Open access peer-reviewed chapter

Magnetic Skyrmions: Theory and Applications

Written By

Lalla Btissam Drissi, El Hassan Saidi, Mosto Bousmina and Omar Fassi-Fehri

Submitted: 08 December 2020 Reviewed: 01 March 2021 Published: 11 May 2021

DOI: 10.5772/intechopen.96927

From the Edited Volume

Magnetic Skyrmions

Edited by Dipti Ranjan Sahu

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Magnetic skyrmions have been subject of growing interest in recent years for their very promising applications in spintronics, quantum computation and future low power information technology devices. In this book chapter, we use the field theory method and coherent spin state ideas to investigate the properties of magnetic solitons in spacetime while focussing on 2D and 3D skyrmions. We also study the case of a rigid skyrmion dissolved in a magnetic background induced by the spin-tronics; and derive the effective rigid skyrmion equation of motion. We examine as well the interaction between electrons and skyrmions; and comment on the modified Landau-Lifshitz-Gilbert equation. Other issues, including emergent electrodynamics and hot applications for next-generation high-density efficient information encoding, are also discussed.


  • Geometric phases
  • magnetic monopoles and topology
  • soliton and holonomy
  • skyrmion dynamics and interactions
  • med-term future applications

1. Introduction

During the last two decades, the magnetic skyrmions and antiskyrmions have been subject to an increasing interest in connection with the topological phase of matter [1, 2, 3, 4], the spin-tronics [5, 6] and quantum computing [7, 8]; as well as in the search for advanced applications such as racetrack memory, microwave oscillators and logic nanodevices making skyrmionic states very promising candidates for future low power information technology devices [9, 10, 11, 12]. Initially proposed by T. Skyrme to describe hadrons in the theory of quantum chromodynamics [13], skyrmions have however been observed in other fields of physics, including quantum Hall systems [14, 15], Bose-Einstein condensates [16] and liquid crystals [17]. In quantum Hall (QH) ferromagnets for example [18, 19], due to the exchange interaction; the electron spins spontaneously form a fully polarized ferromagnet close to the integer filling factor ν1; slightly away, other electrons organize into an intricate spin configuration because of a competitive interplay between the Coulomb and Zeeman interactions [18]. Being quasiparticles, the skyrmions of the QH system condense into a crystalline form leading to the crystallization of the skyrmions [20, 21, 22, 23]; thus opening an important window on promising applications.

In order to overcome the lack of a prototype of a skyrmion-based spintronic devices for a possible fabrication of nanodevices of data storage and logic technologies, intense research has been carried out during the last few years [24, 25]. In this regard, several alternative nano-objects have been identified to host stable skyrmions at room temperature. The first experimental observation of crystalline skyrmionic states was in a three-dimensional metallic ferromagnet MnSi with a B20 structure using small angle neutron scattering [26]. Then, real-space imaging of the skyrmion has been reported using Lorentz transmission electron microscopy in non-centrosymmetric magnetic compounds and in thin films with broken inversion symmetry, including monosilicides, monogermanides, and their alloys, like Fe1–xCoxSi [27], FeGe [28], and MnGe [29].

One of the key parameters in the formation of these topologically protected non-collinear spin textures is the Dzyaloshinskii-Moriya Interaction (DMI) [30, 31, 32]. Originating from the strong spin-orbit coupling (SOC) at the interfaces, the DM exchange between atomic spins controls the size and stability of the induced skyrmions. Depending on the symmetry of the crystal structures and the skyrmion windings number, the internal spins within a single skyrmion envelop a sphere in different arrangements [33]. The in-plane component of the magnetization, in the Néel skyrmion, is always pointed in the radial direction [34], while it is oriented perpendicularly with respect to the position vector in the Bloch skyrmion [26]. Different from these two well-known types of skyrmions are skyrmions with mixed Bloch-Néel topological spin textures observed in Co/Pd multilayers [35]. Magnetic antiskyrmions, having a more complex boundary compared to the chiral magnetic boundaries of skyrmions, exist above room temperature in tetragonal Heusler materials [36]. Higher-order skyrmions should be stabilized in anisotropic frustrated magnet at zero temperature [37] as well as in itinerant magnets with zero magnetic field [38].

In the quest to miniaturize magnetic storage devices, reduction of material’s dimensions as well as preservation of the stability of magnetic nano-scale domains are necessary. One possible route to achieve this goal is the formation of topological protected skyrmions in certain 2D magnetic materials. To induce magnetic order and tune DMIs in 2D crystal structures, their centrosymmetric should first be broken using some efficient ways such a (i) generate one-atom thick hybrids where atoms are mixed in an alternating manner [39, 40, 41], (ii) apply bias voltage or strain [42, 43, 44], (iii) insert adsorbents, impurities and defects [45, 46, 47]. In graphene-like materials, fluorine chemisorption is an exothermic adsorption that gives rise to stable 2D structures [48] and to long-range magnetism [49, 50]. In semi-fluorinated graphene, a strong Dzyaloshinskii-Moriya interaction has been predicted with the presence of ferromagnetic skyrmions [51]. The formation of a nanoskyrmion state in a Sn monolayer on a SiC(0001) surface has been reported on the basis of a generalized Hubbard model [52]. Strong DMI between the first nearest magnetic germanium neighbors in 2D semi-fluorinated germanene results in a potential antiferromagnetic skyrmion [53].

In this bookchapter, we use the coherent spin states approach and the field theory method (continuous limit of lattice magnetic models with DMI) to revisit some basic aspects and properties of magnetic solitons in spacetime while focusing on 1d kinks, 2d and 3d spatial skyrmions/antiskyrmions. We also study the case of a rigid skyrmion dissolved in a magnetic background induced by the electronic spins of magnetic atoms like Mn; and derive the effective rigid skyrmion equation of motion. In this regard, we describe the similarity between, on one hand, electrons in the electromagnetic background; and, on the other hand, rigid skyrmions bathing in a texture of magnetic moments. We also investigate the interaction between electrons and skyrmions as well as the effect of the spin transfer effect.

This bookchapter is organized as follows: In Section 2, we introduce some basic tools on quantum SU(2) spins and review useful aspects of their dynamics. In Section 3, we investigate the topological properties of kinks and 2d space solitons while describing in detail the underside of the topological structure of these low-dimensional solitons. In Section 4, we extend the construction to approach topological properties to 3d skyrmions. In Section 5, we study the dynamics of rigid skyrmions without and with dissipation; and in Section 6, we use emergent gauge potential fields to describe the effective dynamics of electrons interacting with the skyrmion in the presence of a spin transfer torque. We end this study by making comments and describing perspectives in the study of skyrmions.


2. Quantum SU(2) spin dynamics

In this section, we review some useful ingredients on the quantum SU(2) spin operator, its underlying algebra and its time evolution while focussing on the interesting spin 1/2 states, concerning electrons in materials; and on coherent spin states which are at the basis of the study of skyrmions/antiskyrmions. First, we introduce rapidly the SU(2) spin operator S and the implementation of time dependence. Then, we investigate the non dissipative dynamics of the spin by using semi-classical theory approach (coherent states). These tools can be also viewed as a first step towards the topological study of spin induced 1D, 2D and 3D solitons undertaken in next sections.

2.1 Quantum spin 1/2 operator and beyond

We begin by recalling that in non relativistic 3D quantum mechanics, the spin states SzS of spinfull particles are characterised by two half integers SzS, a positive S0 and an Sz taking 2S+1 values bounded as SSzS with integral hoppings. For particles with spin 1/2 like electrons, one distinguishes two basis vector states ±1212 that are eigenvalues of the scaled Pauli matrix 2σz and the quadratic (Casimir) operator 24a=13σa2, here the three 2σa with σa=σ.ea are the three components of the spin 1/2 operator vector1σ. From these ingredients, we learn that the average <Sz,S2σzSz,S>=Sz (for short 2σz) is carried by the z-direction since Sz=S.ez with ez=0,0,1T. For generic values of the SU2 spin S, the spin operator reads as Ja where the three Ja‘s are 2S+1×2S+1 generators of the SU2 group satisfying the usual commutation relations JaJb=iεabcJc with εabc standing for the completely antisymmetric Levi-Civita tensor with non zero value ε123=1; its inverse is εcba with ε123=1. The time evolution of the spin 12 operator σa2 with dynamics governed by a stationary Hamiltonian operator (dH/dt=0) is given by the Heisenberg representation of quantum mechanics. In this non dissipative description, the time dependence of the spin 12 operator Ŝat (the hat is to distinguish the operator Ŝa from classical Sa) is given by


where the Pauli matrices σa obey the usual commutation relations σaσb=2iεabcσc. For a generic value of the SU2 spin S, the above relation extends as Ŝa=eiHtJaeiHt. So, many relations for the spin 1/2 may be straightforwardly generalised for generic values S of the SU2 spin. For example, for a spin value S0, the 2S0+1 states are given by mS0 and are labeled by S0mS0; one of these states namely S0S0 is very special; it is commonly known as the highest weight state (HWS) as it corresponds to the biggest value m=S0; from this state one can generate all other spin states mS0; this feature will be used when describing coherent spin states. Because of the property σa2=I, the square Ŝa2=24I is time independent; and then the time dynamics of Ŝat is rotational in the sense that dŜadt is given by a commutator as follows dŜadt=iHŜaŜaH. For the example where H is a linearly dependent function of Ŝa like for the Zeeman coupling, the Hamiltonian reads as HZ=aωaŜa (for short ωaŜa) with the ωa’s are constants referring to the external source2; then the time evolution of Ŝa reads, after using the commutation relation ŜaŜb=iεabcŜc, as follows


where appears the Levi-Civita εabc which, as we will see throughout this study, turns out to play an important role in the study of topological field theory [54, 55] including solitons and skyrmions we are interested in here [56, 57, 58, 59]. In this regards, notice that, along with this εabc, we will encounter another completely antisymmetric Levi-Civita tensor namely εμ1μD; it is also due to DM interaction which in lattice description is given by Srμ2Srμ1.dμ3μD2εμ1μD; and in continuous limit reads as εabcSbSμ1μ2cdμ3μD2aεμ1μD where, for convenience, we have set Sμ1μ2c=eμ1μ2.Sc with eμ1μ2=eμ2eμ1. To distinguish these two Levi-Civita tensors, we refer to εabc as the target space Levi-Civita with SO3target symmetry; and to εμ1μD as the spacetime Levi-Civita with SO1D1 Lorentz symmetry containing as subsymmetry the usual space rotation group SOD1space. Notice also that for the case where the Hamiltonian HŜ is a general function of the spin, the vector ωa is spin dependent and is given by the gradient HŜa.

2.2 Coherent spin states and semi-classical analysis

To deal with the semi-classical dynamics of Ŝt evolved by a Hamiltonian HŜ, we use the algebra ŜaŜb=iεabcŜc to think of the quantum spin in terms of a coherent spin state [60] described by a (semi) classical vector S=Sn (no hat) of the Euclidean R3; see the Figure 1(a). This “classical” 3-vector has an amplitude S and a direction n related to a given unit vector n0 as n=Rαβγn0; and parameterised by α,β,γ. In the above relation, the n0 is thought of as the north direction of a 2-sphere Sn2 given by the canonical vector 0,0,1T; it is invariant under the proper rotation; i.e. Rzγn0=n0; and consequently the generic n is independent of γ; i.e.: n=Rαβn0. Recall that the 3×3 matrix Rαβγ is an SO3 rotation [SO3SU2] generating all other points of Sn2 parameterised by αβ. In this regards, it is interesting to recall some useful properties that we list here after as three points: 1 the rotation matrix Rαβγ can be factorised like RzαRyβRzγ where each Raψa is a rotation eiψaJa around the a- axis with an angle ψa and generator Ja. 2 As the unit n0 is an eigen vector of eJz; it follows that n reduces to eJzeJyn0; this generic vector obeys as well the constraint nn=1 and is solved as follows.

Figure 1.

(a) Components of the spin orientation n; its time dynamics in presence of a magnetic field is given by Larmor precession. (b) A configuration of several spins in spacetime.


with 0α2π and 0βπ; they parameterise the unit 2-sphere Sn2 which is isomorphic to SU2/U1; the missing angle γ parameterises a circle Sn1, isomorphic to U1, that is fibred over Sn2.3 the coherent spin state representation gives a bridge between quantum spin operator and its classical description; it relies on thinking of the average <Ŝ> in terms of the classical vector S0=Sn0 considered above (S0 HWS S0S0). In this regards, recall that the Ŝa acts on classical 3-vectors Vb through its 3×3 matrix representation like ŜaVb=JcabVc with Jcab given by iεabc; these Jc‘s are precisely the generators of the SU2 matrix representation Rαβγ; by replacing Vb by the operator Ŝb, one discovers the SU2 spin algebra ŜaŜb=iεabcŜc. Notice also that the classical spin vector S=Sn can be also put in correspondence with the usual magnetic moment μ=γS (with γ=ge2m the gyromagnetic ratio); thus leading to μ=μn. So, the magnetization vector describes (up to a sign) a coherent spin state with amplitude ; and a (opposite) time dependent direction nt parameterizing the 2-sphere Sn2.


For explicit calculations, this unit 2-sphere equation will be often expressed like nana=1; this relation leads in turns to the property nadna=0 (indicating that n and dn are normal vectors); by implementing time, the variation n.dn gets mapped into n.ṅ=0 teaching us that the velocity ṅ is carried by u and v; two normal directions to n with components;


and from which we learn that dna=ua+vasinβdα, [(u,v,n) form an orthogonal vector triad). So, the dynamics of μa (and that of S) is brought to the dynamics of the unit na governed by a classical Hamiltonian Hnaαβ. The resulting time evolution is given by the so called Landau-Lifshitz (LL) equation [61]; it reads as dnadt=γμεabcbHnc with bH=Hnb. By using the relations =uadna and sinβdα=vadna with ua=naβ; and vasinβ=naα; as well as the expressions εabcuanc=vb and εabcvanc=ub, the above LL equation splits into two time evolution equations dt=γvbbH and sinβdt=γubbH. These time evolutions can be also put into the form


and can be identified with the Euler- Lagrange equations following from the variation δS=0 of an action S=Ldt. Here, the Lagrangian is related to the Hamiltonian like L=LBHnaαβ where LB is the Berry term [62] known to have the form <n|ṅ>; this relation can be compared with the well known Legendre transform pq̇Hqp. For later interpretation, we scale this hamiltonian as SγH such that the spin lagrangian takes the form Lspin=LBSγH. To determine LB, we identify the Eq. (6) with the extremal variation δS/δβ=0 and δS/δα=0. Straightforward calculations leads to


showing that α and β form a conjugate pair. By substituting sinβdt=vadnadt back into above LB, we find that the Berry term has the form of Aharonov-Bohm coupling LAB=qeAadnadt with magnetic potential vector Aa given by Aa=Sqe1cosβsinβva. However, this potential vector is suggestive as it has the same form as the potential vector Amonopole=Sqer1cosβsinβv of a magnetic monopole. The curl of this potential is given by B=qmrr3 with magnetic charge qm=Sqe located at the centre of the 2-sphere; the flux Φ of this field through the unit sphere is then equal to 4πSqe; and reads as 2SΦ0 with a unit flux quanta Φ0=hqe as indicated by the value S=1/2. So, because 2S=n is an integer, it results that the flux is quantized as Φ=nΦ0.


3. Magnetic solitons in lower dimensions

In previous section, we have considered the time dynamics of coherent spin states with amplitude S and direction described by nt as depicted by the Figure 1(a); this is a 3-vector having with no space coordinate dependence, gradn=0; and as such it can be interpreted as a 1+0D vector field; that is a vector belonging to R1,d with d=0 (no space direction). In this section, we first turn on 1d space coordinate x and promotes the old unit- direction nt to a 1+1D field ntx. After that, we turn on two space directions xy; thus leading to 1+2D field ntxy; a picture is depicted by the Figure 1(b). To deal with the dynamics of these local fields and their topological properties, we use the field theory method while focussing on particular solitons; namely the 1d kinks and the 2d skyrmions. In this extension, one encounters two types of spaces: 1 the target space Rn3 parameterised by na=n1n2n3 with Euclidian metric δab and topological Levi-Civita εabc. 2 the spacetime Rξ1,1 parameterised by ξμ=tx, concerning the 1d kink evolution; and the spacetime Rξ1,2 parameterised by ξμ=txy, regarding the 2d skyrmions dynamics. As we have two kinds of evolutions; time and space; we denote the time variable by ξ0=t; and the space coordinates by ξi=xy. Moreover, the homologue of the tensors δab and εabc are respectively given by the usual Lorentzian spacetime metric gμν, with signature like gμνξμξν=x2+y2t2, and the spacetime Levi-Civita εμνρ with ε012=1.

3.1 One space dimensional solitons

In 1+1D spacetime, the local coordinates parameterising Rξ1,1 are given by ξμ=tx; so the metric is restricted to gμνξμξν=x2t2. The field variable naξ has in general three components n1n2n3 as described previously; but in what follows, we will simplify a little bit the picture by setting n3=0; thus leading to a magnetic 1d soliton with two component field variable n=n1n2 satisfying the constraint equation n.n=1 at each point of spacetime. As this constraint relation plays an important role in the construction, it is interesting to express it as nana=1. Before describing the topological properties of one space dimensional solitons (kinks), we think it interesting to begin by giving first some useful features; in particular the three following ones. 1 The constraint n12+n22=1 is invariant SO2n rotations acting as n'a=Rbanb with orthogonal rotation matrix


The constraint nana=1 can be also presented like N¯N=1 with N standing for the complex field n1+in2 that reads also like e. In this complex notation, the symmetry of the constraint is given by the phase change acting as NUN with U=e and corresponding to the shift αα+ψ. Moreover the correspondence n1n2n1+in2 describes precisely the well known isomorphisms SO2U1Sn1 where Sn1 is a circle; it is precisely the equatorial circle of the 2-sphere Sn2 considered in previous section. 2 As for Eq. (5), the constraint nana=1 leads to nadna=0; and so describes a rotational movement encoded in the relation dna=εabnb where εab is the standard 2D antisymmetric tensor with ε21=ε12=1; this εab is related to the previous 3D Levi-Civita like εzab. Notice also that the constraint nana=1 implies moreover that dn2=nnn2dn1; and consequently the area dn1dn2, to be encountered later on, vanishes identically. In this regards, recall that we have the following transformation


where εμν is the antisymmetric tensor in 1+1 spacetime, and J is the Jacobian of the transformation txn1n2. 3 The condition nana=1 can be dealt in two manners; either by inserting it by help of a Lagrange multiplier; or by solving it in term of a free angular variable like na=cosαsinα from which we deduce the normal direction ua=dna reading as ua=sinαcosα. In term of the complex field; we have N=e and N¯dN=idα. Though interesting, the second way of doing hides an important property in which we are interested in here namely the non linear dynamics and the topological symmetry.

3.1.1 Constrained dynamics

The classical spacetime dynamics of naξ is described by a field action S=dtL with Lagrangian L=dxL and density L; this field density is given by 12μnaμnaVnΛnana1 with μ=ξμ; it reads in terms of the Hamiltonian density as follows


where πa=Lṅa. In the above Lagrangian density, the auxiliary field Λξ (no Kinetic term) is a Lagrange multiplier carrying the constraint relation nana=1. The Vn is a potential energy density which play an important role for describing 1d kinks with finite size. Notice also that the variation δSδΛ=0 gives precisely the constraint nana=1 while the δSδna=0 gives the spacetime dynamics of na described by the spacetime equation μμnaVnaΛna=0. By substituting na=cosαsinα, we obtain L=12μαμαVα. If setting Vα=0, we end up with the free field equation μμα=0 that expands like x2t2α=0; it is invariant under spacetime translations with conserved current symmetry μTμν=0 with Tμν standing for the energy momentum tensor given by the 2×2 symmetric matrix μανα+gμνL. The energy density T00 is given by 12tα2+12xα2 and the momentum density T10 reads as xφxφ. Focussing on T00, the conserved energy E reads then as follows


with minimum corresponding to constant field (α=cte). Notice that general solutions of μμα=0 are given by arbitrary functions fx±t; they include oscillating and non oscillating functions. A typical non vibrating solution that is interesting for the present study is the solitonic solution given (up to a constant c) by the following expression


where λ is a positive parameter representing the width where the soliton αtx acquires a significant variation. Notice that for a given t, the field varies from αt=π to αt+=π regardless the value of λ. These limits are related to each other by a period 2π.

3.1.2 Topological current and charge

To start, notice that as far as conserved symmetries of (10) are concerned, there exists an exotic invariance generated by a conserved Jμtx going beyond the spacetime translations generated by the energy momentum tensor Tμν. The conserved spacetime current Jμ=J0J1 of this exotic symmetry can be introduced in two different, but equivalent, manners; either by using the free degree of freedom α; or by working with the constrained field na. In the first way, we think of the charge density J0 like 12π1α and of the current density as J1=12π0α. This conserved current is a topological 1+1D spacetime vector Jμ that is manifestly conserved; this feature follows from the relation between Jμ and the antisymmetric εμν as follows [57],


Because of the εμν; the continuity relation μJμ=12πεμνμνα vanishes identically due to the antisymmetry property of εμν. The particularity of the above conserved Jμ is its topological nature; it is due to the constraint nana=1 without recourse to the solution na=cosαsinα. Indeed, Eq. (13) can be derived by computing the Jacobian J=detnaξμ of the mapping from the 2d spacetime coordinates tx to the target space fields (n1,n2). Recall that the spacetime area dtdx can be written in terms of εμν like 12εμνdξμdξν and, similarly, the target space area dn1dn2 can be expressed in terms of as εab follows 12εabdnadnb. The Jacobian J is precisely given by (9); and can be presented into a covariant form like J=12εμνμnaνnbεab. This expression of the Jacobian J captures important informations; in particular the three following ones. 1 It can be expressed as a total divergence like μπJμ with spacetime vector


and where 1π is a normalisation; it is introduced for the interpretation of the topological charged as just the usual winding number of the circle [encoded in the homotopy group relation π1S1=]. 2 Because of the constraint dn2=nnn2dn1 following from nana=1, the Jacobian J vanishes identically; thus leading to the conservation law μJμ=0; i.e. J=0 and then μJμ=0. 3 The conserved charge Q associated with the topological current is given by +dxJ0tx; it is time independent despite the apparent t- variable in the integral (dQ/dt=0). By using (13), this charge reads also as 12π+dxxαtx and after integration leads to


Moreover, seen that αt is an angular variable parameterising Sn1; it may be subject to a boundary condition like for instance the periodic αt=αt+2πN with N an integer; this leads to an integral topological charge Q=N interpreted as the winding number of the circle. In this regards, notice that: i the winding interpretation can be justified by observing that under compactification of the space variable x, the infinite space line Rx=+ gets mapped into a circle Sx1 with angular coordinate πφπ; so, the integral 12π+dxxαtx gets replaced by 12ππ+παφ; and then the mapping αt:φαtφ is a mapping between two circles namely Sx1Sn1; the field αtφ then describes a soliton (one space extended object) wrapping the circle Sx1 N times; this propery is captured by πSx1Sn1=, a homotopy group property [63]. ii The charge Q is independent of the Lagrangian of the system as it follows completely from the field constraint without any reference to the field action. iii Under a scale transformation ξ'=ξ/λ with a scaling parameter λ>0, the topological charge of the field (12) is invariant; but its total energy (11) get scaled as follows


This energy transformation shows that stable solitons with minimal energy correspond to λ; and then to a trivial soliton spreading along the real axis. However, one can have non trivial solitonic configurations that are topologically protected and energetically stable with non diverging λ. This can done by turning on an appropriate potential energy density Vn in Eq. (10). An example of such potential is the one given by g8n14+n241, with positive g=M2, breaking SO2n; by using the constraint n12+n22=1, it can be put g4n12n22. In terms of the angular field α, it reads as Vα=g161cos4α leading to the well known sine-Gordon Eq. [64, 65] namely μμαg4sin4α=0 with the symmetry property αα+π2. So, the solitonic solution is periodic with period π2; that is the quarter of the old 2π period of the free field case. For static field αx, the sine Gordon equation reduces to d2αdx2M24sin4α=0; its solution for M>0 is given by arctanexpMx representing a sine- Gordon field evolving from 0 to π2 and describing a kink with topological charge Q=14. For M<0, the soliton is an anti-kink evolving from π2 to 0 with charge Q=14. Time dependent solutions can be obtained by help of boost transformations xx±vt1v2.

3.2 Skyrmions in 2d space dimensions

In this subsection, we investigate the topological properties of 2d Skyrmions by extending the field theory study we have done above for 1d kinks to two space dimensions. For that, we proceed as follows: First, we turn on the component n3 so that the skyrmion field n is a real 3-vector with three components n1n2n3 constrained as in Eqs. (4) and (5); see Figure 2. Second, here we have n=ntxy; that is a 3-component field living in the 2+1 space time with Lorentzian metric and coordinates ξμ=txy. This means that dn=μndξμ; explicitly dn=ntdt+nxdx+nydy.

Figure 2.

On left: a spin configuration with n12+n22+n32=1 dispatched on a 2-sphere. On right: a two space dimensional magnetic skyrmion given by the stereographic projection of S2 to plane.

3.2.1 Dzyaloshinskii-Moriya potential

The field action S3D=dtL3D describing the space time dynamics of ntxy has the same structure as Eq. (10); except that here the Lagrangian L3D involves two space variable like dxdyL3D and the density L3D=12μn2VnΛn.n1; this is a function of the constrained 3-vector n and its space time gradient μn; it reads in term of the Hamiltonian density as follows.


In this expression, the H3Dn is the continuous limit of a lattice Hamiltonian Hlatt(narμ involving, amongst others, the Heisenberg term, the Dyaloshinskii- Moriya (DM) interaction and the Zeeman coupling. The Vn in the first expression of L3D is the scalar potential energy density; it models the continuous limit of the interactions that include the DM and Zeeman ones [see Eq. (1.22) for its explicit relation]. The field Λξ is an auxiliary 3D spacetime field; it is a Lagrange multiplier that carries the constraint n.n=1 which plays the same role as in subSection 3.1. By variating this action with respect to the fields n and Λ; we get from δS3DδΛ=0 precisely the field constraint n.n=1; and from δS3Dδn=0 the following Euler–Lagrange equation Wn=Vn+Λn. For later use, we express this field equation like


The interest into this (18) is twice; first it can be put into the equivalent form μμna=εabcDbnc where Db is an operator acting on nc to be derived later on [see Eq. (22) given below]; and second, it can be used to give the relation between the scalar potential and the operator Db. To that purpose, we start by noticing that there are two manners to deal with the field constraint nana=1; either by using the Lagrange multiplier Λ; or by solving it in terms of two angular field variables as given by Eq. (5). In the second case, we have the triad na=sinβcosαsinβsinαcosβ and


but now β=βtxy and α=αtxy with 0βπ and 0α2π. Notice also that the variation of the filed constraint leads to nadna=0 teaching us interesting informations, in particular the two following useful ones. 1 the movement of na in the target space is a rotational movement; and so can be expressed like


where the 1-form ωb is the rotation vector to be derived below. By substituting (20) back into nadna, we obtain εbcaωbncna which vanishes identically due to the property εbcancna=0. 2 Having two degrees of freedom α and β, we can expand the differential dna like ua+vasinαdα with the two vector fields ua=naβ and va=naα as given above. Notice that the three unit fields nuv plays an important role in this study; they form a vector basis of the field space; they obey the usual cross products namely n=uv and its homologue which given by cyclic permutations; for example,


Putting these Eq. (21) back into the expansion of dna in terms of ,; and comparing with Eq. (20), we end up with the explicit expression of the 1-form angular “speed” vector ωb; it reads as follows ωb=vbubsinαdα. Notice that by using the space time coordinates ξ, we can also express Eq. (20) like μna=εabcωμbnc with ωμb given by vbμβubsinαμα. From this expression, we can compute the Laplacian μμna; which, by using the above relations; is equal to εabcμωμbnc reading explicitly as εabcμωμbnc+ωμbμnc or equivalently like μμna=εabcDbnc with operator Db=ωμbμ+μωμb. Notice that the above operator has an interesting geometric interpretation; by factorising ωμb, we can put it in the form ωμdDμdb where Dμdb appears as a gauge covariant derivative Dμdb=δdbμ+Aμdb with a non trivial gauge potential Aμdb given by ωdμνωνb. Comparing with (18) with μμna=εabcDbnc, we obtain Vna=εabcDbncΛna; and then a scalar potential energy V given by εabcdnaDbncΛnadna. The second term in this relation vanishes identically because nadna=0; thus reducing to


containing εabcnaDbnc as a sub-term. In the end of this analysis, let us compare this sub-term with the εabcnbnμ1μ2cΔaμ1μ2 with Δaμ1μ2=dμ3μD2aεμ1μD giving the general structure of the DM coupling (see end of subSection 2.1). For 1+2D spacetime, the general structure of DM interaction reads εabcd0anbnc.eμνε0μν; by setting e0=eμνε0μν and da=d0ae0 as well as Da=da., one brings it to the form εabcnaDbnc which is the same as the one following from (22).

3.2.2 From kinks to 2d Skyrmions

Here, we study the topological properties of the 2d Skyrmion with dynamics governed by the Lagrangian density (17). From the expression of the 1+1D topological current Jμ2D discussed in subSection 2.1, which reads as 12πεμνnaνnbεab, one can wonder the structure of the 1+2D topological current Jμ3D that is associated with the 2d Skyrmion described by the 3-vector field naξ. It is given by


where εabc is as before and where εμνρ is the completely antisymmetric Levi-Civita tensor in the 1+2D spacetime. The divergence μJμ3D of the above spacetime vector vanishes identically; it has two remarkable properties that we want to comment before proceeding. 1 The μJμ3D is nothing but the determinant of the 3×3 Jacobian matrix naξμ relating the three field variables na to the three spacetime coordinates ξμ; this Jacobian detnaξμ is generally given by 13!εμνρμnaνnbνnbεabc; it maps the spacetime volume d3ξ=dtdxdy into the target space volume d3n=dn1dn2dn3. In this regards, recall that these two 3D volumes can be expressed in covariant manners by using the completely antisymmetric tensors εμνρ and εabc introduced earlier; and as noticed before play a central role in topology. The target space volume d3n can be expressed like 13!εabcdnadnbdnc; and a similar relation can be also written down for the spacetime volume d3ξ. Notice also that by substituting the differentials dna by their expansions naξμdξμ; and putting back into d3n, we obtain the relation d3n=J3Dd3ξ where J3D is precisely the Jacobian detnaξμ. 2 The conservation law μJμ3D=0 has a geometric origin; it follows from the field constraint relation n12+n22+n32=1 degenerating the volume of the 3D target space down to a surface. This constraint relation describes a unit 2-sphere Sn2; and so a vanishing volume d3nSn2=0; thus leading to J3D=0 and then to the above continuity equation. Having the explicit expression (23) of the topological current Jμ in terms of the magnetic texture field nξ, we turn to determine the associated topological charge Q=dxdyJ0 with charge density J0 given by 18πεabcε0ijinbjncna. Substituting ε0ijdxdy by dξidξj, we have J0dxdy=18πεabcnadnbdnc. Moreover using the differentials dnb=ub+vbsinαdα, we can calculate the area dnbdnc in terms of the angles α and β; we find 2nasinα where we have used εabcubvcucvb=2na. So, the topological charge Q reads as 14πSn2sinβdαdβ which is equal to 1. In fact this value is just the unit charge; the general value is an integer Q=N with N being the winding number π2Sn2; see below. Notice that J0 can be also presented like


Replacing na by their expression in terms of the angles sinβcosαsinβsinαcosβ, we can bring the above charge density J0 into two equivalent relations; first into the form like sinβ4πβxαyβyβx; and second as 14παcosβxy which is nothing but the Jacobian of the transformation from the xy space to the unit 2-sphere with angular variables αβ. The explicit expression of n1n2n3 in terms of the xy space variables is given by


but this is nothing but the stereographic projection of the 2-sphere Sξ2 on the real plane. So, the field na defines a mapping between Sξ2 towards Sn2 with topological charge given by the winding number Sn2 around Sn2; this corresponds just to the homotopy property π2Sn2=N.


4. Three dimensional magnetic skyrmions

In this section, we study the dynamics of the 3d skyrmion and its topological properties both in target space Rn4 (with euclidian metric δAB) and in 4D spacetime Rξ1,3 parameterised by ξμ=txyz (with Lorentzian metric gμν). The spacetime dynamics of the 3d skyrmion is described by a four component field nAξ obeying a constraint relation fn=1; here the fn is given by the quadratic form nAnA invariant under SO4 transformations isomorphic to SU2×SU2. The structure of the topological current of the 3d skyrmion is encoded in two types of Levi-Civita tensors namely the target space εABCD and the spacetime εμνρτ extending their homologue concerning the kinks and 2d skyrmions.

4.1 From 2d skyrmion to 3d homologue

As for the 1d and 2d solitons considered previous section, the spacetime dynamics of the 3d skyrmion in R1,3 is described by a field action S4D=dtL4D with Lagrangian realized as the space integral dxdydzL4D. Generally, the Lagrangian density L4D is a function of the soliton ntxyz which is a real 4-component field [n=n1n2n3n4] constrained like fnξ=1. For self ineracting field, the typical field expression of L4D is given by 12μn2VnΛfn1 where Vn is a scalar potential; and where the auxiliary field Λξ is a Lagrange multiplier carrying the field constraint. This density L4D reads in terms of the Hamiltonian as Π.ntHn. Below, we consider a 4-component skyrmionic field constrained as n.n=1; and focuss on a simple Lagrangian density L=12μnμnΛn.n1 to describe the degrees of freedom of n. Being a unit 4-component vector, we can solve the constraint n.n=1 in terms of three angular angles αβγ; by setting


where m is a unit 3-vector parameterising the unit sphere Sα2. Putting this field realisation back into L, we obtain cos2γ2μγ21cos2γ4μm2Λm.m1. Notice that by restricting the 4D spacetime R1,3 to the 3D hyperplane z=const; and by fixing the component field γ to π2, the above Lagrangian density reduces to the one describing the spacetime dynamics of the 2d skyrmion. Notice also that we can expand the differential dnA in terms of ,,; we find the following


For convenience, we sometimes refer to the three αβγ collectively like αa=α1α2α3; so we have dnA=EaAdαb with EaA=nAαa.

4.2 Conserved topological current

First, we investigate the topological properties of the 3d skyrmion from the target space view; that is without using the spacetine variables txyz=ξμ. Then, we turn to study the induced topological properties of the 3d skyrmion viewed from the side of the 4D space time R1,3.

4.2.1 Topological current in target space

The 3d skyrmion field is described by a real four component vector nA subject to the constraint relation nAnA=1; so the soliton has SO4SO31×SO32 symmetry leaving invariant the condition nAnA=1 that reads explicitly as n12+n22+n32+n42=1. The algebraic condition fn=1 induces in turns the constraint equation df=0 leading to nAdnA=0 and showing that nA and dnA orthogonal 4-vectors in Rn4. From this constraint, we can construct nAdnBnBdnA/2 which is a 4×4 antisymmetric matrix ΩAB generating the SO4 rotations; this ΩAB contains 3+3 degrees of freedom generating the two SO31 and SO32 making SO4; the first three degrres are given by Ωab with a,b=1,2,3; and the other three concern Ωa4. Notice also that, from the view of the target space, the algebraic relation nAnA=1 describes a unit 3-sphere Sn3 sitting in Rn4; as such its volume 4-form d4n, which reads as 14!εABCDdnAdnBdnCdnD, vanishes identically when restricted to the 3-sphere; i.e.: d4nSn3=0. This vanishing property of d4n on Sn3 is a key ingredient in the derivation of the topological current J of the 3D skyrmion and its conservation dJ=0. Indeed, because of the property d2=0 (where we have hidden the wedge product ), it follows that d4n can be expressed as dJ with the 3-form J given by


This 3-form describes precisely the topological current in the target space; this is because on Sn3, the 4-form d4n vanishes; and then dJ vanishes. By solving, the skyrmion field constraint nAnA=1 in terms of three angles αa as given by Eq. (26); with these angular coordinates, we have mapping f:Rn4Sn3 with Sn3Sα3. By expanding the differentials like dnA=EaAdαa with EaA=nAαa; then the conserved current on the 3-sphere Sα3 reads as follows


where we have substituted the 3-form dαbdαcdαd on the 3-sphere Sα3 by the volume 3-form εabcd3α. In this regards, recall that the volume of the 3-sphere is Sn3d3α=π22.

4.2.2 Topological symmetry in spacetime

In the spacetime R1,3 with coordinates ξμ=txyz, the 3d skyrmion is described by a four component field nAξ and is subject to the local constraint relation nAnA=1. A typical static configuration of the 3d skyrmion is obtained by solving the field sonctraint in terms of the space coordinates; it is given by Eq. (26) with the local space time fields mξ and γξ thought of as follows


with r=x2+y2+z2, giving the radius of Sξ2, and R associated with the circle Sξ1 fibered over Sξ2; the value R=r corresponds to γ=π2 and R>>r to γ=π. Notice that γξ in Eq. (30) has a spherical symmetry as it is a function only of r (no angles α,β,γ). Moreover, as this configuration obeys sinγ0=0 and sinγ=0; we assume γ0=n0π and γ=nπ. Putting these relations back into (26), we obtain the following configuration


describing a compactification of the space Rξ3 into Sξ3 which is homotopic to Sn3. From this view, the n˜:ξn˜ξ is then a mapping from Sξ3 into Sn3 with topological charge given by the winding number characterising the wrapping Sn3 on Sξ3; and for which we have the property π3Sn3=Z. In this regards, recall that 3-spheres S3 have a Hopf fibration given by a circle S1 sitting over S2 (for short S3S1S2); this non trivial fibration can be viewed from the relation S3SU2 and the factorisation U1×SU2/U1 with the coset SU2/U1 identified with S2; and U1 with S1. Applying this fibration to Sξ3 and Sn3, it follows that n˜:Sξ3Sn3; and the same thing for the bases Sξ2Sn2 and for the fibers Sξ1Sn1. Returning to the topological current and the conserved topological charge Q=R3d3rJ0tr, notice that in space time the differential dnA expands like μnAdξμ; then using the duality relation Jνρτ=εμνρτJμ, we find, up to a normalisation by the volume of the 3-sphere π2/2, the expression of the topological current Jμξ in terms of the 3D skyrmion field


In terms of the angular variables, this current reads like Nναρβτγεμνρτ with N=12π2sinβsinγ2. From this current expression, we can determine the associated topological charge Q by space integration over the charge density


Because of its spherical symmetry, the space volume d3r can be substituted by 4πr2dr; then the charge Q reads as the integral 4π2π2γ0γsin2γdγ whose integration leads to the sum of two terms coming from the integration of sin2γ=1212cos2γ. The integral first reads as 1πγ0γ; by substituting γ0=n0π, it contributes like . The integral of the second tem gives 12πsin2γ0sin2γ; it vanishes identically. So the topological charge is given by


5. Effective dynamics of skyrmions

In this section, we investigate the effective dynamics of a point-like skyrmion in a ferromagnetic background field while focussing on the 2d configuration. First, we derive the effective equation of a rigid skyrmion and comment on the underlying effective Lagrangian. We also describe the similarity with the dynamics of an electron in a background electromagnetic field. Then, we study the effect of dissipation on the skyrmion dynamics.

5.1 Equation of a rigid skyrmion

To get the effective equation of motion of a rigid skyrmion, we start by the spin 0+1D action Sspin=dtLspin describing the time evolution of a coherent spin vector modeled by a rotating magnetic moment nt with velocity ṅ=dndt; and make some accommodations. For that, recall that the Lagrangian Lspin has the structure LBSγH where LB is the Berry term having the form LB=qeA.ṅ with geometric (Berry) potential Anṅ; and where H is the Hamiltonian of the magnetic moment nt obeying the constraint n2=1. This magnetisation constraint is solved by two free angles αt,βt; they appear in the Berry term LB=S1cosβdt. Below, we think of the above magnetisation as a ferromagnetic background nr filling the spatial region of Rξ1,d with coordinates ξ=tr; and of the skymion as a massive point- like particle Rt moving in this background.

5.1.1 Rigid skyrmion

We begin by introducing the variables describing the skyrmion in the magnetic background field nr. We denote by Ms the mass of the skyrmion, and by R and Ṙ its space position and its velocity. For concreteness, we restrict the investigation to the spacetime Rξ1,2 and refer to R by the components Xi=XY and to r by the components xi=xy. Because of the Euclidean metric δij; we often we use both notations Xi and Xi=δijXj without referring to δij. Furthermore; we limit the discussion to the interesting case where the only source of displacements in Rξ1,2 is due to the skyrmion Rt (rigid skyrmion). In this picture, the description of the skyrmion Rt dissolved in the background magnet nr is given by


In this representation, the velocity ṅ of the skyrmion dissolved in the background magnet can be expressed into manners; either like ẊinXi; or as Ẋinxi; this is because Xi=xi. With this parametrisation, the dynamics of the skyrmion is described by an action Ss=dtLs with Lagrangian given by a space integral Ls=sa2d2rLs and spacetime density as follows


In this relation, the density LB=1cosβαt where now the angular variables are spacetime fields βtr and αtr. Similarly, the density H is the Hamiltonian density with arguments as Hnμnr and magnetic n as in Eq. (35). In this field action Ss, the prefactor a2 is required by the continuum limit of lattice Hamiltonians Hlatt living on a square lattice with spacing parameter a. Recall that for these Hlatt‘s, one generally has discrete sums like μ, μ,ν and so on; in the limit where a is too small, these sums turn into 2D space integrals like a2d2r. To fix ideas, we illustrate this limit on the typical hamiltonian HHDMZ, it describes the Heisenberg model on the lattice Z2 augmented by the Dzyaloshinskii-Moriya and the Zeeman interactions [66, 67]


with rν=rμ+aeνμ; that is eνμ=rνrμ/a where a is the square lattice parameter. So, the continuum limit H of this lattice Hamiltonian involves the target space metric δab and the topological Levi-Civita tensor εabc of the target space Rn3; it involves as well the metric gμν and the completely antisymmetry εμνρ of the space time Rξ1,2. In terms of δab and εabc tensors, the continuous hamiltonian density reads as follows


with νρ=eνμ.. Below, we set J=1 and, to factorise out the normalisation factor a2, scale the parameters of the model like dμa=ad˜μa and B=a2B˜. For simplicity, we sometimes set as well a=1.

5.1.2 Skyrmion equation without dissipation

To get the effective field equation of motion of the point-like skyrmion without dissipation, we calculate the vanishing condition of the functional variation of the action; that is δdtd2rL=0. General arguments indicate that the effective equation of the skyrmion with topological charge qs in the background magnet has the form


from which one can wonder the effective Lagrangian describing the effective dynamics of the skyrmion. It is given by


Notice that the right hand of Eq. (39) looks like the usual Lorentz force (qeE+qeṙB) of a moving electron with qe in an external electromagnetic field EiBi; the corresponding Lagrangian is m2ṙ2+qeB.rṙqeE.r. This similarity between the skyrmion and the electron in background fields is because the skyrmion has a topological charge qs that can be put in correspondence with qe; and, in the same way, the background field magnet Ei,Bi can be also put in correspondence with the electromagnetic field EiBi. To rigourously derive the spacetime Eqs. (39) and (40), we need to perform some manipulations relying on computing the effective expression of Ss=Sdtd2rLs and its time variation δSs=0. However, as Ls has two terms like γSHSLB, the calculations can be split in two stages; the first stage concerns the block γSd2rH with Hnμnr which is a function of the magnetic texture (35); that is nrR. The second stage regards the determination of the integral Sd2rLB. The computation of the first term is straightforwardly identified; by performing a space shift rr+R, the Hamiltonian density becomes Hnμnr+R with nr and where the dependence in R becomes explicit; thus allowing to think of the integral γSd2rH as nothing but the scalar energy potential VR=d2rHtrR. So, we have


Concerning the calculation of the Sd2rLB, the situation is somehow subtle; we do it in two steps; first we calculate the δd2rLB because we know the variation δLBδna which is equal to 12εabcnbṅc. Then, we turn backward to determine Sd2rLB by integration. To that purpose, recall also that the Berry term LB is given by 1cosβαt; and its variation δLBδnanaXjXj is equal to 12εabcnbṅc. To determine the time variation δLB=δd2rδLB, we first expand it like d2rδLδnaδna; and use δna=naXjXj to put it into the form -d2rδLBδnanaXjXj. Then, substituting δLBδna by its expression 12εabcnbṅc with ṅc expanded like -ncXiẊi, we end up with


Next, using the relation εijd2r=dxidxj, the first factor becomes 12εabcnadnbdnc gives precisely the skyrmion topological charge qs. So, the resulting δLB reduces to 2qsSẊδYẎδX that reads also like


This variation is very remarkable because it is contained in the variation of the effective coupling LBint=2qsSεijẊiXj which can be presented like LBint=qsAiẊi where we have set Ai=2SεzijXj; this vector can be interpreted as the vector potential of an effective magnetic field Bz=12εzijiAi. By adding the kinetic term Ms2ẊiẊi, we end up with an effective Lagrangian LB associated with the Berry term; it reads as follows LB=Ms2δijẊiẊjqsAiẊi. So, the effective Lagrangian Leff describing the rigid 2d skyrmion in a ferromagnet is


From this Lagrangian, we learn the equation of the motion of the rigid skyrmion namely MsX¨j=fj+4qsSεzijẊi; for the limit Ms=0, it reduces to Ẋi=14qsSεzjifj.

5.2 Implementing dissipation

So far we have considered magnetic moment obeying the constraint n2=1 with time evolution given by the LLequation ṅ=γfn where the force f=Hn. Using this equation, we deduce the typical properties n.ṅ=f.ṅ=0 from which we learn that the time variation dHdt of the Hamiltonian, which reads as d2rHnaṅa, vanishes identically as explicitly exhibited below,


In presence of dissipation, we loose energy; and so one expects that dHdt<0; indicating that the rigid skyrmion has a damped dynamics. In what follows, we study the effect of dissipation in the ferromagnet and derive the damped skyrmion equation.

5.2.1 Landau-Lifshitz-Gilbert equation

Due to dissipation, the force F acting on the rigid skyrmion Rt has two terms, the old conservative f=Hn; and an extra force δf linearly dependent in magnetisation velocity ṅ. Due to this extra force δf=αγṅ, the LL equation gets modified; its deformed expression is obtained by shifting the old force f like F=fαγṅ with α a positive damping parameter (Gilbert parameter). As such, the previous LL relation gives the so called Landau-Lifschitz-Gilbert (LLG) equation [68, 69]


where its both sides have ṅ. From this generalised relation, we still have n.ṅ=0 (ensuring n2=1); however f.ṅ0 as it is equal to the Gilbert term namely αfn.ṅ. Notice that Eq. (46) still describe a rotating magnetic moment in the target space (dn=0); but with a different angular velocity Ω which, in addition to f, depends moreover on the Gilbert parameter and the magnetisation n. By factorising Eq. (46) like ṅ=Ωn, we find


Notice that in presence of dissipation (α0), the variation of the hamiltonian dHdt given by (45) is no longer non vanishing; by first replacing f.ṅ=αfn.ṅ and putting back in it, we get


then, substituting γfn=αṅnṅ; we find that dHdt is given by αsd2r.ṅ2 indicating that dHdt<0; and consequently a decreasing energy Ht (loss of energy) while increasing time.

5.2.2 Damped skyrmion equation

To obtain the damped skyrmion equation due to the Gilbert term, we consider the rigid magnetic moment nrRt; and compute the expression of the skyrmion velocity Ṙ in terms of the conservative force f and the parameter α. To that purpose we start from Eq. (46) and multiply both equation sides by dn while assuming f.n=0 (the conservative force transverse to magnetisation), we get ṅdn=γf.dnn+αdn.ṅn. Then, multiply scalarly by n, which corresponds to a projection along the magnetisation, we obtain


Substituting dn and ṅ by their expansions dxiin and Ẋiin, then multiplying by dxl; we end up with a relation involving dxjdxl (which reads as εzjld2r); so we have


where we have set J0=12πεzijn.injn, defining the magnetization density, and where we have replaced in.jn by δijkn2. By integrating over the 2d space while using J0d2r=4πqs and setting ηj=14πd2rjn2η, we arrive at the relation


with εzxy=εzxy=1. The remaining step is to replace the conservative force f by Hn and proceeds in performing the integral over f.jn. Because of the explicit dependence into r, the f.jn can be expressed like jexpHjtotH; the explicit derivation term jexpH has been added because the Hamiltonian density has an explicit dependence Hnμnr. Recall that jtotH is given by jexpH+Hn.jn which is equal to jexpHf.jn. Notice also that the term jexpH can be also expressed like HR. Therefore, the integral f.jnd2r has two contributions namely the jtotHd2r which, being a total derivative, vanishes identically; and the term jexpHd2r that gives VR. Putting this value back into (51), we end up with


Implementing the kinetic term of the skyrmion, we obtain the equation with dissipation MsR¨=VR+GzṘηαqsṘ where the constant G=4πSqsa2 stands for the gyrostropic constant.


6. Electron-skyrmion interaction

In this section, we investigate the interacting dynamics between electrons and skyrmions with spin transfer torque (STT) [70]. The electron-skyrmion interaction is given by Hund coupling JHΨσaΨ.na which leads to emergent SU2 gauge potential that mediate the interaction between the spin texture ntr and the two spin states ΨΨ of the electron. We also study other aspects of electron/skyrmion system like the limit of large Hund coupling; and the derivation of the effective equation of motion of rigid skyrmions with STT.

6.1 Hund coupling

We start by recalling that a magnetic atom (like iron, manganese, …) can be modeled by a localized magnetic moment ntr and mobile carriers represented by a two spin component field Ψtr; the components of the fields n and Ψ are respectively given by natr with a=1,2,3; and by Ψαtr with α=. Using the electronic vector density je=ΨσΨ, the interaction between localised and itinerant electrons of the magnetic atom are bound by the Hund coupling reading as with Hund parameter JH>0 promoting alignment of n and je. So, the dynamics of the interacting electron with the backround n is given by the Lagrangian density Le=ΨitΨHen expanding as follows


where P=i and σ.n=σxnx+σyny+σznz.

6.1.1 Emergent gauge potential

Because of the ferromagnetic Hund coupling (JH>0), the spin observable Ŝez=2σz of the conduction electron tends to align with the orientation σn=σ.n of the magnetisation n — with angle θ=ez,n̂; this alinement is accompanied by a local phase change of the electronic wave function Ψ which becomes ψ=UΨ where Utr=eiΘtr is a unitary SU2 transformation mapping σz into σn; that is σn=UσzU. For later use, we refer to the new two components of the electronic field like ψ+n,ψn (for short ψα̇ with label α̇=±) such that the gauge transformation reads as ψα̇=Uα̇αΨα; that is ψ±=U±Ψ+U±Ψ. This local rotation of the electronic spin wave induces a non abelian gauge potential with components Aμ=iUμU mediating the interaction between the electron and the magnetic texture. Indeed, putting the unitary change into LeΨn, we end up with an equivalent Lagrangian density; but now with new field variables as follows


Here, the vector potential matrix Aμ is valued in the SU2 Lie algebra generated by the Pauli matrices σa; so it can be expanded as Aμxσx+Aμyσy+Aμzσz with components Aμa=12TrσaAμ. Notice that in going from the old LeΨn to the new L˜eψAμ, the spin texture n has disappeared; but not completely as it is manifested by an emergent non abelian gauge potential Aμ; so everything is as if we have an electron interacting with an external field Aμ. To get the explicit relation between the gauge potential and the magnetisation, we use the isomorphism SU2S3 and the Hopf fibration S1×S2 to write the unitary matrix U as follows


where the factor e describes S1 and where, for later use, we have set Wμ±=Aμ1±iAμ2 and μ=Aμ3. So, a specific realisation of the gauge transformation is given by fixing γ=cst (say γ=0); it corresponds to restricting S3 down to S2 and SU2 reduces down to SU2/U1. In this parametrisation, we can also express the unitary matrix U like m.σ with magnetic vector m=sinθ2cosφsinθ2sinφcosθ2 obeying the property m2=1; the same constraint as before. By putting back into .nU, and using some algebraic relations like εabdεdce=δacδbeδbcδae, we obtain 2m.nmn.σ. Then, substituting n by its expression sinθcosφsinθsinφcosθ, we end up with the desired direction σz appearing in Eq. (54). On the other hand, by putting U=m.σ back into iUμU, we obtain an explicit relation between the gauge potential and the magnetic texture namely Aμa=εabcmbμmc. From this expression, we learn the entries of the potential matrix Aμ of Eq. (55); the relation with the texture n is given in what follows seen that mθ=nθ/2.

6.1.2 Large Hund coupling limit

We start by noticing that the non abelian gauge potential Aμa obtained above can be expressed in a condensed form like εabcmaμmb (for short mμm); so it is normal to m; and then it can be expanded as follows


where we have used the local basis vectors mθ,eθ and fθ. This is an orthogonal triad which turn out to be intimately related with the triad vectors given by Eq. (5); the relationships read respectively like nθ/2,uθ/2 and vθ/2 involving θ/2 angle instead of θ. Substituting these basis vectors by their angular values, we obtain


from which we learn that the two first components combine in a complex gauge field Wμ±=Aμ1±iAμ2 which is equal to i2ew±μn with w±=u±iv; and the third component Aμ3 has the remarkable form 121cosθμφ whose structure recalls the geometric Berry term (7). Below, we set Aμ3=μ as in Eq. (55); it contains the temporal component 0 and the three spatial ones i — denoted in Section 2 respectively as a0 and ai—.

In the large Hund coupling (JH>>1), the spin of the electron is quasi- aligned with the magnetisation n; so the electronic dynamics is mainly described by the chiral wave function ψ+0 denoted below as χ=χ0. Thus, the effective properties of the interaction between the electron and the skyrmion can be obtained by restricting the above relations to the polarised electronic spin wave χ. By setting ψ=0 into Eq. (54) and using χσxχ=χσyχ=0 and χσzχ=χ¯χ as well as replacing Aμxσx2+Aμyσy2 by 14μn2, the Lagrangian (54) reduces to the polarised Lepol=LeχnZμ given by


where 0i define the four components of the emergent abelian gauge prepotential μ associated with the Pauli matrix σz; their explicit expressions are given by 0=121cosθφ̇ and i=121cosθiφ; their variation with respect to the magnetic texture are related to the magnetisation field like δμδn=12μnn.

6.2 Skyrmion with spin transfer torque

Here, we investigate the full dynamics of the electron/skyrmion system en described by the Lagrangian density Ltot containing the parts Ln+Len; the electronic Lagrangian Len is given by Eq. (54). The Lagrangian Ln, describing the skyrmion dynamics, is as in eqs (5)(7) namely S0Hn with 0=121cosθφ̇. By setting H˜n=Hn+28mμn2ψψ, the full Lagrangian density Ltot with can be then presented like L˜ψμH˜n like


with Pi+iσz2 expanding as Pi2+i22+Pii+iPiσz. The equations of motion of ψ and n are obtained as usual by computing the extremisation of this Lagrangian density with respect to the corresponding field variables. In general, we have δLtot=δLtot/δn.δn+δLtot/δψ.δψ+hc which vanishes for δLtot/δn=0 and δLtot/δψ=0.

6.2.1 Modified Landau-Lifshitz equation

Regarding the spin texture n, the associated field equation of motion is given by δLtot/δn=0; the contributions to this equation of motion come from the variations L˜ and H˜n with respect to δn namely


The variation δHnδn depends on the structure of the skyrmion Hamiltonian density H˜; its contribution to the equation of motion can be presented like λμμn=F with some factor λ. However, the variation δLδμδμδn describes skyrmion-electron interaction; and can be done explicitly into two steps; the first step concerns the calculation of the time like component δLδ0δ0δn; it gives 22S+ψσzψṅn; it is normal to n and to velocity ṅ and involves the eletron spin density ρez=ψσzψ.

The second step deals with the calculation of the space like component δLδiδiδn; the factor δLδi gives Ji with a 3-component current vector density reading as follows


This vector two remarkable properties: 1 it is given by the sum of two contributions as it it reads like Ji+n+Jin with


These vectors are respectively interpreted as two spin polarised currents; the Ji+n is associated with the ψ+n wave function as it points in the same direction as n; the Jin is however associated with ψn pointing in the opposite direction of n. 2 Each one of the two J+n and Jn are in turn given by the sum of two contributions as they can be respectively split like mψ¯+nψ+n+jψ+n and mψ¯nψn+jψn with vector density jψ standing for the usual current vector jψ=12mψ¯Pψ. The contribution mψ¯ψ is proportional to the emergent gauge field ; it defines a spin torque transfert to the vector current density Ji.

Regarding the factor δiδn, it gives 12inn; by substituting, the total contribution of δLδiδiδn leads to 2Jiinn that reads in a condensed form like 2J.nn. Putting back into Eq. (60), we end up with the following modified LL equation


To compare this equation with the usual LL equation (Sṅ=δHnδnn) in absence of Hund coupling (which corresponds to putting ψ to zero), we multiply Eq. (63) by n in order to bring it to a comparable relation with LL equation. By setting ρez=ψσzψ, describing the electronic spin density ψ+n2ψn2; we find


where, due to n2=1, the space gradient J.n is normal to n; and so it can be set as Ωen with Ωe=Jiωie. The above equation is a modified LL equation; it describes the dynamics of the spin texture interacting with electrons through Hund coupling. Notice that for ψ0, this equation reduces to Sadṅ=ωnn showing that the vector n rotates with ωn=δHnδn. By turning on ψ, we have ṅωn+Ωen indicating that the LL rotation is drifted by Ωe coming from two sources: i the term J.n which deforms LL vector ωn drifted by the nJ.n; and ii the electronic spin density ρez=Nead; this term adds to the density Sad of the magnetic texture per unit volume; it involves the number Ne=Ne+nNen with Ne±n standing for the filling factor of polarized conduction electrons. Moreover, if assuming ntr=nrVst with a uniform Vs, then the drift velocity ṅa=inaVsi and Jeiina=Jea. Putting back into the modified LLG equation, we end up with the following relation between the Vs and ve velocities S+ne2vsa=nevea where we have set inaVsi=vsa and Jea=nevea.

6.2.2 Rigid skyrmion under spin transfer torque

Here, we investigate the dynamics of a 2D rigid skyrmion [n=nrR] under a spin transfer torque (STT) induced by itinerant electrons. For that, we apply the method, used in sub-subSection 5.1.2 to derive Ls from the computation space integral of d2rLs and Eq. (36). To begin, recall that in absence of the STT effect, the Lagrangian Ls of the 2D skyrmion’s point- particle, with position R=XY and velocity Ṙ=ẊẎ, is given by Ms2Ṙ2G2z.RṘVR with effective scalar energy potential VR=d2rHrR and a constant G=4πa2qsS. Under STT induced by Hund coupling, the Lagrangian Ls gets deformed into L˜s=Ls+ΔLs, that is


To determine ΔLs, we start from L˜s=d2rL˜tot with Lagrangian density as L˜tot=L˜H˜n with L˜ given by Eq. (59). For convenience, we set L˜=S0+L˜en and set


The deviation ΔLs with respect to Ls in (65) comes from those terms in Eq. (66). Notice that this expression involves the wave function ψ coupled to the emergent gauge potential field μ=0i; that is d2rψσzψ0 and 12md2rψPi+iσz2ψ. Thus, to obtain ΔLs, we first calculate the variation δΔLsδμδμ and put δμ=δμδR.δR. Once, we have the explicit expression of this variation, we turn backward to deduce the value of ΔLs. To that purpose, we proceed in two steps as follows: i We calculate the temporal contribution δΔLsδ0δ0δR.δR; and ii we compute the spatial δΔLsδiδiδR.δR. Using the variation δ0=12δn.njnẊj, the contribution of the first term can be put as follows


where we have set ρz=ψσzψ and J0z=d2rρz2εzkln.knln. Notice that the right hand side in above relation can be also put into the form 2J0zεzijδẊiXjδ2εzijJ0zẊiXj indicating that ΔLs must contain the term 2εzijJ0zẊiXj which reads as well like 2J0z.ṘR. Regarding the spatial part δΔLsδi.δiδXlδXl, we have quite similar calculations allowing to put it in the following form


where we have set Jzit=d2rJzitr with Jzitr given by Eq. (61). Here also notice that the right hand of above equation can be put as well like δεzijJziXj indicating that ΔLs contains in addition to 2J0z.ṘR, the term εzijJziXj which reads also as z.JR with two component vector J=JzxJzy. Thus, we have the following modified skyrmion equation


from which we determine the modified equation of motion of the rigid skyrmion in presence of spin transfer torque.


7. Comments and perspectives

In this bookchapter, we have studied the basic aspects of the solitons dynamics in various 1+d spacetime dimensions with d=1,2,3; while emphasizing the analysis of their topological properties and their interaction with the environment. After having introduced the quantum SU2 spins, their coherent vector representation S=RαβγS0 with S0 standing for the highest weight spin state; and their link with the magnetic moments μSn, we have revisited the time evolution of coherent spin states; and proceeded by investigating their spatial distribution while focusing on kinks, 2d and 3d skyrmions. We have also considered the rigid skyrmions dissolved in the magnetic texture without and with dissipation. Moreover, we explored the interaction between electrons and skyrmions and analyzed the effect of the spin transfer torque. In this regard, we have refined the results concerning the modified LL equation for the rigid skyrmion in connection with emergent non abelian SU2 gauge fields. It is found that the magnetic skyrmions, existing in a ferromagnetic (FM) medium, show interesting behaviors such as emergent electrodynamics [71] and current-driven motion at low current densities [72, 73]. Consequently, the attractive properties of ferromagnetic skyrmions make them promising candidates for high-density and low-power spintronic technology. Besides, ferromagnetic skyrmions have the potential to encode bits in low-power magnetic storage devices. Therefore, alternative technology of forming and controlling skyrmions is necessary for their use in device engineering. This investigation was performed by using the field theory method based on coherent spin states described by a constrained spacetime field captured by fn=1. Such condition supports the topological symmetry of magnetic solitons which is found to be characterised by integral topological charges Q that are interpreted in terms of magnetic skyrmions and antiskyrmion; these topological states can be imagined as (winding) quasiparticle excitations with Q>0 and Q<0 respectively.

Regarding these two skyrmionic configurations, it is interesting to notice that, unlike magnetic skyrmions, the missing rotational symmetry of antiskyrmions leads to anisotropic DMI, which is highly relevant for racetrack applications. It follows that antiskyrmions exist in certain Heusler materials having a particular type of DMI, including MnPtPdSn [36] and MnRhIrSn [74]. It is then deduced that stabilized antiskyrmions can be observed in materials exhibiting D2d symmetry such as layered systems with heavy metal atoms. Furthermore, the antiskyrmion show some interesting features, namely long lifetimes at room temperature and a parallel motion to the applied current [75]. Thus, antiskyrmions are easy to detect using conventional experimental techniques and can be considered as the carriers of information in racetrack devices.

To lift the limitations associated with ferromagnetic skyrmions for low-power spintronic devices, recent trends combine multiple subparticles in different magnetic surroundings. Stable room-temperature antiferromagnetic skyrmions in synthetic Pt/Co/Ru antiferromagnets result from the combination of two FM nano-objects coupled antiferromagnetically [76]. Compared to their ferromagnetic analogs, antiferromagnetic skyrmions exhibit different dynamics and are driven with several kilometers per second by currents. Coupling two subsystems with mutually reversed spins, gives rise to ferrimagnetic skyrmions as detected in GdFeCo films using scanning transmission X-ray microscopy [77]. At ambient temperature, these skyrmions move at a speed of 50m/s with a reduced skyrmion Hall angle of 20°. Characterized by uncompensated magnetization, the vanishing angular momentum line can be utilized as a self-focusing racetrack for skyrmions. Another technologically promising object is generated by the coexistence of skyrmions and antiskyrmions in materials with D2d symmetry. The resulting spin textures constitute information bits ‘0’ and ‘1’ generalizing the concept of racetrack device. Insensitive to the repulsive interaction between the two distinct nano-objects, such emergent devices are promising solution for racetrack storage applications.



L. B. Drissi would like to acknowledge “Académie Hassan II des Sciences et Techniques-Morocco”. She also acknowledges the Alexander von Humboldt Foundation for financial support via the George Forster Research Fellowship for experienced scientists (Ref. 3.4 - MAR - 1202992).


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  • For convenience, we often refer to σ→,e→i,σ→.e→i=σi respectively by bold symbols as σ, ei,σ.ei=σi.
  • For an electron with Zeeman field Ba, we have ωa=−gqe2meBa with g=2 and qe=−e.

Written By

Lalla Btissam Drissi, El Hassan Saidi, Mosto Bousmina and Omar Fassi-Fehri

Submitted: 08 December 2020 Reviewed: 01 March 2021 Published: 11 May 2021