SSpin waves scattering on a Bloch point
R. G. Elías , V. L. Carvalho-Santos , , A. S. Núñez , A. D. Verga (1) Departamento de Física, Universidad de Santiago de Chile, Avda. Ecuador 3493, Santiago, Chile.(2) Instituto Federal de Educação, Ciência e Tecnologia Baiano - Campus Senhor do Bonfim,Km 04 Estrada da Igara, 48970-000 Senhor do Bonfim, Bahia, Brazil.(3) Departamento de Física, Facultad de Ciencias Físicas y Matemáticas,Universidad de Chile, Casilla 487-3, Santiago, Chile and(4) Université d’Aix-Marseille, IM2NP-CNRS, Campus St. Jérôme, Case 142, 13397 Marseille, France (Dated: October 16, 2018)We show that, after a transformation, the dynamics of linear perturbations (spin waves) around asingular Bloch point soliton is formally equivalent to a quantum system of an electron in a magneticmonopole field. The analytical solution to this problem is known and allows us to find the spectrumand the scattering of a wave in a Bloch point field. I. INTRODUCTION
Bloch points (BPs) are topological solitons found inthree-dimensional magnets. They have been observed orinferred in different contexts, such as in the transition re-gion between Bloch lines embedded in Bloch walls and innumerical simulations of quasi-two-dimensional systemsduring the process of reversal of vortex cores. Morerecently, BPs have been identified as required sourcesand sinks for the unwinding of a skyrmion lattice. Re-markably, there is also a recent experimental work inwhich a static BP was observed in cylindrical magneticnanowires. The defining property of BPs is that in aclosed surface around its center the direction of the mag-netization field covers the whole solid angle an integernumber of times. When the norm of the magnetizationfield is preserved this property turns into a topologicalprotection (in which case the BP center is a singular pointwhere ferromagnetic order is destroyed). Actually, BPscan be seen as the equivalent of two-dimensional Belavin-Polyakov solitons (skyrmions) if we fold the physicalplane over the surface of a sphere by means of the stere-ographic projection. In this sense, the simplest BPs aresolitons with unitary topological charge and it is for thisreason that they are implicated in topological transitionswhere topological charge always changes by steps of ± .Even in strictly two-dimensional system we can observethe appearance of a BP-like configuration given by thesuperposition of two magnetic vortices (a vortex and ananti-vortex) with the same topological charge (Pontrya-gin invariant) but opposite vorticity. The control andmanipulation of topological solitons (principally vorticesand skyrmions) by means of electrical currents in thehope to find new alternatives for the information stor-age has relaunched in recent years the investigation onBPs. Another potential utilization of BPs is in the field ofmagnonics that pretends to manipulate magnetic solitonsby means of the spin waves generated in the material, preventing in this way the Joule effect produced by cur-rents. In any case, the knowledge of spin waves behavioris of paramount importance to understand and to controlBPs dynamics.Considering these facts it is worth to know the dy- namical and stability properties of BPs in a ferromag-netic materials. For this purpose, in this paper we studythe spin waves (SWs) around a singular BP describedby the exchange energy that is the most important termaround the singularity. Exchange interaction is respon-sible of ferromagnetic order and is the most divergentterm around the center of the BP, giving the topologicalstructure to BPs. In this paper we will concentrate in ex-change energy that is a geometry-independent term , andso it can give us the universal results that can be consid-ered as a first order contribution to spin wave dynamics.By performing a transformation of magnetization fieldinto the complex plane we are able to calculate the spec-trum of oscillations around the BP that turns to be thesame as those of a quantum system of an electrical chargein a monopolar magnetic field. The mathematical anal-ogy between Dirac monopole and spin waves dynamicsaround the BP allows us to calculate with ease the scat-tering of a SW in a BP field, opening a new possibility toBP detection and localization, by means of the observa-tion of the interference pattern and the intensity profileof scattered spin waves. It is interesting to note that insome particular cases the same method give as result aSchrödinger-like equation for SWs; for example for singleskyrmions, vortex domain walls and one-dimensionaldomain wall. It is worth to note, however, that there aresituations in which the equation for SWs are more com-plicated, giving sets of two coupled Schrödinger equation,as in the case of magnetic vortices. The paper is organized as follows: in the second sec-tion we present the physical system and the equations ofmotion, with the BP as a solution of them. In the thirdsection we perform a change of variables into the complexplane and we perturb the equations of motion around theBP solution showing that the resulting linear equation isa Schrödinger-like equation for the interaction betweenan electron and a magnetic monopole. In the forth sec-tion we show the analytical solution for the oscillationsand the functional form of the SWs. In the fifth sectionwe study the scattering of a plane wave produced by theBP using the results of the previous sections and classicalresults on magnetic monopoles. a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov q = 1 , γ = 0 q = 1 , γ = π/ q = 2 , γ = 0 FIG. 1. Three kind of BPs depending on the vorticity q andthe phase in the azimuthal coordinate γ . II. MODEL AND BLOCH POINT SOLUTION
We consider a system composed of dimensionless clas-sical spins parameterized by field coordinates ( S, Θ , Φ) as S = S (cos Φ sin Θ , sin Φ sin Θ , cos Θ) where (cid:126) S is themolecular spin proportional to the magnetization. Theenergy of the spin configuration is dominated, in thevicinity of the BP, by the exchange contribution E = (cid:82) (d V / a ) J ( ∇ S ) , with J the exchange energy constantand a the lattice parameter. In this frame, there is anatural cut-off for the wave-vectors norm correspond-ing to a minimum wavelength of the order of the ex-change length (cid:96) e , defined by (cid:96) e = (cid:112) J/ M s a , with M s the saturation magnetization. This exchange length turnto be of around six nanometers in Permalloy. The ki-netic term for such a system is the so called Berry term S B = (cid:82) (d V /a ) (cid:126) S cos Θ ˙Φ . The magnetic texture pre-dicted by this action acquires the form of a twisted BPand can be simply written as Θ = pθ and Φ = qφ + γ where θ and φ are the usual spherical coordinates in space r = ( r, θ, φ ) . Since the magnetization field is single val-ued we need p and q to be integers (see Fig. 1). The freeparameter γ represents an azimuthal tilt with respectto the radial direction, making the BP twist around z.Its optimal value depends on additional terms in the en-ergy, particularly the dipolar energy, giving as resulta twisted BP with a definite angle γ . Works found inthe literature showed that if we allow the variationof the magnetization norm it is possible to estimate thesize of the singular region for prototype magnet (for ex-ample permalloy); this singularity region is found to beof a few nanometers. The behavior of the magnetizationin the vicinity of the BP is, nevertheless, fill with sub-tleties. Recent simulations reveal that to give a properassessment of the the singular behavior it is necessary toresolve the magnetic degrees of freedom down to atomicresolution. This is consistent with some micromagneticsimulations that have shown that it is possible to stabilizea BP in a spherical domain of a few nanometers.
Inthis work we focus on the topological BP with topologicalcharge Q = pq (also called the Pontryagin invariant thatit is nothing but than the number of times that magne-tization on a closed surface around the BP center coversthe whole solid angle). In this article we will concen-trate us in the case p = 1 so the vorticity q is the same FIG. 2. The magnetic field B = q ˆ r r − πq ˆ r δ ( x ) δ ( y ) generatedby the vector potential A ( r ) = − q cot θr ˆ φ . The black arrowsare the singularity line where potential is infinity and field hasopposite direction. In this case the singularity line is alongthe whole z axis (this is called symmetric potential). as the topological charge Q . BPs (as vortices) cannotbe considered localized solutions because the spin field isnot homogeneous at infinite, and so, the BP energy E B calculated for a spherical domain is proportional to theradius of the sphere R as E B = 8 JS Q ( R/a ) . III. SPIN WAVES EXCITATIONS IN THEVICINITY OF A BLOCH POINT
In order to calculate the SW excitations around theBP, we start from a given stationary solution parame-terized by the spherical coordinates field Θ and Φ andconsider a small distortion of the magnetization texturecharacterized by a local change δ Θ and δ Φ . This distor-tion is readily associated with a change in the magne-tization vector equal to δ S = δ Θ ˆΘ + sin Θ δ Φ ˆΦ . Asexpected the distortion lies in the tangent plane to themagnetization sphere, within this plane we follow anduse complex notation: Ψ = δ Θ + iδ Φ sin Θ . The squareof variations of the spin vector around a particular con-figuration are therefore related with the norm of Ψ by δ S = S | Ψ | , allowing us to interpret the "density ofprobability" | Ψ | as the density of SWs. Expanding themagnetic action up until the second order in the pertur-bation Ψ we obtain: S (2) = 12 (cid:90) Ψ (cid:16) i (cid:126) ∂ t − ˆ H B (cid:17) Ψ dtdV /a , (1)with ˆ H B = − (cid:126) m ∗ (cid:18) ∇ + 2 iq cot θr ˆ φ · ∇ − q cos 2 θr sin θ (cid:19) (2)where we have defined the equivalent mass m ∗ = (cid:126) / JSa .The equations of motion for the spin waves are ob-tained from the Euler-Lagrange equations of the lin-earized action, δ S δ Ψ = 0 , giving i (cid:126) ∂ t Ψ = ˆ H B Ψ . (3) Y (1)1 − ( θ, φ ) Y (1)10 ( θ, φ ) Y (1)11 ( θ, φ ) Y (1)2 − ( θ, φ ) Y (1)2 − ( θ, φ ) Y (1)20 ( θ, φ ) Y (1)21 ( θ, φ ) Y (1)22 ( θ, φ ) FIG. 3. Angular dependence of the eigenfunctions Eq. (9) for q=1, for different (l,m) modes; polar plot of the absolute value |Y ( q ) lm ( θ, φ ) | with, in color, their argument. The effective Hamiltonian for the linear oscillationsaround the soliton can be written as ˆ H B = (cid:126) m ∗ [ − i ∇ + A ( r )] + V ( r ) , A ( r ) = q cot θr ˆ φ, and V ( r ) = − (cid:126) m ∗ q r . The effective spin wave Hamiltonian is formally equiva-lent to that one describing a quantum mechanical chargedparticle moving under the influence of the magnetic fieldcreated by a magnetic monopole located at the singu-larity, and in a scalar attractive isotropic potential V .Away from the BP we have B = ∇ × A = − q ˆ r /r +4 πq ˆ r δ ( x ) δ ( y ) (see Fig. 2).The vector potential satisfies ∇ · A = 0 , the so calledCoulomb gauge. The magnetic field corresponding to po-tential vector A is the famous Dirac monopole field that is radial but having zero divergence (here and inwhat follows we adopt the formalism of Schwinger etal. ). The absence of a Coulomb term in the Hamilto-nian means that the interaction is between two particleshaving one an electric charge and the other a magneticcharge, but not both kind of charges in the same particle(there are no “dyons” implied). IV. STATIONARY SOLUTIONS
To solve this system we work in the standard waysearching for the eigenvalues E of the Hamiltonian ˆ H B Ψ( r ) = E Ψ( r ) . From the classical version of the mag-netic monopole problem it is known that this Hamilto-nian conserves a generalized version of the classical an-gular momentum J = r × p + q ˆ r , that in the quantumformalism give rise to the operators ˆ J and ˆ J z , which explicit form are ˆ J = − (cid:126) (cid:20) θ ∂∂θ (cid:18) sin θ ∂∂θ (cid:19) + 1sin θ ∂ ∂φ (cid:21) + 2 i (cid:126) q cos θ sin θ ∂∂φ + (cid:126) q sin θ (4) ˆ J = − i (cid:126) ∂∂φ . (5)Their eigenfunctions are given by the so called gener-alized spherical harmonics Y ( q ) lm ( θ, φ ) (see Appendix A).The eigenvalues problem is solved in a way similar to thestandard angular momentum problem: ˆ J Y ( q ) lm ( θ, φ ) = l ( l + 1) (cid:126) Y ( q ) lm ( θ, φ ) , (6)and ˆ J z Y ( q ) lm ( θ, φ ) = m (cid:126) Y ( q ) lm ( θ, φ ) (7)where l ≥ | q | , and − l ≥ m ≥ l are integers. Thegeneralized spherical harmonics share a series of prop-erties with the usual spherical harmonics and have theimportant property of reducing to them when q = 0 , Y (0) lm ( θ, φ ) = Y lm ( θ, φ ) .Using the separation of variables Ψ( r ) = R ( r ) Y ( q ) lm ( θ, φ ) and defining (cid:96) ( (cid:96) + 1) ≡ l ( l + 1) − q , theradial differential equation becomes: (cid:20) r ∂∂r (cid:18) r ∂∂r (cid:19) + k − (cid:96) ( (cid:96) + 1) r (cid:21) R ( r ) = 0 (8)where we have the dispersion relation k = 2 m ∗ E/ (cid:126) pre-dicting a group velocity v = (2 JSa / (cid:126) ) k . This equation Θ (cid:72) d Σ d (cid:87) (cid:76) (cid:144) (cid:72) d Σ d (cid:87) (cid:76) FIG. 4. Ratio of the differential cross section with respect tothe small angles one, for q=1, as a function of the scatteringangle. is solved by spherical Bessel function j (cid:96) ( kr ) of order (cid:96) .With this we can write the general solution for the spinwave excitations in the form of an expansion: Ψ( r ) = ∞ (cid:88) l = q l (cid:88) m = − l A klm j (cid:96) ( kr ) Y ( q ) lm ( θ, φ ) . (9)The condition l ≥ q shows that there is only one mode ofoscillation being zero at the origin: the one correspondingto q = 1 and l = 1 (see the relation previous to theEq. (8)), which gives a Bessel function of order (cid:96) = 0 .In all the other cases the modes are different from zeroat the origin. In this way, the singularity at r = 0 isnaturally avoid by most of the modes. We show in theFig. (3) some representative modes of the angular partof the solution.An important feature of these solutions is the ab-sence of local modes. In fact, in our system there isno Coulomb-like potential, making that the radial partof the equation is equivalent to a free particle (aside therelation between the quantum numbers (cid:96) and m ). Thisis closely related to the fact that, from the classical pointof view, the angular momentum conservation in the in-teraction of the monopole and the electron gives an opentrajectory over the surface of a cone in which the closestdistance between the particles is the impact parameter.In general, solitons breaking the continuous translationalsymmetry should have special zero-modes that are usu-ally quasi-local ones. These zero-modes do not appearin our treatment because we do not consider the rigidmotion of the BP, fixing its position at the origin of co-ordinates.
V. SCATTERING OF SPIN WAVES BY THEBLOCH POINT
Let us now consider the problem of scattering of SWsby a BP. We have a straightforward relation between theprobability density | Ψ | and the intensity of SWs. Letus assume that we are far away enough from the BP in regions where magnetization is almost homogeneous.There the Hamiltonian reduces to a free particle whosesimplest solutions are in the form of plane waves. We canexpand a planar wave into a series with the eigenfunctionsof Eqs. (6) and (7). Recognizing and setting apart theterms that represents an incident plane wave from thoseassociated with an outgoing spherical wave we can findthe amplitude of the spherical scattered wave f ( θ ) , where θ is the angle between the incident wave and the point weare looking at (the scattering amplitude is independent ofthe choice of the singularity line). Explicitly we have: ikf ( θ ) = ∞ (cid:88) l = q (2 l + 1) Y ( q ) lq ( π − θ, φ = 0) e − iπ(cid:96) . (10)There is no explicit expression for this series and we needto evaluate it numerically. In the limit of small angles θ (cid:28) the differential cross section becomes functionallyequivalent to a Rutherford scattering: (cid:18) dσd Ω (cid:19) = (cid:16) q k (cid:17) ( θ/ , (11)We plot numerically the differential cross section di-vided by the small-angles limit, Fig. (4), for the topolog-ical charge q = 1 .The scattering of a spin wave moving across a BP sin-gularity can be used to alter its location. Just like ithas been proposed in the context of domain wall dynam-ics where by means of the spin waves generated in thematerial a force is found to over the domain wall.The result that we are presenting, concerning the be-haviour of spin waves around BPs is of paramount im-portance to understand and to control BPs dynamics. VI. DISCUSSION AND CONCLUSION
In this paper we study a static and singular BP andthe SWs in its presence. We use the simplest model thatgives rises to a BP solution and its topological proper-ties, that is, the exchange energy. By doing a very simpletransformation that put the constant norm magnetiza-tion field into a complex variable, we are able to write aSchrödinger equation for the perturbations around theBP. This equation give us the dynamics of the SWsand it is found to be equivalent to the dynamics of thequantum interaction between an electron and a magneticmonopole, where the product of the electric charge andmagnetic charge of the particles is the topological charge.We take advantage of the enormous understanding cu-mulated through the years on the subject to solve theSchrödinger equation (and found in this way the dynam-ics of the SWs) and to reinterpret the previous results onquantum scattering as the problem of free SW that findin their way a BP. The formula predict a Rutherford-likescattering for small angles, and a complex behavior forangles larger than π/ , especially when the topologicalcharge of the BP is increased and we are close to thebackscattering ( θ → π ).The properties of the scattering on a BP field open newpossibilities for its detection by means of the study of itsspectrum, and can be considered as the first order effectscoming from the topology of the soliton. The differencein scattering of different charge BP can also be used tomeasure the topological charge. Analytical calculationson BP with other terms in the energy (as dipolar energy,anisotropies or external fields) are intrinsically complexand it is not yet clear what are the stability regions, evenif it is commonly accepted that it is unstable in the pres-ence of external field and anisotropies (considering formanisotropies and crystallographic ones). But even con-sidering additional terms in energy, it is quite possiblethat the new situation follows the analogy between SWsin the BP field and the quantum interaction between anelectron and a magnetic monopole because of topologicalreasons. The existence of a region with reduced mag-netization rises a difficulty for the exact solution we areproposing. At a first glance the applicability of our solu-tion is restricted to wavelengths larger than characteristicsize of this region. New calculations must include mod-ifications in the radial part that could support boundstates, absent in our approach. It could then be veryinteresting to test these kind of considerations in futureworks and the lost of stability of BPs by spin waves aswell. ACKNOWLEDGMENTS
R.G.E thanks Conicyt Pai/Concurso Nacional deApoyo al Retorno de Investigadores/as desde el Ex-tranjero Folio 821320024. V.L.C.S. thanks the Brazil-ian agency CNPq (Grant No. 229053/2013-0), for fi-nancial support. The authors acknowledge funding from Proyecto Fondecyt numbers 11070008 and 1110271,Proyecto Basal FB0807-CEDENNA, Anillo de Ciencia yTecnonología ACT 1117, and by Núcleo Científico Mile-nio P06022-F.
Appendix A: Generalized spherical harmonics
For completeness we show here the definition of thefunctions Y ( q ) lm ( θ, φ ) .Giving the eigenfunctions of the operator ˆ J z be Φ( φ ) = e imφ (with m integer), the polar part of the operator ˆ J gives the equation: − (cid:20) θ ∂∂θ (cid:18) sin θ ∂∂θ (cid:19) − m + 2 qm cos θ + q sin θ (cid:21) Θ = l ( l + 1) Θ . (A1)This equation is solved using the general rotation func-tions U ( q ) lm ( θ ) . These functions are related to the Jacobipolynomials P ( b,c ) n ( x ) (for n integer) as: U ( q ) lm ( θ ) = (cid:20) ( l + q )!( l − q )!( l + m )!( l − m )! (cid:21) / × (cid:18) − x (cid:19) ( q − m ) / (cid:18) x (cid:19) ( q + m ) / P ( q − m,q + m ) l − q ( x ) , (A2)where x = cos θ . The general rotations functions U ( q ) lm ( θ ) are used to define the generalized spherical harmonics Y ( q ) lm ( θ, φ ) ≡ √ l + 1 U ( q ) lm ( θ ) e imφ , (A3)that are the eigenfunctions of the whole angular operatorwith the properties already mentioned in the text. A. Hubert and R. Schäfer,
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