Open access peer-reviewed chapter

# Magnetic Skyrmions: Theory and Applications

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

Submitted: December 8th 2020Reviewed: March 1st 2021Published: May 11th 2021

DOI: 10.5772/intechopen.96927

## Abstract

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.

### Keywords

• 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 Sand 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 SzSof spinfull particles are characterised by two half integers SzS, a positive S0and an Sztaking 2S+1values bounded as SSzSwith integral hoppings. For particles with spin 1/2 like electrons, one distinguishes two basis vector states ±1212that are eigenvalues of the scaled Pauli matrix 2σzand the quadratic (Casimir) operator 24a=13σa2, here the three 2σawith σa=σ.eaare 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.ezwith ez=0,0,1T. For generic values of the SU2spin S, the spin operator reads as Jawhere the three Ja‘s are 2S+1×2S+1generators of the SU2group satisfying the usual commutation relations JaJb=iεabcJcwith εabcstanding for the completely antisymmetric Levi-Civita tensor with non zero value ε123=1; its inverse is εcbawith ε123=1. The time evolution of the spin 12operator σa2with 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 12operator Ŝat(the hat is to distinguish the operator Ŝafrom classical Sa) is given by

Ŝa=eiHtσa2eiHtE1

where the Pauli matrices σaobey the usual commutation relations σaσb=2iεabcσc. For a generic value of the SU2spin S, the above relation extends as Ŝa=eiHtJaeiHt. So, many relations for the spin 1/2may be straightforwardly generalised for generic values Sof the SU2spin. For example, for a spin value S0, the 2S0+1states are given by mS0and are labeled by S0mS0; one of these states namely S0S0is 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=24Iis time independent; and then the time dynamics of Ŝatis rotational in the sense that dŜadtis given by a commutator as follows dŜadt=iHŜaŜaH.For the example where His a linearly dependent function of Ŝalike 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 Ŝareads, after using the commutation relation ŜaŜb=iεabcŜc, as follows

where appears the Levi-Civita εabcwhich, 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 εabcSbSμ1μ2cdμ3μD2aεμ1μDwhere, for convenience, we have set Sμ1μ2c=eμ1μ2.Scwith eμ1μ2=eμ2eμ1.To distinguish these two Levi-Civita tensors, we refer to εabcas the target space Levi-Civita with SO3targetsymmetry; and to εμ1μDas the spacetime Levi-Civita with SO1D1Lorentz symmetry containing as subsymmetry the usual space rotation group SOD1space. Notice also that for the case where the Hamiltonian HŜis a general function of the spin, the vector ωais 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 Ŝtevolved by a Hamiltonian HŜ, we use the algebra ŜaŜb=iεabcŜcto 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 Sand a direction nrelated to a given unit vector n0as n=Rαβγn0;and parameterised by α,β,γ. In the above relation, the n0is thought of as the north direction of a 2-sphere Sn2given by the canonical vector 0,0,1T; it is invariant under the proper rotation; i.e. Rzγn0=n0; and consequently the generic nis independent of γ; i.e.: n=Rαβn0.Recall that the 3×3 matrix Rαβγis an SO3rotation [SO3SU2] generating all other points of Sn2parameterised by αβ. In this regards, it is interesting to recall some useful properties that we list here after as three points: 1the rotation matrix Rαβγcan be factorised like RzαRyβRzγwhere each Raψais a rotation eiψaJaaround the a- axis with an angle ψaand generator Ja. 2As the unit n0is an eigen vector of eJz; it follows that nreduces to eJzeJyn0; this generic vector obeys as well the constraint nn=1and is solved as follows.

n=sinβcosαsinβsinαcosβE3

with 0α2πand 0βπ; they parameterise the unit 2-sphere Sn2which is isomorphic to SU2/U1; the missing angle γparameterises a circle Sn1,isomorphic to U1,that is fibred over Sn2.3the 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=Sn0considered above (S0HWS S0S0). In this regards, recall that the Ŝaacts on classical 3-vectors Vbthrough its 3×3 matrix representation like ŜaVb=JcabVcwith Jcabgiven by iεabc; these Jc‘s are precisely the generators of the SU2matrix representation Rαβγ; by replacing Vbby the operator Ŝb, one discovers the SU2spin algebra ŜaŜb=iεabcŜc. Notice also that the classical spin vector S=Sncan be also put in correspondence with the usual magnetic moment μ=γS(with γ=ge2mthe 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 ntparameterizing the 2-sphere Sn2.

nx2t+ny2t+nz2t=1nt=1E4

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 nand dnare normal vectors); by implementing time, the variation n.dngets mapped into n.ṅ=0teaching us that the velocity ṅis carried by uand v; two normal directions to nwith components;

ua=cosβcosαcosβsinαsinβ,va=sinαcosα0E5

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 nagoverned by a classical Hamiltonian Hnaαβ. The resulting time evolution is given by the so called Landau-Lifshitz (LL) equation [61]; it reads as dnadt=γμεabcbHncwith bH=Hnb.By using the relations =uadnaand sinβdα=vadnawith ua=naβ; and vasinβ=naα; as well as the expressions εabcuanc=vband εabcvanc=ub, the above LL equation splits into two time evolution equations dt=γvbbHand sinβdt=γubbH. These time evolutions can be also put into the form

sinβdt+γHα=0,sinβdtγHβ=0E6

and can be identified with the Euler- Lagrange equations following from the variation δS=0of an action S=Ldt. Here, the Lagrangian is related to the Hamiltonian like L=LBHnaαβwhere LBis 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 SγHsuch that the spin lagrangian takes the form Lspin=LBSγH. To determine LB, we identify the Eq. (6) with the extremal variation δS/δβ=0and δS/δα=0. Straightforward calculations leads to

LB=S1cosβdtE7

showing that αand βform a conjugate pair. By substituting sinβdt=vadnadtback into above LB, we find that the Berry term has the form of Aharonov-Bohm coupling LAB=qeAadnadtwith magnetic potential vector Aagiven by Aa=Sqe1cosβsinβva.However, this potential vector is suggestive as it has the same form as the potential vector Amonopole=Sqer1cosβsinβvof a magnetic monopole. The curl of this potential is given by B=qmrr3with magnetic charge qm=Sqelocated at the centre of the 2-sphere; the flux Φof this field through the unit sphere is then equal to 4πSqe; and reads as 2SΦ0with a unit flux quanta Φ0=hqeas indicated by the value S=1/2. So, because 2S=nis 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 Sand direction described by ntas 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+0Dvector field; that is a vector belonging to R1,dwith d=0(no space direction). In this section, we first turn on 1d space coordinate xand promotes the old unit- direction ntto 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: 1the target space Rn3parameterised by na=n1n2n3with Euclidian metric δaband topological Levi-Civita εabc. 2the spacetime Rξ1,1parameterised by ξμ=tx,concerning the 1d kink evolution; and the spacetime Rξ1,2parameterised 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 δaband εabcare respectively given by the usual Lorentzian spacetime metric gμν, with signature like gμνξμξν=x2+y2t2, and the spacetime Levi-Civita εμνρwith ε012=1.

### 3.1 One space dimensional solitons

In 1+1D spacetime, the local coordinates parameterising Rξ1,1are given by ξμ=tx; so the metric is restricted to gμνξμξν=x2t2. The field variable naξhas in general three components n1n2n3as 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=n1n2satisfying the constraint equation n.n=1at 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. 1The constraint n12+n22=1is invariant SO2nrotations acting as n'a=Rbanbwith orthogonal rotation matrix

Rba=cosψsinψsinψcosψ,RTR=IE8

The constraint nana=1can be also presented like N¯N=1with Nstanding for the complex field n1+in2that reads also like e. In this complex notation, the symmetry of the constraint is given by the phase change acting as NUNwith U=eand corresponding to the shift αα+ψ. Moreover the correspondence n1n2n1+in2describes precisely the well known isomorphisms SO2U1Sn1where Sn1is a circle; it is precisely the equatorial circle of the 2-sphere Sn2considered in previous section. 2As for Eq. (5), the constraint nana=1leads to nadna=0; and so describes a rotational movement encoded in the relation dna=εabnbwhere εabis the standard 2D antisymmetric tensor with ε21=ε12=1; this εabis related to the previous 3D Levi-Civita like εzab. Notice also that the constraint nana=1implies 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

dn1dn2=Jdtdx,J=εμνμn1νn2E9

where εμνis the antisymmetric tensor in 1+1 spacetime, and Jis the Jacobian of the transformation txn1n2. 3The condition nana=1can 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=cosαsinαfrom which we deduce the normal direction ua=dnareading as ua=sinαcosα. In term of the complex field; we have N=eand 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 naξis described by a field action S=dtLwith Lagrangian L=dxLand density L; this field density is given by 12μnaμnaVnΛnana1with μ=ξμ; it reads in terms of the Hamiltonian density as follows

L=πaṅaHE10

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 Vnis a potential energy density which play an important role for describing 1d kinks with finite size. Notice also that the variation δSδΛ=0gives precisely the constraint nana=1while the δSδna=0gives the spacetime dynamics of nadescribed 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=0that expands like x2t2α=0; it is invariant under spacetime translations with conserved current symmetry μTμν=0with Tμνstanding for the energy momentum tensor given by the 2×2 symmetric matrix μανα+gμνL. The energy density T00is given by 12tα2+12xα2and the momentum density T10reads as xφxφ. Focussing on T00, the conserved energy Ereads then as follows

E=12+dxtα2+xα20E11

with minimum corresponding to constant field (α=cte). Notice that general solutions of μμα=0are 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

φtx=πtanhx+tλE12

where λis a positive parameter representing the width where the soliton αtxacquires a significant variation. Notice that for a given t, the field varies from αt=πto αt+=πregardless 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 Jμtxgoing beyond the spacetime translations generated by the energy momentum tensor Tμν. The conserved spacetime current Jμ=J0J1of 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 J0like 12π1αand of the current density as J1=12π0α. This conserved current is a topological 1+1Dspacetime vector Jμthat is manifestly conserved; this feature follows from the relation between Jμand the antisymmetric εμνas follows [57],

Jμ=12πεμνναE13

Because of the εμν; the continuity relation μJμ=12πεμνμναvanishes identically due to the antisymmetry property of εμν. The particularity of the above conserved Jμis its topological nature; it is due to the constraint nana=1without recourse to the solution na=cosαsinα. Indeed, Eq. (13) can be derived by computing the Jacobian J=detnaξμof the mapping from the 2d spacetime coordinates txto the target space fields (n1,n2). Recall that the spacetime area dtdxcan be written in terms of εμνlike 12εμνdξμdξνand, similarly, the target space area dn1dn2can be expressed in terms of as εabfollows 12εabdnadnb. The Jacobian Jis precisely given by (9); and can be presented into a covariant form like J=12εμνμnaνnbεab. This expression of the Jacobian Jcaptures important informations; in particular the three following ones. 1It can be expressed as a total divergence like μπJμwith spacetime vector

Jμ=12πεμνnaνnbεabE14

and where 1πis 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=]. 2Because of the constraint dn2=nnn2dn1following from nana=1, the Jacobian Jvanishes identically; thus leading to the conservation law μJμ=0; i.e. J=0and then μJμ=0. 3The conserved charge Qassociated 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+dxxαtxand after integration leads to

Q=12παtαtE15

Moreover, seen that αtis an angular variable parameterising Sn1; it may be subject to a boundary condition like for instance the periodic αt=αt+2πNwith N an integer; this leads to an integral topological charge Q=Ninterpreted as the winding number of the circle. In this regards, notice that: ithe 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 Sx1with angular coordinate πφπ; so, the integral 12π+dxxαtxgets replaced by 12ππ+παφ; and then the mapping αt:φαtφis a mapping between two circles namely Sx1Sn1; the field αtφthen describes a soliton (one space extended object) wrapping the circle Sx1N times; this propery is captured by πSx1Sn1=, a homotopy group property [63]. iiThe charge Qis independent of the Lagrangian of the system as it follows completely from the field constraint without any reference to the field action. iiiUnder a scale transformation ξ'=ξ/λwith a scaling parameter λ>0, the topological charge of the field (12) is invariant; but its total energy (11) get scaled as follows

Q=Q,E=1λEE16

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 Vnin 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=g161cos4αleading to the well known sine-Gordon Eq. [64, 65] namely μμαg4sin4α=0with the symmetry property αα+π2. So, the solitonic solution is periodic with period π2; that is the quarter of the old 2πperiod of the free field case. For static field αx, the sine Gordon equation reduces to d2αdx2M24sin4α=0; its solution for M>0is given by arctanexpMxrepresenting a sine- Gordon field evolving from 0to π2and describing a kink with topological charge Q=14.For M<0, the soliton is an anti-kink evolving from π2to 0with 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 n3so that the skyrmion field nis a real 3-vector with three components n1n2n3constrained 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+1space time with Lorentzian metric and coordinates ξμ=txy. This means that dn=μndξμ; explicitly dn=ntdt+nxdx+nydy.

#### 3.2.1 Dzyaloshinskii-Moriya potential

The field action S3D=dtL3Ddescribing the space time dynamics of ntxyhas the same structure as Eq. (10); except that here the Lagrangian L3Dinvolves two space variable like dxdyL3Dand the density L3D=12μn2VnΛn.n1; this is a function of the constrained 3-vector nand its space time gradient μn; it reads in term of the Hamiltonian density as follows.

L3D=π.ṅH3DnE17

In this expression, the H3Dnis the continuous limit of a lattice Hamiltonian Hlatt(narμinvolving, amongst others, the Heisenberg term, the Dyaloshinskii- Moriya (DM) interaction and the Zeeman coupling. The Vnin the first expression of L3Dis 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 Λξis an auxiliary 3D spacetime field; it is a Lagrange multiplier that carries the constraint n.n=1which plays the same role as in subSection 3.1. By variating this action with respect to the fields nand Λ; we get from δS3DδΛ=0precisely the field constraint n.n=1; and from δS3Dδn=0the following Euler–Lagrange equation Wn=Vn+Λn.For later use, we express this field equation like

μμna=Vna+ΛnaE18

The interest into this (18) is twice; first it can be put into the equivalent form μμna=εabcDbncwhere Dbis an operator acting on ncto 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=sinβcosαsinβsinαcosβand

ua=cosβcosαcosβsinαsinβ,va=sinαcosα0E19

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

dna=εabcωbncdn=ωnωndnE20

where the 1-form ωbis the rotation vector to be derived below. By substituting (20) back into nadna,we obtain εbcaωbncnawhich vanishes identically due to the property εbcancna=0. 2Having two degrees of freedom αand β, we can expand the differential dnalike ua+vasinαdαwith the two vector fields ua=naβand va=naαas given above. Notice that the three unit fields nuvplays an important role in this study; they form a vector basis of the field space; they obey the usual cross products namely n=uvand its homologue which given by cyclic permutations; for example,

ua=εabcvbnc,va=εabcubncE21

Putting these Eq. (21) back into the expansion of dnain 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=εabcωμbncwith ωμbgiven by vbμβubsinαμα.From this expression, we can compute the Laplacian μμna; which, by using the above relations; is equal to εabcμωμbncreading explicitly as εabcμωμbnc+ωμbμncor equivalently like μμna=εabcDbncwith operator Db=ωμbμ+μωμb. Notice that the above operator has an interesting geometric interpretation; by factorising ωμb, we can put it in the form ωμdDμdbwhere Dμdbappears as a gauge covariant derivative Dμdb=δdbμ+Aμdbwith a non trivial gauge potential Aμdbgiven by ωdμνωνb.Comparing with (18) with μμna=εabcDbnc, we obtain Vna=εabcDbncΛna;and then a scalar potential energy Vgiven by εabcdnaDbncΛnadna. The second term in this relation vanishes identically because nadna=0; thus reducing to

containing εabcnaDbncas a sub-term. In the end of this analysis, let us compare this sub-term with the εabcnbnμ1μ2cΔaμ1μ2with Δaμ1μ2=dμ3μD2aεμ1μDgiving 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=eμνε0μνand da=d0ae0as well as Da=da., one brings it to the form εabcnaDbncwhich 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 Jμ2Ddiscussed in subSection 2.1, which reads as 12πεμνnaνnbεab, one can wonder the structure of the 1+2D topological current Jμ3Dthat is associated with the 2d Skyrmion described by the 3-vector field naξ. It is given by

Jμ3D=18πεμνρnaνnbνncεabcE23

where εabcis as before and where εμνρis the completely antisymmetric Levi-Civita tensor in the 1+2D spacetime. The divergence μJμ3Dof the above spacetime vector vanishes identically; it has two remarkable properties that we want to comment before proceeding. 1The μJμ3Dis nothing but the determinant of the 3×3Jacobian matrix naξμrelating the three field variables nato the three spacetime coordinates ξμ; this Jacobian detnaξμis generally given by 13!εμνρμnaνnbνnbεabc;it maps the spacetime volume d3ξ=dtdxdyinto 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 εμνρand εabcintroduced earlier; and as noticed before play a central role in topology. The target space volume d3ncan 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 dnaby their expansions naξμdξμ; and putting back into d3n,we obtain the relation d3n=J3Dd3ξwhere J3Dis precisely the Jacobian detnaξμ. 2The conservation law μJμ3D=0has a geometric origin; it follows from the field constraint relation n12+n22+n32=1degenerating 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=0and then to the above continuity equation. Having the explicit expression (23) of the topological current Jμin terms of the magnetic texture field nξ, we turn to determine the associated topological charge Q=dxdyJ0with charge density J0given by 18πεabcε0ijinbjncna. Substituting ε0ijdxdyby dξidξj, we have J0dxdy=18πεabcnadnbdnc. Moreover using the differentials dnb=ub+vbsinαdα, we can calculate the area dnbdncin terms of the angles αand β;we find 2nasinαwhere we have used εabcubvcucvb=2na.So, the topological charge Qreads as 14πSn2sinβdαdβwhich is equal to 1. In fact this value is just the unit charge; the general value is an integer Q=Nwith N being the winding number π2Sn2;see below. Notice that J0can be also presented like

J0=εabc8πnanbxncynbyncxE24

Replacing naby their expression in terms of the angles sinβcosαsinβsinαcosβ, we can bring the above charge density J0into two equivalent relations; first into the form like sinβ4πβxαyβyβx; and second as 14παcosβxywhich is nothing but the Jacobian of the transformation from the xyspace to the unit 2-sphere with angular variables αβ. The explicit expression of n1n2n3in terms of the xyspace variables is given by

n1=2xx2+y2+1,n2=2yx2+y2+1,n3=x2+y21x2+y2+1E25

but this is nothing but the stereographic projection of the 2-sphere Sξ2on the real plane. So, the field nadefines a mapping between Sξ2towards Sn2with topological charge given by the winding number Sn2around 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 Rξ1,3parameterised by ξμ=txyz(with Lorentzian metric gμν). The spacetime dynamics of the 3d skyrmion is described by a four component field nAξobeying a constraint relation fn=1; here the fnis given by the quadratic form nAnAinvariant under SO4transformations 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 εABCDand the spacetime εμνρτextending 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 R1,3is described by a field action S4D=dtL4Dwith Lagrangian realized as the space integral dxdydzL4D. Generally, the Lagrangian density L4Dis a function of the soliton ntxyzwhich is a real 4-component field [n=n1n2n3n4] constrained like fnξ=1.For self ineracting field, the typical field expression of L4Dis given by 12μn2VnΛfn1where Vnis a scalar potential; and where the auxiliary field Λξis a Lagrange multiplier carrying the field constraint. This density L4Dreads 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.n1to describe the degrees of freedom of n. Being a unit 4-component vector, we can solve the constraint n.n=1in terms of three angular angles αβγ;by setting

n=msinγcosγ,m=sinβcosαsinβsinαcosβE26

where mis 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 R1,3to the 3D hyperplane z=const; and by fixing the component field γto 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 dnAin terms of ,,; we find the following

dna=macosγdγ+sinγua+vasinβdα,dn4=sinγdγE27

For convenience, we sometimes refer to the three αβγcollectively like αa=α1α2α3;so we have dnA=EaAdαbwith 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 nAsubject to the constraint relation nAnA=1; so the soliton has SO4SO31×SO32symmetry leaving invariant the condition nAnA=1that reads explicitly as n12+n22+n32+n42=1.The algebraic condition fn=1induces in turns the constraint equation df=0leading to nAdnA=0and showing that nAand dnAorthogonal 4-vectors in Rn4. From this constraint, we can construct nAdnBnBdnA/2which is a 4×4 antisymmetric matrix ΩABgenerating the SO4rotations; this ΩABcontains 3+3 degrees of freedom generating the two SO31and SO32making SO4; the first three degrres are given by Ωabwith 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=1describes a unit 3-sphere Sn3sitting 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 d4non Sn3is a key ingredient in the derivation of the topological current Jof 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 d4ncan be expressed as dJwith the 3-form Jgiven by

This 3-form describes precisely the topological current in the target space; this is because on Sn3, the 4-form d4nvanishes; and then dJvanishes. By solving, the skyrmion field constraint nAnA=1in terms of three angles αaas given by Eq. (26); with these angular coordinates, we have mapping f:Rn4Sn3with Sn3Sα3. By expanding the differentials like dnA=EaAdαawith EaA=nAαa; then the conserved current on the 3-sphere Sα3reads as follows

J=14!3!εABCDnAEbBEcCEdDεabcd3αE29

where we have substituted the 3-form dαbdαcdαdon the 3-sphere Sα3by 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 R1,3with coordinates ξμ=txyz, the 3d skyrmion is described by a four component field nAξand 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 mξand γξthought of as follows

mξ=xryrzr,γξ=arcsin2rRr2+R2=arccosr2R2r2+R2E30

with r=x2+y2+z2,giving the radius of Sξ2,and Rassociated with the circle Sξ1fibered over Sξ2; the value R=rcorresponds to γ=π2and R>>rto γ=π. Notice that γξin 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=0and sinγ=0; we assume γ0=n0πand γ=nπ. Putting these relations back into (26), we obtain the following configuration

n˜=2xRr2+R22yRr2+R22zRr2+R2r2R2r2+R2E31

describing a compactification of the space Rξ3into Sξ3which is homotopic to Sn3. From this view, the n˜:ξn˜ξis then a mapping from Sξ3into Sn3with topological charge given by the winding number characterising the wrapping Sn3on Sξ3; and for which we have the property π3Sn3=Z. In this regards, recall that 3-spheres S3have a Hopf fibration given by a circle S1sitting over S2(for short S3S1S2); this non trivial fibration can be viewed from the relation S3SU2and the factorisation U1×SU2/U1with the coset SU2/U1identified with S2; and U1with S1. Applying this fibration to Sξ3and Sn3, it follows that n˜:Sξ3Sn3;and the same thing for the bases Sξ2Sn2and for the fibers Sξ1Sn1. Returning to the topological current and the conserved topological charge Q=R3d3rJ0tr, notice that in space time the differential dnAexpands 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 Jμξin terms of the 3D skyrmion field

Jμ=112π2εμνρτnaνnbρncτncεabcdE32

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

J0tr=sin2γ2π2r2drE33

Because of its spherical symmetry, the space volume d3rcan be substituted by 4πr2dr; then the charge Qreads as the integral 4π2π2γ0γsin2γdγwhose 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

Q=γ0γπ=NE34

## 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+1Daction Sspin=dtLspindescribing the time evolution of a coherent spin vector modeled by a rotating magnetic moment ntwith velocity ṅ=dndt; and make some accommodations. For that, recall that the Lagrangian Lspinhas the structure LBSγHwhere LBis the Berry term having the form LB=qeA.ṅwith geometric (Berry) potential Anṅ; and where His the Hamiltonian of the magnetic moment ntobeying 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 nrfilling the spatial region of Rξ1,dwith coordinates ξ=tr; and of the skymion as a massive point- like particle Rtmoving 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 Msthe mass of the skyrmion, and by Rand Ṙits space position and its velocity. For concreteness, we restrict the investigation to the spacetime Rξ1,2and refer to Rby the components Xi=XYand to rby the components xi=xy. Because of the Euclidean metric δij; we often we use both notations Xiand Xi=δijXjwithout referring to δij. Furthermore; we limit the discussion to the interesting case where the only source of displacements in Rξ1,2is due to the skyrmion Rt(rigid skyrmion). In this picture, the description of the skyrmion Rtdissolved in the background magnet nris given by

nrt=nrRtE35

In this representation, the velocity ṅof 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=dtLswith Lagrangian given by a space integral Ls=sa2d2rLsand spacetime density as follows

Ls=γSHSLBE36

In this relation, the density LB=1cosβαtwhere now the angular variables are spacetime fields βtrand αtr. Similarly, the density His the Hamiltonian density with arguments as Hnμnrand magnetic nas in Eq. (35). In this field action Ss, the prefactor a2is required by the continuum limit of lattice Hamiltonians Hlattliving on a square lattice with spacing parameter a. Recall that for these Hlatt‘s, one generally has discrete sums like μ, μ,νand 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 Z2augmented by the Dzyaloshinskii-Moriya and the Zeeman interactions [66, 67]

HHDMZ=JμνnrμnrνDμ,ν,ρdμ.nrνnrρεμνρμB.nrμE37

with rν=rμ+aeνμ; that is eνμ=rνrμ/awhere ais the square lattice parameter. So, the continuum limit Hof this lattice Hamiltonian involves the target space metric δaband the topological Levi-Civita tensor εabcof the target space Rn3; it involves as well the metric gμνand the completely antisymmetry εμνρof the space time Rξ1,2. In terms of δaband εabctensors, the continuous hamiltonian density reads as follows

H=Ja22δabinainb+aεabcdμanbDνρncεμνρB.nE38

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˜μaand 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 qsin the background magnet has the form

MsX¨i=qsEi+qsεijzẊjBzE39

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

Ls=Ms2δijẊiẊjqsεzijBzXiẊjqsVXE40

Notice that the right hand of Eq. (39) looks like the usual Lorentz force (qeE+qeṙB) of a moving electron with qein 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 qsthat can be put in correspondence with qe; and, in the same way, the background field magnet Ei,Bican 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=Sdtd2rLsand its time variation δSs=0. However, as Lshas two terms like γSHSLB, the calculations can be split in two stages; the first stage concerns the block γSd2rHwith Hnμnrwhich 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+Rwith nrand where the dependence in Rbecomes explicit; thus allowing to think of the integral γSd2rHas 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 δd2rLBbecause we know the variation δLBδnawhich is equal to 12εabcnbṅc. Then, we turn backward to determine Sd2rLBby integration. To that purpose, recall also that the Berry term LBis given by 1cosβαt; and its variation δLBδnanaXjXjis equal to 12εabcnbṅc. To determine the time variation δLB=δd2rδLB, we first expand it like d2rδLδnaδna; and use δna=naXjXjto put it into the form -d2rδLBδnanaXjXj. Then, substituting δLBδnaby its expression 12εabcnbṅcwith ṅcexpanded like -ncXiẊi, we end up with

δLB=2Sd2r12εabcnbncxinaxjεijẊδYẎδXE42

Next, using the relation εijd2r=dxidxj, the first factor becomes 12εabcnadnbdncgives precisely the skyrmion topological charge qs.So, the resulting δLBreduces to 2qsSẊδYẎδXthat reads also like

δLB=2qsSεijẊiδXjE43

This variation is very remarkable because it is contained in the variation of the effective coupling LBint=2qsSεijẊiXjwhich can be presented like LBint=qsAiẊiwhere 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 LBassociated with the Berry term; it reads as follows LB=Ms2δijẊiẊjqsAiẊi.So, the effective Lagrangian Leffdescribing the rigid 2d skyrmion in a ferromagnet is

Leff=Ms2Ṙ2qsA.ṘVRE44

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=1with time evolution given by the LLequation ṅ=γfnwhere the force f=Hn. Using this equation, we deduce the typical properties n.ṅ=f.ṅ=0from which we learn that the time variation dHdtof the Hamiltonian, which reads as d2rHnaṅa, vanishes identically as explicitly exhibited below,

dHdt=d2rf.ṅE45

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 Facting on the rigid skyrmion Rthas two terms, the old conservative f=Hn;and an extra force δflinearly 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 flike F=fαγṅwith αa positive damping parameter (Gilbert parameter). As such, the previous LL relation gives the so called Landau-Lifschitz-Gilbert (LLG) equation [68, 69]

ṅ=γfn+αṅnE46

where its both sides have ṅ. From this generalised relation, we still have n.ṅ=0(ensuring n2=1); however f.ṅ0as 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 Ωwhich, in addition to f, depends moreover on the Gilbert parameter and the magnetisation n. By factorising Eq. (46) like ṅ=Ωn,we find

Ω=γ1+α2f+αfnE47

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

dHdt=αSd2rγfn.ṅE48

then, substituting γfn=αṅnṅ; we find that dHdtis given by αsd2r.ṅ2indicating 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 Ṙin terms of the conservative force fand the parameter α. To that purpose we start from Eq. (46) and multiply both equation sides by dnwhile 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

n.ṅdn=γf.dn+αdn.ṅE49

Substituting dnand ṅby their expansions dxiinand Ẋiin,then multiplying by dxl; we end up with a relation involving dxjdxl(which reads as εzjld2r); so we have

Ẋl4πJ0d2r=γε0ljf.jnd2r+αε0ljẊjjn2d2rE50

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

4πqsẊl=γε0ljf.jnd2r4πηαε0ljẊlE51

with εzxy=εzxy=1. The remaining step is to replace the conservative force fby Hnand proceeds in performing the integral over f.jn. Because of the explicit dependence into r, the f.jncan be expressed like jexpHjtotH; the explicit derivation term jexpHhas been added because the Hamiltonian density has an explicit dependence Hnμnr.Recall that jtotHis given by jexpH+Hn.jnwhich is equal to jexpHf.jn.Notice also that the term jexpHcan be also expressed like HR. Therefore, the integral f.jnd2rhas two contributions namely the jtotHd2rwhich, being a total derivative, vanishes identically; and the term jexpHd2rthat gives VR.Putting this value back into (51), we end up with

4πqsẊl=γεzljVXj+4πηαγẊlE52

Implementing the kinetic term of the skyrmion, we obtain the equation with dissipation MsR¨=VR+GzṘηαqsṘwhere the constant G=4πSqsa2stands 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.nawhich leads to emergent SU2gauge potential that mediate the interaction between the spin texture ntrand the two spin states ΨΨof 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 ntrand mobile carriers represented by a two spin component field Ψtr; the components of the fields nand Ψare respectively given by natrwith a=1,2,3; and by Ψαtrwith 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 Hen=JHn.jewith Hund parameter JH>0promoting alignment of nand je. So, the dynamics of the interacting electron with the backround nis given by the Lagrangian density Le=ΨitΨHenexpanding as follows

LeΨn=ΨitΨΨP22mJHσ.nΨE53

where P=iand σ.n=σxnx+σyny+σznz.

#### 6.1.1 Emergent gauge potential

Because of the ferromagnetic Hund coupling (JH>0), the spin observable Ŝez=2σzof the conduction electron tends to align with the orientation σn=σ.nof the magnetisation n— with angle θ=ez,n̂;this alinement is accompanied by a local phase change of the electronic wave function Ψwhich becomes ψ=UΨwhere Utr=eiΘtris a unitary SU2transformation mapping σzinto 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 ψα̇with 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=iUμUmediating 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

LeψAμ=ψi0A0aσaψψP+Aaσa22mJHσzψE54

Here, the vector potential matrix Aμis valued in the SU2Lie algebra generated by the Pauli matrices σa; so it can be expanded as Aμxσx+Aμyσy+Aμzσzwith components Aμa=12TrσaAμ. Notice that in going from the old LeΨnto the new L˜eψAμ, the spin texture nhas 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 SU2S3and the Hopf fibration S1×S2to write the unitary matrix Uas follows

U=ecosθ2esinθ2e+sinθ2cosθ2,Aμ=μWμWμ+μE55

where the factor edescribes S1and where, for later use, we have set Wμ±=Aμ1±iAμ2and μ=Aμ3. So, a specific realisation of the gauge transformation is given by fixing γ=cst(say γ=0);it corresponds to restricting S3down to S2and SU2reduces down to SU2/U1. In this parametrisation, we can also express the unitary matrix U like m.σwith magnetic vector m=sinθ2cosφsinθ2sinφcosθ2obeying 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 nby its expression sinθcosφsinθsinφcosθ, we end up with the desired direction σzappearing in Eq. (54). On the other hand, by putting U=m.σback 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 Aμof Eq. (55); the relation with the texture nis 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 Aμaobtained 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

Aμa=12eaμθfasinθ2μφE56

where we have used the local basis vectors mθ,eθand 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/2and vθ/2involving θ/2angle instead of θ. Substituting these basis vectors by their angular values, we obtain

Aμ1Aμ2Aμ3=12sinφcosφ0μθ12sinθcosφsinθsinφcosθ1μφE57

from which we learn that the two first components combine in a complex gauge field Wμ±=Aμ1±iAμ2which is equal to i2ew±μnwith w±=u±iv; and the third component Aμ3has the remarkable form 121cosθμφwhose structure recalls the geometric Berry term (7). Below, we set Aμ3=μas in Eq. (55); it contains the temporal component 0and the three spatial ones i— denoted in Section 2 respectively as a0and 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+0denoted 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=0into Eq. (54) and using χσxχ=χσyχ=0and χσzχ=χ¯χas well as replacing Aμxσx2+Aμyσy2by 14μn2, the Lagrangian (54) reduces to the polarised Lepol=LeχnZμgiven by

Lepol=χi00σzχχPi+iaσa22m+28mμn2JHσzχE58

where 0idefine the four components of the emergent abelian gauge prepotential μassociated with the Pauli matrix σz; their explicit expressions are given by 0=121cosθφ̇and 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 endescribed by the Lagrangian density Ltotcontaining the parts Ln+Len;the electronic Lagrangian Lenis given by Eq. (54). The Lagrangian Ln, describing the skyrmion dynamics, is as in eqs (5)(7) namely S0Hnwith 0=121cosθφ̇.By setting H˜n=Hn+28mμn2ψψ, the full Lagrangian density Ltotwith can be then presented like L˜ψμH˜nlike

L˜ψμ=S0+ψit0σzψψPi+iσz22mψE59

with Pi+iσz2expanding as Pi2+i22+Pii+iPiσz. The equations of motion of ψand nare 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+hcwhich vanishes for δLtot/δn=0and 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˜nwith respect to δnnamely

δH˜δnδL˜δμδμδn=0E60

The variation δHnδndepends on the structure of the skyrmion Hamiltonian density H˜; its contribution to the equation of motion can be presented like λμμn=Fwith some factor λ. However, the variation δLδμδμδndescribes 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 nand to velocity ṅand involves the eletron spin density ρez=ψσzψ.

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

Ji=12mψσzPiψPiψσzψ+mψψiE61

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

Jl+n=mψ¯+nψ+nl+12mψ¯+nilψ+nilψ¯+nψ+nJln=mψ¯nψnl2imψ¯nilψnilψ¯nψnE62

These vectors are respectively interpreted as two spin polarised currents; the Ji+nis associated with the ψ+nwave function as it points in the same direction as n; the Jinis however associated with ψnpointing in the opposite direction of n. 2Each one of the two J+nand Jnare in turn given by the sum of two contributions as they can be respectively split like mψ¯+nψ+n+jψ+nand mψ¯nψn+jψnwith vector density jψstanding for the usual current vector jψ=12mψ¯Pψ. The contribution mψ¯ψis 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 δLδiδiδnleads to 2Jiinnthat reads in a condensed form like 2J.nn. Putting back into Eq. (60), we end up with the following modified LL equation

22S+ψσzψṅn+2J.nnδHnδn=0E63

To compare this equation with the usual LL equation (Sṅ=δHnδnn)in absence of Hund coupling (which corresponds to putting ψto zero), we multiply Eq. (63) by nin 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.nis normal to n; and so it can be set as Ωenwith 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=ωnnshowing that the vector nrotates with ωn=δHnδn. By turning on ψ, we have ṅωn+Ωenindicating that the LL rotation is drifted by Ωecoming from two sources: ithe term J.nwhich deforms LL vector ωndrifted by the nJ.n; and iithe electronic spin density ρez=Nead; this term adds to the density Sadof the magnetic texture per unit volume; it involves the number Ne=Ne+nNenwith Ne±nstanding for the filling factor of polarized conduction electrons. Moreover, if assuming ntr=nrVstwith a uniform Vs, then the drift velocity ṅa=inaVsiand Jeiina=Jea. Putting back into the modified LLG equation, we end up with the following relation between the Vsand vevelocities S+ne2vsa=neveawhere we have set inaVsi=vsaand 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 Lsfrom the computation space integral of d2rLsand Eq. (36). To begin, recall that in absence of the STT effect, the Lagrangian Lsof the 2D skyrmion’s point- particle, with position R=XYand velocity Ṙ=ẊẎ,is given by Ms2Ṙ2G2z.RṘVRwith effective scalar energy potential VR=d2rHrRand a constant G=4πa2qsS.Under STT induced by Hund coupling, the Lagrangian Lsgets deformed into L˜s=Ls+ΔLs, that is

L˜s=Ms2Ṙ2G2z.RṘVR+ΔLsE65

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

L˜en=ψit0σzψψPi+iσz22mψE66

The deviation ΔLswith respect to Lsin (65) comes from those terms in Eq. (66). Notice that this expression involves the wave function ψcoupled to the emergent gauge potential field μ=0i; that is d2rψσzψ0and 12md2rψPi+iσz2ψ. Thus, to obtain ΔLs, we first calculate the variation δΔLsδμδμand 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: iWe calculate the temporal contribution δΔLsδ0δ0δR.δR; and iiwe compute the spatial δΔLsδiδiδR.δR.Using the variation δ0=12δn.njnẊj, the contribution of the first term can be put as follows

δΔLsδ0δ0δXlδXl=2J0zεzijẊiδXjE67

where we have set ρz=ψσzψand J0z=d2rρz2εzkln.knln.Notice that the right hand side in above relation can be also put into the form 2J0zεzijδẊiXjδ2εzijJ0zẊiXjindicating that ΔLsmust contain the term 2εzijJ0zẊiXjwhich 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

δΔLsδi.δiδXlδXl=εzijJziδXjE68

where we have set Jzit=d2rJzitrwith Jzitrgiven by Eq. (61). Here also notice that the right hand of above equation can be put as well like δεzijJziXjindicating that ΔLscontains in addition to 2J0z.ṘR, the term εzijJziXjwhich reads also as z.JRwith two component vector J=JzxJzy. Thus, we have the following modified skyrmion equation

L˜s=Ms2Ṙ212G+J0z.ṘR+z.JRVRE69

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

In this bookchapter, we have studied the basic aspects of the solitons dynamics in various 1+dspacetime 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 SU2spins, their coherent vector representation S=RαβγS0with S0standing 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 SU2gauge 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 Qthat are interpreted in terms of magnetic skyrmions and antiskyrmion; these topological states can be imagined as (winding) quasiparticle excitations with Q>0and Q<0respectively.

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 D2dsymmetry 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/swith 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 D2dsymmetry. 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.

## Acknowledgments

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).

## Notes

• 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.

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Lalla Btissam Drissi, El Hassan Saidi, Mosto Bousmina and Omar Fassi-Fehri (May 11th 2021). Magnetic Skyrmions: Theory and Applications, Magnetic Skyrmions, Dipti Ranjan Sahu, IntechOpen, DOI: 10.5772/intechopen.96927. Available from:

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