical cross-section bymeans of Monte-Carlo integration.Further, we have introduced a Gaussian profile centred at = 0.01 meV to describe the incoherent scattering to-


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FIG. 2: Typical constant-Q scans at selected locations of the three trajectories shown in Fig. 1. Note that the energy andmomentum resolution is at the technical limit currently available. The strong elastic peak at 0 meV is due to incoherentscattering. The inset in each panel shows the precise location in reciprocal space as a black spot where data were recorded. Thecurves represent the intensity calculated in the model described in the text, where all data are accounted for by the same valuesof the ferromagnetic spin wave stiffness, the helical wavelength and the intensity. (a), (b): Data for the trajectory with q k1;these scans are dominated by a very broad maximum. (c), (d): Data for q k1; these scans show very broad, essentiallyfeatureless intensity that decreases for increasing energy. (e), (f): Data for an arbitrary trajectory emanating from k1; data forthis trajectory are characterized by almost featureless intensity over an extremely wide range of energies and a distinct peak in

a small range. The open data p oints b elow -0.4 meV in (f) represent a spurious signal that arises from additional incoherentscattering of neutrons from the monochromator crystals of the triple axis spectrometer.

gether with a constant background for the fit of all datapresented. The background was carefully measured atpositions in reciprocal space that were not contaminatedby inelastic scattering around the (1, 1, 0) Bragg reflec-tion and was estimated as 20 Counts/25 min.Finally, some of the energy scans measured during thecourse of our experiments were contaminated by spuri-ous scattering that was effectively caused by neutronsthat were scattered incoherently from the monochroma-tor crystals of the used triple-axis spectrometer (see e.g.

open symbols Fig. 2(f) at E 0.4 meV). When thespectrometer is set to a non-zero energy transfer, i.e.|ki| = |kf|, but the angle between ki and kf as wellas the orientation of the sample correspond to a geom-etry that allows for Bragg scattering from the sample,the incoherently scattered neutrons from the monochro-mator that have a wave vector |ki,inc| = |kf| lead to theobservation of accidental Bragg scattering. This type ofspurious scattering is well-known as Currat-Axe peaksin the literature41. For the analysis we considered onlyscans where the spurious scattering could be separatedwell from the inelastic signal.

III. EXPERIMENTAL RESULTS

All scans shown in the following were performed at atemperature of 20 K. This way the intensity was strongwhile still being significantly below Tc deep inside thehelical phase. Summarized in Fig. 1(d) are the locationsin reciprocal space where we have recorded excitationspectra. A total of 24 spectra were recorded. To coverthe regime of interest near the magnetic ordering wave

vector we have mostly carried out inelastic scans at re-ciprocal lattice positions along three traces in reciprocalspace as shown in Fig. 1(d). Two traces (square andrhombus symbols) were selected such that they are par-allel and perpendicular to k1, while the third trace (circlesymbols) was recorded along an arbitrary direction. Thelatter may be seen as the most stringent test of the theo-retical model used to describe the experimental data. Allspectra were recorded in energy scans at fixed momen-tum Q. The scattering intensity observed experimentally

is hence the result of contributions from each of the do-mains, that is, most of the scattering wave vectors wereneither perfectly parallel nor perfectly perpendicular tothe ordering wave vector.Prior to our study we expected spectra that are stronglyreminiscent of the excitations of the ferromagnetic limit.In stark contrast, we observed highly anomalous lineshapes that appear to be inconsistent with any conven-tional scenario. For instance, shown in Fig. 2(a) and (b)are typical data for q k1 (squares in Fig. 1). Here thedata are characterized by fairly broad dispersive maxima,but a naive interpretation of the data suggests an extremeform of broadening. This may be compared with data

recorded for q k1 shown in Fig. 2(c) and (d) (rhombusin Fig. 1), which is essentially featureless and more like abackground. Finally, the data for the arbitrary directionshown in Fig. 2(e) and (f) display yet another type ofcharacteristic, namely distinct maxima suggesting a welldefined intense mode on a large background of scattering.The remarkable variety of seemingly anomalous spectrawe observe experimentally certainly does not look remi-niscent of ferromagnetic spin waves at all. In a first at-tempt we tried to fit each scan individually with one or


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several Gaussians. However, this procedure failed. Rec-ognizing that our data was taken for wave vectors muchsmaller than the Fermi momentum, q kF, in princi-ple, the spectra should be insensitive to most microscopicdetails. We have therefore set up an effective model tocalculate the excitation spectra based on two physicallytransparent and meaningful parameters, namely the fer-romagnetic spin wave stiffness c(20 K) = 37 meVA2 de-

rived from previous experiments33,35

and the pitch of thehelical modulation kh = 0.035 A1. A detailed descrip-

tion of the theoretical model will be given in the follow-ing section IV. To compare the calculated spectra withour data we folded them with the energy and momen-tum resolution of our measurements (which is about thebest currently available) and adjusted the absolute valueof the intensity. We emphasize that the scaling of theintensity of all spectra by the same value represents theonly free parameter of our model.Our theoretical model gives complete and precise accountof all measured features and lineshapes for all momentaas illustrated in Fig. 2 for six of the 24 spectra we havemeasured. The agreement between the calculated spec-tra and the experimental data can be quantified by theremarkably small value of the reduced 2 value of 1.3.We thereby note that we used a rather conservative es-timate of the instrumental resolution, which tentativelyaverages out small details that may possibly exist in thedata. These underestimates of the experimental resolu-tion are quite typical and observed in many inelastic neu-tron scattering studies. Further, when artificially mod-ifying the values of c and kh by more than 5% thequality of the agreement as measured by 2 deterioratesrapidly. This suggests strongly that our model sensitivelycaptures the entire physics of all excitation spectra mea-sured.

IV. THEORY

As a starting point of a detailed account of our theoret-ical description we emphasize again, that the weaknessof spin-orbit coupling in MnSi leads to well-separated en-ergy and length scales. The underlying physics is basedon the properties of a ferromagnet. At energies below 3 meV, where the spin-flip Stoner continuum sets in33,and for corresponding small momenta, q < 0.3 A

1, the

low-energy excitations of the ferromagnet in the absenceof spin-orbit interactions are described with high preci-sion by a simple quadratic dispersion EFMq = cq2. Atthe small wave vectors relevant for our experiment thedamping of those modes may be neglected.Accordingly, the starting point of our theoretical descrip-tion is a rotationally invariant non-linear model (in-cluding the appropriate Berry-phase term) to describethe low-energy excitations of this ferromagnet. Follow-ing Ref. 42,43, we add to this model the leading spin-orbit coupling effect, which is given by the rotation-ally invariant Dzyaloshinskii-Moriya (DM) interaction

FIG. 3: Illustration of characteristic features of helimagnonbands. In the regime investigated, q kh, a crossover to aferromagnetic dispersion with little weight in side bands wasgenerally expected (see text). Instead, for finite q multi-ple bands with approximately equal weight are excited due tosignificant Umklapp scattering. (a) For wave vectors strictlyparallel to the helix a special symmetry prohibits higher orderUmklapp processes: a translation of the helix can be com-pensated by a rotation of all spins. As a consequence onlythree modes are excited, two modes with minima at kh andan additional mode centered directly at the position of thenuclear Bragg peak with zero momentum but vanishing in-tensity for q 0. For comparison the dispersion of a fer-romagnetic mode = cq2 is given (red solid line). (b) The

strong Umklapp scattering for finite perpendicular momen-tum, q = 4kh, stops the motion of spin excitations withsmall q leading to flat bands (see text and Eq. (6)). In bothpanels the spectral weight of the corresponding modes is pro-portional to the area of the points where we use a maximalsize for better visibility. For the calculation of the dispersionwe used c(20K) = 37 meVA2 and kh = 0.035 A

1. For clarityonly a single domain is shown, namely k1.

SDM = g ( ). Being linear in momentum,

the DM interactions necessarily lead to an instability ofthe ferromagnet and stabilize a chiral helix of the form

h

(x) = n1

cos(kh x) + n

2sin(k

h x) (1)

where the unit vectors n1 and n2 are perpendicular toeach other and to the ordering vector kh. In our MnSicrystals the sign of g is such that n1, n2 and kh definea left-handed coordinate system. Corrections to (1) aresmall as they arise only from tiny forth order terms in theweak spin-orbit coupling neglected in our calculation. Ina ferromagnetic state these corrections lead to the forma-tion of a gap. However, due to the spontanteously brokentranslation invariance and the Goldstone theorem there


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FIG. 4: Dispersion curves for various directions as calculatedin the model described in the text. Contributions from differ-ent domains are color coded as shown in Fig. 1. The area ofeach point is proportional to the weight of the correspondingpeak in neutron scattering (only for the most intense peaksclose to the reciprocal lattice vectors a fixed point size wasused for better visibility). The complex lineshapes observedexperimentally (cf. Fig. 2) result from the large number ofexcited bands. The gray shaded bars show the direction and

parameter range of the constant Q scans recorded in our mea-surements, where the width in Q of the gray area representsthe resolution in Q. Data for the gray bars labeled a throughd are shown in Fig. 2. (a) Dispersion for the trajectory withq k1. (b) Dispersion for the trajectory with q k1.

cannot be a gap for an incommensurate helimagnet. Theonly effect of the higher-order spin-orbit coupling termsis a modification of the spin wave spectrum for momentamuch smaller than kp |kh|, i.e., far below the experi-mental resolution available to date21,22.To describe the spin-excitations of the helix we expand around the mean field, = h+. As discussed be-low the physical situation becomes transparent by usinga comoving coordinate system, where the order param-eter at point x is rotated by the angle kh x aroundthe kh axis using the space-dependent rotation matrix Rwith = R. In the comoving coordinate system thehelix is mapped on a ferromagnetic solution h = n1.Using length and energy units such that |kh| = 1 andEFMkh = Ekh = ck

2h = 1, all free parameters are fixed.

The long wavelength of the helix with the crystallo-graphic lattice (which breaks translation invariance), im-

plies a tiny magnetic Brillouin zone and the formationof an abundance of bands. For the further discussion itis helpful to split the momentum q parallel to kh intoq = nkh + q introducing a band index n while usingkh/2 < q kh/2. Denoting by q both q and q,the Gaussian fluctuations around the mean field are de-scribed by the action

S =1

2

n,n,q,in

nq(n) Mnn(q, in) nq(n) (2)

where each entry of Mnn is a two by two matrix describ-ing the fluctuations in two spin-directions perpendicularto the mean field h. For a ferromagnet one needs onlya single matrix

MFM(q, ) =

q2 i

i q2.

(3)

From the condition detM = 0 one obtains the well-knowndispersion, = q2, of a ferromagnet.The fluctuations around the helimagnetic state are incontrast described by an infinite dimensional matrix,where the DM interaction yields two changes with re-spect to the ferromagnetic state. First, one obtains anadditional 1 in the diagonal entries Mnn(q, ), whichare given by:

Mnn(q, ) =

q2 i

i 1 + q2

(4)

where q2 = q2 + (q + n)2. Second, one obtains non-

vanishing off-diagonal matrix elements given by:

Mn,n1(q, ) = 0 iq

iq 0

(5)with q = qxiqy for a helix oriented in z direction. Thesimple matrix defined by Eqns. (4,5) leads to the physicsexplained below and describes the neutron scattering ex-periments with high precision and no other parametersbesides the (measured) pitch of the helix, the measuredferromagnetic spin wave stiffness and the overall intensityas the only free parameter.

V. DISCUSSION

To compare the calculated spectra with the neu-tron scattering results, one determines the momentum-diagonal 3 3 susceptibility matrix q() qby inverting first the matrix M and undoing the basischange due to the rotation R defined above. The scat-tering cross section for neutrons with a given momentumtransfer Q = G + q close to a reciprocal lattice vec-tor G of the MnSi crystalline lattice is proportional to

(1 + nB())Im

Tr(q( + i0)) Qq( + i0)Q

, where

nB() = 1/(e/kBT 1) is the Bose function and where


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we have taken into account, that the neutrons couple onlyto the component of perpendicular to Q. Within theGaussian approximation one obtains sharp modes. For acomparison with experiment these theoretical results areconvoluted by the experimental resolution.We first consider the excitations with vanishing momen-tum perpendicular to the propagation vector, q = 0,predicted in this model. In this limit all off-diagonal

matrices vanish and one obtains a single mode with thedispersion relation = cq

k2h + q

2, well known from

the physics of Bose condensed atoms. It describes thecrossover from a mode with linear dispersion21,32 at smallmomentum to a ferromagnetic mode with quadratic dis-persion. This crossover is also manifest in the eigen-modes as illustrated in Fig. 1. In the ferromagnetic limitkc q kh the magnetization displays the typical pre-cession (Fig. 1b), while it oscillates only perpendicular toq for q kh (Fig. 1c).By transforming back to the physical coordinate system,one realizes that generically three copies of this mode canbe observed in neutron scattering with unpolarized neu-

trons as shown in Fig. 3(a). Besides the two copies withminima at kh one also obtains generically a mode di-rectly at the Bragg spot with zero momentum. However,in contrast to the dispersion of a ferromagnetic mode ofidentical spin stiffness c shown in red, in the helimagneticcase the intensity vanishes for q 0. The fact that onlythree modes rather than a large number of bands canbe observed, originates in the special symmetry of thehelical state: a translation of the helix can be compen-sated by an appropriate rotation of the spins. Theoreti-cally this property has actually long been known44. Onlyhigher-order spin-orbit coupling terms which break thissymmetry may lead to the excitation of further modes,

however, with a tiny weight.The more important second example, which reveals anentirely unexpected property, concerns spectra for fi-nite perpendicular momentum as shown in Fig. 3(b).Here the off-diagonal terms (5) lead to a mixing of themodes. For vanishing q and q kh one obtains21,32E(q) =

3/8cq2 which can, however, not be observed

directly within our experimental resolution. In the limitq kh one finds that more and more modes are ex-cited. Mapping the matrix to a harmonic oscillator on alattice (see e.g. Ref. 45) one finds that typically of the

order of

q/kh bands are excited with equally spacedenergies

En(q) c

q2 2khq + 2

qk3h(n +12

)

,

n = 0, 1, 2,...

q/kh. (6)

Here two aspects are remarkable. First, for increasingperpendicular momentum an increasing number of bandsare excited. Second, the dispersion is essentially indepen-dent of q for q

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M. Janoschek et al- Helimagnon Bands as Universal Excitations of Chiral Magnets