Dynamics of magnon fluid in Dzyaloshinskii-Moriya magnet and its manifestation in magnon-Skyrmion scattering Yun-Tak Oh,1 Hyunyong Lee,1 Jin-Hong Park,2 and Jung Hoon Han1,∗ 1Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea 2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan (Dated: January 30, 2015) We construct Holstein-Primakoff Hamiltonian for magnons in arbitrary slowly varying spin 5 background, for a microscopic spin Hamiltonian consisting of ferromagnetic spin exchange, 1 0 Dzyaloshinskii-Moriya exchange, and the Zeeman term. The Gross-Pitaevskii-type equation for 2 magnon dynamics contains several background gauge fields pertaining to local spin chirality, in- homogeneous potential, and anomalous scattering that violates the boson number conservation. n Non-trivial corrections to previous formulas derived in the literature are given. Subsequent map- a ping to hydrodynamic fields yields the continuity equation and the Euler equation of the magnon J fluid dynamics. Magnon wave scattering off a localized Skyrmion is examined numerically based 6 onourGross-Pitaevskiiformulation. Dependenceoftheeffectivefluxexperiencedbytheimpinging 2 magnon on the Skyrmion radius is pointed out, and compared with analysis of the same problem using theLandau-Lifshitz-Gilbert equation. ] l l PACSnumbers: 75.78.n,75.30.Ds,12.39.Dc,75.78.Cd a h - s I. INTRODUCTION number. In these approaches, however, the relevant de- e scriptioniswithinthemomentumspacepictureassuming m Magnons are quantized waves of spin fluctuations translationallyinvariantspinbackgroundoversomelarge . t aroundanorderedstateofspinsinamagnet. Thetheory unitcell. Wetryinthisarticletopresentathorough,self- a m oftheirdynamicswasformulatedinthe1940sbyHolstein containedformulationofmagnondynamicsinreal space, andPrimakoff[1]andforthisreasontheyarealsoknown suited in particular for adressing the scattering process d- as Holstein-Primakoff(HP) bosons. Magnon dispersion off a localized object such as a Skyrmion. The work is n intheferromagneticortheanti-ferromagneticspinback- thus complementary to both strands of existing theories o ground can be worked out easily from the HP theory as ofspindynamicsbasedonLLGequation[7,8],andthose c explained in textbooks on quantum magnetism[2]. formulated in momentum space[9–13]. [ Modern experimental and theoretical advances call In section II we present a general HP Hamilto- 1 for refinements of the HP theory. On the experimen- nian assuming arbitrary smooth spin background of v tal front abundant sightings of the Skyrmionic spin tex- the ferromagnetic exchange Hamiltonian supplemented 7 ture in chiral magnets demand a well-defined theory of by Dzyaloshinskii-Moriya (DM) exchange and Zeeman 6 spin waves in the textured spin background[3]. Exist- terms. While this formulation was attempted in some 4 6 ing attempts[4–8] for theories of spin dynamics in the earlier works [14, 15] and applied in various context of 0 Skyrmion background are done in terms of the Landau- spintronics,notallofthetermswederiveherehavebeen . Lifshitz-Gilbert (LLG) equation for the magnetization coherentlypresentedthusfar. Inparticularthetermsdi- 1 unit vector n(r,t). Despite its obviousstrength in direct rectlyfollowingfromtheDMspininteractionhavenever 0 5 simulation of the magnetization dynamics in such com- beenconsistentlyderivedtoourknowledge. Theninsec. 1 plex spin background, LLG approach is not well suited III we show how the magnon equation of motion pre- : foranintuitive understanding ofthe low-energyspindy- sentedintheprevioussectioncanbeequivalentlyviewed v namics basedon the particle picture. Moreover,magnon as a hydrodynamics problem, with a set of hydrody- i X scatteringoffthelocalizedSkyrmionisasubjectofgrow- namicequationspertainingtomagnondensityandveloc- r ing importance and interest as addressed by several re- ityflows. NumericalsimulationsbasedonHPtheoryare a cent works[7, 8]. Here again, a formulation of the prob- infactequivalenttosolvingthehydrodynamicsversionof lem in terms of the well-established HP boson theory is the magnon problem. We present such numerical result surprisingly lacking. insec. IV,intheparticularcaseofmagnonscatteringoff As amply demonstrated in several recent theories of a single, localized Skyrmion. Our results partially over- magnon Hall effect[9–13], HP theory conveniently cap- lap with an earlier simulation of the same problem [8]. tures magnondynamics ina non-trivialspinbackground Herewefindasurprisingdependenceofthemagnonscat- inamannercompletelyanalogoustothatofelectrondy- tering angle (and even its sign!) on the Skyrmion radius namics in a non-trivialflux backgroundand band Chern normalized by the spiral wavelength. This unexpected feature of the magnon-Skyrmion scattering follows from takingthoroughconsiderationoftheDMtermintheHP formulation. We carry out a numerical check of the pre- ∗ Electronicaddress: [email protected] diction using the more conventional LLG approach. It 2 turns out that LLG approach embodies a more complex where eˆ is the unit vector in each µ-direction. Hence- µ pictureofthescatteringdynamicsduetothefactthatthe forth we choose the unit of energy J = 1 and re-scale Skyrmiontendstoresponddynamicallytotheimpinging the field strength B accordingly. Restriction to d = 2 magnonwaveinthe LLGtheory,whereasthe HPtheory is easily achieved by deleting all terms from Eq. (2.4) in effect treats it as a static object of infinite mass and pertaining to the spatial direction z. internalexcitationenergies. Despitethesedifferences,we Letusdenotethenewgroundstateorsomemetastable were able to deduce features of the subtle dependence of spin configuration of the Hamiltonian H by n . Given 0 the magnon scattering on the Skyrmion radius that was such unexpected in previous simulations [7, 8]. We conclude with a summary and possible future applications of our formulation in sec. V. n0 =(sinθcosφ, sinθsinφ, cosθ) onecanconstructanorthogonalmatrixRwhoseelements are II. HAMILTONIAN FORMULATION OF MAGNON DYNAMICS R =2m m δ , αβ α β αβ − θ θ θ m= sin cosφ, sin sinφ, cos . (2.5) Thelow-energyspindynamicsofaferromagnetiscap- 2 2 2 tured by the nonlinear sigma model (NLσM) (cid:16) (cid:17) With this R one can show Rzˆ= n and Rn = zˆ. The 0 0 strategy is to introduce HP bosons in the rotated frame of reference, as obtained by replacing n in Eq.(2.4) by d J Rn. H = (∂ n) (∂ n) (2.1) 0 2 µ · µ AftersomeextensivealgebraonefindstheHamiltonian µ=1 X in the new spin basis with the spatial index µ running over 1 through d in d-dimensional space. When a smooth perturbation is 1 d H = (∂ n a n)2 SBn n (2.6) added to the NLσM the new ground state will gener- µ µ 0 2 − × − · allybecomeaslowlyvaryingspintexture. Inlightofthe µX=1 recent surge of interests in textured ground states of the where, component-wise, the vector potentials are chiral magnet[3], we include the DM interaction, a1 = n0(∂ φ) sinφ(∂ θ)+κf (θ,φ), µ − x µ − µ µ HDM =Dn·∇×n, (2.2) a2µ =cosφ(∂µθ)−n0y(∂µφ)+κgµ(θ,φ), a3 =(1 cosθ)(∂ φ)+κn0. (2.7) as well as the Zeeman term µ − µ µ The two functions f (θ,φ) and g (θ,φ) tied to the DM µ µ interaction are H = SB n (2.3) Z − · cos2 θ +cos2φsin2 θ (µ=x) (S = size ofspin) as the perturbation. Space integration fµ(θ,φ)= − si2n2φsin2 θ2 2 (µ=y) , isimplicitlyassumedintheaboveexpressions. Itisuseful ( n0 (µ=z) x for further technical development to point out that the sin2φsin2 θ (µ=x) combined spin Hamiltonian H = H0 +HDM +HZ can g (θ,φ)= cos2 θ cos2φ2sin2 θ (µ=y) .(2.8) be organized in the compact form, up to an irrelevant µ ( − 2 −n0 2 (µ=z) constant and taking κ=D/J, y All the κ-dependent terms in the emergent gauge fields a were missed in the literature. µ d Quadratic magnon Hamiltonian follows from the HP H 1 S = (∂ n κeˆ n)2 B n, (2.4) substitution in the rotated spin Hamiltonian (2.6): µ µ J 2 − × − J · µ=1 X 3 d d H 1 1 = (∂ +ia3)b† (∂ ia3)b + (∂ ia3)b (∂ +ia3)b† S 2 µ µ r µ− µ r 2 µ− µ r µ µ r µ=1 µ=1 X(cid:2) (cid:3)(cid:2) (cid:3) X(cid:2) (cid:3)(cid:2) (cid:3) d d d 1 1 1 + B n [(a1)2+(a2)2] b†b (a1 ia2)2b b (a1+ia2)2b†b†. (2.9) · 0− 2 µ µ ! r r− 4 µ− µ r r− 4 µ µ r r µ=1 µ=1 µ=1 X X X ThesizeofspinS thatservesasanoverallconstantinthe quadratic theory can be set to one. Heisenberg equation of motion for the boson operator is d d d ∂b 1 1 i~ r = (∂ ia3)2b + B n [(a1)2+(a2)2] b (a1 +ia2)2 b†. (2.10) ∂t − µ− µ r · 0− 2 µ µ ! r− 2 µ µ ! r µ=1 µ=1 µ=1 X X X Thisequationofmotionforthemagnonoperatorb ,com- The meaning of the anomalous term becomes clear r binedwithEqs.(2.7)and(2.8)forthebackgroundgauge with the exemplary spiral spin n = (cosκz,sinκz,0) 0 fields, gives the governing dynamics of magnons moving - a right-handed spiral with the propagation vector k = inthe arbitrarysmoothspinbackgroundn . Relationof (0,0,κ). We find that Eq. (2.12)reducesto κ2e2iκz, and 0 the gauge field a3 to the background spin n is the whole magnon Hamiltonian written in momentum 0 space becomes 1 (∇ a3) = ε n (∂ n ∂ n )+κ(∇ n ) × α 2 αµν 0· µ 0× ν 0 × 0 α 1 b† T bq+k +(1−cosθ)(∇×∇φ)α. (2.11) H = 4 q (cid:18)b−qq+−k3k(cid:19) Hq(cid:18)b†−q−3k(cid:19), X The prefactor in the first term on the r.h.s. shows that 2q2+κ2 κ2 = . (2.13) one Skyrmion acts as two units of flux quanta for the Hq κ2 2q2+κ2 bosons. Interestingly,the spinvorticity∇ n alsocon- (cid:18) (cid:19) 0 tributes to ∇ a3 due to the DM interac×tion. We will One recovers the well-known magnon dispersion of the × see later that this extra contribution has an interesting spiral spin, ω = q q2+κ2. It is through the anoma- q consequence for the magnon-Skyrmion scattering prob- lous potential tha|t|the correct magnon dispersion is re- lem. The final expression on the r.h.s. might be ignored covered,whileitsompissionmighthaveledtoanincorrect on the ground that ∇ ∇φ = 0, but as we will soon dispersion ω =q2+κ2/2. × q see this is not the case with the Skyrmion or the anti- Finally, there is an inhomogeneous on-site potential Skyrmion configuration we have in mind. The anomalous potential, which violates the boson number conservation, has the connection to the back- d d ground spin [(a1µ)2+(a2µ)2]= (∂µn0−κeˆµ×n0)2 (2.14) µ=1 µ=1 X X d arising from local deviation of the spin structure from (a1+ia2)2 that of a simple spiral. − µ µ µ=1 The formulation presented in this section is com- X d pletely general, and applies to arbitrary ground state or 2 =e2iφ (θˆ+iφˆ) (∂ n κeˆ n ) . (2.12) metastable spin configurations allowed in the Hamilto- µ 0 µ 0 · − × µX=1h i nian (2.4). Other unit vector fields in the above formula are III. HYDRODYNAMIC FORMULATION OF MAGNON DYNAMICS θˆ=(cosθcosφ,cosθsinφ, sinθ), − φˆ=( sinφ,cosφ,0), Magnon dynamics is typically viewed as the solution − of Eq. (2.10) with some characteristic frequency cor- both locally orthogonalto n . responding to the energy of magnon excitation. In an 0 4 inhomogeneous environment of the spin n , however, it the real-space magnon dynamics, presumably based on 0 will be useful to have a complementary picture in real the observation that they appear in the magnon Hamil- space,quite like that providedby the hydrodynamicfor- tonian with one more spatialderivative than a3. On the µ mulationof superfluid dynamics [16], which follows from other hand, we already saw that their omission leads to writingdownthecorrespondingmagnonLagrangianand an erroneousresulteven in the well-knowntest case. In- substituting stead the relevance of the anomalous term has to be ad- dressed carefully, for each specific background spin con- figuration. b=√ρeiη. (3.1) In the next section we take up in particular the prob- lem of magnonscattering off a single localized Skyrmion Variation of the Lagrangian with respect to the density based on our HP formulation. This problem is of great ρ and the phase η yields hydrodynamic equations. An- topical interest and has already received some attention other way, paralleling the derivation of Gross-Pitaevskii in the literature [7, 8]. The direct numerical approach equation in superfluids[16], is to replace the boson op- based on LLG equation in these papers however makes erator b by its average b = √ρeiη in the Heisenberg it difficult to interpret the scattering process in terms of h i equationofmotion(2.10). Samehydrodynamicrelations magnonparticles. Solvingthescatteringproblemanalyt- are obtained from both approaches. ically together with the anomalous term is impossible8. The boson number is no longer conserved in the con- Our method of choice is the numerical integrationof the tinuity equation due to the anomalous potential, magnon dynamics, as given in Eq.(2.10). ∂ ρ+∇ (ρv)=ρFi, (3.2) IV. SIMULATION OF MAGNON SCATTERING ∂t · where Fi follows from A. Simulation based on HP theory FornumericalsimulationbasedonEq. (2.10)we work d e−2iη (a1+ia2)2 F +iF . (3.3) in two dimensions, external field B = Bzˆ, B > 0, and − µ µ !≡ r i the DM interaction constant κ > 0 that favors a right- µ=1 X handed spiral. Under these parameter values the stable TheotherpieceofhydrodynamicsisprovidedbytheEu- Skyrmion configuration is the one whose spins point up ler equation of the flow vector v = 2(∇η a3) given farfromthecore,executearight-handedspiralturnupon − by approachtotheorigin,whereitpointsdownward, zˆ. A − model spin configuration fulfilling all these requirements is (r2 =x2+y2) D v=v ∇p. (3.4) t ×B− The material derivative D ∂ +v ∇ is familiar from 2Ry 2Rx r2 R2 hydrodynamics. The emertg≡enttmagn·etic field B respon- n0 = − r2+R2, r2+R2, r2−+R2!. (4.1) sible for the Lorentz force on the magnon is defined by The Skyrmion number of this configuration is N = 1, − qualifying it as an anti-Skyrmion. The HP Hamiltonian 1 ∇ v=∇ a3 ∇ ∇η. (3.5) in the presence of a single anti-Skyrmion is B ≡−2 × × − × Readers can refer to Eq. (2.11)for the definition of ∇ 2 a3. We will not immediately identify with the cu×rl H = [(∂ +ia3)b†][(∂ ia3)b ] G(r)b†b ∇ a3 because in some situations the pBhase singularity µ=1 µ µ r µ− µ r − r r in t×he boson field can lead to ∇ ∇η =0. Allowing for X1 1 × 6 + e−2iφF(r)b b + e2iφF(r)b†b†, (4.2) the time-dependent background spin (which we do not 2 r r 2 r r do in this paper) will induce the emergent electric field where onthe r.h.s. ofthe Eulerequationaswell. Thequantum pressure p is given by 2κR(κR 2)r2 F(r)= − , (R2+r2)2 ∇2√ρ d p= +2Bnz+F (a1)2+(a2)2 .(3.6) R2(κR 2)2+κ2r4 R2 r2 − √ρ 0 r− µ µ G(r)= − +B − . (4.3) µ=1 (R2+r2)2 R2+r2 X(cid:2) (cid:3) Thereisatendencyintheexistingliterature[8,14,15, The a3 and its curl is tricky as it contains a singular 17,18]toignoreeffectsarisingfroma1 anda2 inviewing contribution, µ µ 5 (a) (b) (c) ρ/ρ0 2yR2 2xR2 a3 = − , 3 r2(r2+R2) r2(r2+R2) (cid:18) (cid:19) 2Ry 2Rx +κ − , , r2+R2 r2+R2 1 (cid:18) (cid:19) 4R2(κR 1) ∇ a3 = − +4πδ(r). (4.4) × z (r2+R2)2 (cid:0) (cid:1) FIG. 1. (color online) Snapshots of scattered magnon waves The delta-function flux can be removedthrough the sin- for κ˜ = κR equal to (a) 0.5, (b) 1.0, and (c) 1.5. Shown are gularphasetransformationofthebosonfieldbr →bre2iφ color plots of the magnon density ρ=|b|2 normalized by the whereφistheazimuthalangleintheplane. Thenewbo- incoming magnon density ρ0. Skyrmion isindicated byared son field obeys the Hamiltonian circle of radius R. The magnon trajectory changes direction at κ˜ = 1 as anticipated from the total flux consideration in theHP theory. 2 H = [(∂ +ia3)b†][(∂ ia3)b ] G(r)b†b µ µ r µ− µ r − r r µ=1 X 1 1 +2e2iφF(r)brbr+ 2e−2iφF(r)b†rb†r, (4.5) i∂b˜r = (∂˜ ia˜3)2+G(r˜) b +e−2iφF(r˜)b∗, ∂t˜ −" µ− µ # ˜r ˜r where F and G remain the same as before, but a3 and (4.8) its curl becomes where the rescaled time t˜ = t/~ is also dimensionless (Recall that we already set J =1 earlier). The tilde can 2y(κR 1) 2x(κR 1) a3 = − , − , be removed now without confusion. − r2+R2 r2+R2 (cid:18) (cid:19) Hydrodynamic relations following from Eq. (4.8) are 4R2(κR 1) ∇ a3 = − . (4.6) × z (r2+R2)2 ∂ ρ+∇ (ρv)= 2ρFsin[2(η+φ)], (4.9) (cid:0) (cid:1) t · − In the numerical treatment we will work in this and singularity-free gauge. It is customaryto regardthe two-dimensionalintegral of (∇×a3)z divided by 2π as the effective flux experi- Dtv=v×B−∇p, enced by the magnonquasiparticle. In suchpicture each ∇2√ρ Skyrmion serves as p=2 G+Fcos[2(η+φ)] . (4.10) − √ρ − (cid:18) (cid:19) We solved Eq. (4.8) using the Runge-Kutta method. Φ=2(κR 1) Forced spin wave is generatedat the bottom of the rect- − angular simulation grid with the space-time dependence units of flux quanta, spread over the distance of the ra- dius R. In contrast to electron motion which sees the Skyrmion as a quantized flux object carrying one flux b(x,y <y0,t)=b0eiky−iωkt. (4.11) quantum[3],magnonsseeitcarryingavariablefluxthat The frequency ω =k2+B matches the dispersion in free can even change sign with the Skyrmion radius! k space far from the Skyrmion, and y is a suitable cutoff Let us rescale all the fields into dimensionless form 0 much less than the total length of the grid. We then convenient for numerical analysis, closely monitor the scattering processes as the magnons collide with the (static) Skyrmion. 2κ˜(κ˜ 2)r˜2 By far the most interesting findings of our investiga- F(r˜)= − , (1+r˜2)2 tion was the variation of the scattering angle as κ˜ =κR crosses the threshold value 1, whereupon the scattering (κ˜ 2)2+κ˜2r˜4 1 r˜2 G(r˜)= − +B˜ − , directionchangessignasshowninFig. 1. Thisisastark (1+r˜2)2 1+r˜2 prediction of our HP theory, in departure from previous 2y˜(κ˜ 1) 2x˜(κ˜ 1) considerations based either on LLG theory [7, 8] or on a˜3 = − , − , (4.7) − 1+r˜2 1+r˜2 the magnon theory without taking into careful account (cid:18) (cid:19) of the DM term [14, 15]. In the following subsection we where x˜ = x/R, y˜ = y/R, r˜= r/R, and κ˜ = κR and return to the conventional LLG calculation to verify to B˜ = BR2 are the two dimensionless parameters. The whatextent, ifatall, the predictions ofthe HP theoryis equation of motion in dimensionless form becomes born out. 6 B. Simulation based on LLG theory to obtain δn =n n . The resulting profile has only i i i −h i planar components δn = (δn ,δn ) whose magnitude i x y Lattice version of the LLG equation, and phase are plotted in Fig. 2. A small amount of displacement of the Skyrmion [8] during the cycle does not affect the average n significantly. i n˙i =ni×Beiff −αni× ni×Beiff , At the smaller radiuhs κiR = 0.9, overall scattering be- havior are analogous to that of magnons for κR < 1. Beiff = (nj +aκnj ×eˆ(cid:0)j)+a2B,(cid:1) (4.12) At κR = 1.5, we see a second scattering channel begin- j∈i X ning to open up shown as a white, vertical jet in Fig. is solved on a large grid of lattice spacing a, with the 2(b). At κR = 2.0, this second channel is pointing at initialspinconfigurationgivenbyEq. (4.1). Itwasfound an angle opposite to the first one (Fig. 2(c)), while the useful to first run the simulation with a large Gilbert phase profile (Fig. 2(f)) becomes nearly isotropic with constantα oforderone to relaxthe Skyrmionto its true respecttotheincomingspinwavedirection. Anattempt equilibriumconfiguration,whichdiffersindetailfromthe to carry out LLG calculation at even larger radius R, model form (4.1), at a given B and κ. Once an optimal in the hope of finding the reversed scattering direction, Skyrmion configuration is reached in this way, we turn failed because stabilizing such a large Skyrmion requires on the spin waveat the bottom of the boundary to start B forwhichSkyrmionis nolongerthe moststable phase the scattering process. (instead spiral spin state wins out). To further compli- Several different Skyrmion radii R were considered cate the analysis,low-energyexcitation modes are easily in the simulation while simultaneously maintaining the createdforlargerSkyrmionswhichsubsequentlyinteract sameratiotothemagnonwavelength,kR. Adetailedde- with the incoming spin wave. pendenceofthespinwavescatteringangleonthisdimen- In short, our LLG simulation does give a hint that sionlessquantitykRwasreportedearlier[8],whereasour spin wave scattering angle has a significant dependence focushereiswithanotheranotherdimensionlessnumber, on the reduced Skyrmion radius κR, in accordance with κR. Variationofthisnumbercanbeachievedinpractice anticipations of the HP theory. To observe the proposed withthemagneticfieldB,whichwouldchooseadifferent effect one can tune the magnetic field over a range in optimal Skyrmion radius R. whichisolatedSkyrmionsarethemostfavoredstates[3]. Field-dependentvariationsofthemagnonHalleffectand (a) (b) (c) possible reversal of its sign will be a telltale sign of our effect. ρ 0.02 V. SUMMARY AND DISCUSSION 0 Much of this paper dealt with the formal theory of magnon dynamics from Hamiltonian and hydrodynam- (d) (e) (f) ics perspectives. Despite the long history of the theory φ of magnon dynamics dating back to 1940s,its real-space 2π formulation with regard to the underlying spin Hamil- tonian containing the Dzyaloshinskii-Moriya interaction has never been rigorously derived in the form presented in Sec. II. 0 ThespinHamiltonian(2.4)usedinthispaperisknown to host interesting topological phases of spins called the Skyrmion lattice in two dimensions[19, 20] and the FIG. 2. (color online) Snapshots of scattered spin waves in hedgehog lattice in three[21, 22]. The dynamical equa- the LLG calculation for κR equal to (a,d) 0.9, (b,e) 1.5, and tionderivedinthispapercanbeusedforlarge-scalesim- (c,f) 2.0. In the top row are plots of the spin wave den- sity ρ = p(δnx)2+(δny)2. See the main text for the def- ulation of magnon dynamics in these topologicalphases. inition of δn = (δnx,δny). Bottom row shows the phase ThehydrodynamicformulationdevelopedinSec. IIIcan φ=arctan(δny/δnx). Skyrmion positions and their sizes are subsequently provide intuitive interpretation of magnon indicated by red circles. A clear sign of skew scattering at dynamics simulation as movements of magnon fluid. κR=0.9 is completely gone at κR=2.0. In this paper we studied one such problem - magnon scattering off a localized Skyrmion - numerically in de- Figure 2 shows patterns of scattered spin waves for tail. Magnons do see a localized Skyrmion as a source varying Skyrmion radii R. The background spin texture of flux quanta responsible for skew scattering, in accor- has been removed by taking the time average n over dance with severalearlierobservations [7, 8, 15], but the i h i a single period of the incoming spin wave, which is then net flux seen by the magnondeviates from the two units subtracted from the snapshot at a particular instant n of flux quanta anticipated in earlier theories due to an i 7 important correctionfrom the DM interaction, as in Eq. the electrons. (2.11). Our analysis suggests that the net flux seen by theincidentmagnonmayevenchangeitssignatthecrit- ical Skyrmion radius κR = 1. Indeed magnon dynam- c icssimulationbasedonourHPHamiltoniandidrevealed ACKNOWLEDGMENTS sucheffect. Unfortunately,theLLGsimulationpresented a more complex picture due to various low-energy exci- tations of the Skyrmion induced by the very magnons This work is supported by the NRF grant(No. which scatter off of it. Nevertheless we were able to 2013R1A2A1A01006430). YTO was supported by show that a Skyrmion of increasingly larger radius has Global PhD Fellowship Program through the National less pronouncedskewscatteringofmagnonsinroughac- ResearchFoundationofKorea(NRF)fundedbytheMin- cordancewiththepredictionoftheHPtheory. Weantic- istry of Education (2014H1A2A1018320). 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