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
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 , skyrmions have however been observed in other fields of physics, including quantum Hall systems [14, 15], Bose-Einstein condensates  and liquid crystals . 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 ; slightly away, other electrons organize into an intricate spin configuration because of a competitive interplay between the Coulomb and Zeeman interactions . 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 . 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–
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 . The in-plane component of the magnetization, in the Néel skyrmion, is always pointed in the radial direction , while it is oriented perpendicularly with respect to the position vector in the Bloch skyrmion . Different from these two well-known types of skyrmions are skyrmions with mixed Bloch-Néel topological spin textures observed in Co/Pd multilayers . Magnetic antiskyrmions, having a more complex boundary compared to the chiral magnetic boundaries of skyrmions, exist above room temperature in tetragonal Heusler materials . Higher-order skyrmions should be stabilized in anisotropic frustrated magnet at zero temperature  as well as in itinerant magnets with zero magnetic field .
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  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 . 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 . Strong DMI between the first nearest magnetic germanium neighbors in 2D semi-fluorinated germanene results in a potential antiferromagnetic skyrmion .
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 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 of spinfull particles are characterised by two half integers , a positive and an taking values bounded as with integral hoppings. For particles with spin 1/2 like electrons, one distinguishes two basis vector states that are eigenvalues of the scaled Pauli matrix and the quadratic (Casimir) operator , here the three with are the three components of the spin 1/2 operator vector1. From these ingredients, we learn that the average (for short ) is carried by the z-direction since with . For generic values of the SUspin , the spin operator reads as where the three ‘s are generators of the SUgroup satisfying the usual commutation relations with standing for the completely antisymmetric Levi-Civita tensor with non zero value ; its inverse is with . The time evolution of the spin operator with dynamics governed by a stationary Hamiltonian operator () is given by the Heisenberg representation of quantum mechanics. In this non dissipative description, the time dependence of the spin operator (the hat is to distinguish the operator from classical ) is given by
where the Pauli matrices obey the usual commutation relations . For a generic value of the SUspin , the above relation extends as . So, many relations for the spin may be straightforwardly generalised for generic values of the SUspin. For example, for a spin value , the states are given by and are labeled by ; one of these states namely is very special; it is commonly known as the highest weight state (HWS) as it corresponds to the biggest value ; from this state one can generate all other spin states ; this feature will be used when describing coherent spin states. Because of the property the square is time independent; and then the time dynamics of is rotational in the sense that is given by a commutator as follows For the example where is a linearly dependent function of like for the Zeeman coupling, the Hamiltonian reads as (for short ) with the ’s are constants referring to the external source2; then the time evolution of reads, after using the commutation relation , as follows
where appears the Levi-Civita 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 , we will encounter another completely antisymmetric Levi-Civita tensor namely ; it is also due to DM interaction which in lattice description is given by ; and in continuous limit reads as where, for convenience, we have set with To distinguish these two Levi-Civita tensors, we refer to as the target space Levi-Civita with SOsymmetry; and to as the spacetime Levi-Civita with SOLorentz symmetry containing as subsymmetry the usual space rotation group SO. Notice also that for the case where the Hamiltonian is a general function of the spin, the vector is spin dependent and is given by the gradient
2.2 Coherent spin states and semi-classical analysis
To deal with the semi-classical dynamics of evolved by a Hamiltonian , we use the algebra to think of the quantum spin in terms of a coherent spin state  described by a (semi) classical vector (no hat) of the Euclidean see the Figure 1(a). This “classical” 3-vector has an amplitude and a direction related to a given unit vector as and parameterised by . In the above relation, the is thought of as the north direction of a 2-sphere given by the canonical vector ; it is invariant under the proper rotation; i.e. ; and consequently the generic is independent of ; i.e.: Recall that the 33 matrix is an SOrotation [SO] generating all other points of parameterised by . In this regards, it is interesting to recall some useful properties that we list here after as three points: the rotation matrix can be factorised like where each is a rotation around the a- axis with an angle and generator . As the unit is an eigen vector of ; it follows that reduces to ; this generic vector obeys as well the constraint and is solved as follows.
with and ; they parameterise the unit 2-sphere which is isomorphic to ; the missing angle parameterises a circle isomorphic to that is fibred over .the coherent spin state representation gives a bridge between quantum spin operator and its classical description; it relies on thinking of the average in terms of the classical vector considered above (HWS ). In this regards, recall that the acts on classical 3-vectors through its 33 matrix representation like with given by ; these ‘s are precisely the generators of the matrix representation ; by replacing by the operator , one discovers the SUspin algebra . Notice also that the classical spin vector can be also put in correspondence with the usual magnetic moment (with the gyromagnetic ratio); thus leading to . So, the magnetization vector describes (up to a sign) a coherent spin state with amplitude ; and a (opposite) time dependent direction parameterizing the 2-sphere .
For explicit calculations, this unit 2-sphere equation will be often expressed like ; this relation leads in turns to the property (indicating that and are normal vectors); by implementing time, the variation gets mapped into teaching us that the velocity is carried by and ; two normal directions to with components;
and from which we learn that , [() form an orthogonal vector triad). So, the dynamics of (and that of ) is brought to the dynamics of the unit governed by a classical Hamiltonian H. The resulting time evolution is given by the so called Landau-Lifshitz (LL) equation ; it reads as with By using the relations and with ; and ; as well as the expressions and , the above LL equation splits into two time evolution equations and . These time evolutions can be also put into the form
and can be identified with the Euler- Lagrange equations following from the variation of an action . Here, the Lagrangian is related to the Hamiltonian like where is the Berry term  known to have the form <
showing that and form a conjugate pair. By substituting back into above , we find that the Berry term has the form of Aharonov-Bohm coupling with magnetic potential vector given by However, this potential vector is suggestive as it has the same form as the potential vector of a magnetic monopole. The curl of this potential is given by with magnetic charge located at the centre of the 2-sphere; the flux of this field through the unit sphere is then equal to ; and reads as with a unit flux quanta as indicated by the value . So, because is an integer, it results that the flux is quantized as
3. Magnetic solitons in lower dimensions
In previous section, we have considered the time dynamics of coherent spin states with amplitude and direction described by as depicted by the Figure 1(a); this is a 3-vector having with no space coordinate dependence, ; and as such it can be interpreted as a vector field; that is a vector belonging to with (no space direction). In this section, we first turn on 1d space coordinate and promotes the old unit- direction to a D field . After that, we turn on two space directions ; thus leading to D field 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: the target space parameterised by with Euclidian metric and topological Levi-Civita . the spacetime parameterised by concerning the 1d kink evolution; and the spacetime parameterised by , regarding the 2d skyrmions dynamics. As we have two kinds of evolutions; time and space; we denote the time variable by ; and the space coordinates by . Moreover, the homologue of the tensors and are respectively given by the usual Lorentzian spacetime metric , with signature like , and the spacetime Levi-Civita with .
3.1 One space dimensional solitons
In D spacetime, the local coordinates parameterising are given by ; so the metric is restricted to . The field variable has in general three components as described previously; but in what follows, we will simplify a little bit the picture by setting ; thus leading to a magnetic 1d soliton with two component field variable satisfying the constraint equation at each point of spacetime. As this constraint relation plays an important role in the construction, it is interesting to express it as . 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. The constraint is invariant rotations acting as with orthogonal rotation matrix
The constraint can be also presented like with standing for the complex field that reads also like . In this complex notation, the symmetry of the constraint is given by the phase change acting as with and corresponding to the shift . Moreover the correspondence describes precisely the well known isomorphisms where is a circle; it is precisely the equatorial circle of the 2-sphere considered in previous section. As for Eq. (5), the constraint leads to ; and so describes a rotational movement encoded in the relation where is the standard 2D antisymmetric tensor with ; this is related to the previous 3D Levi-Civita like . Notice also that the constraint implies moreover that and consequently the area , to be encountered later on, vanishes identically. In this regards, recall that we have the following transformation
where is the antisymmetric tensor in 1+1 spacetime, and is the Jacobian of the transformation . The condition 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 from which we deduce the normal direction reading as . In term of the complex field; we have and . 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 is described by a field action with Lagrangian and density ; this field density is given by with ; it reads in terms of the Hamiltonian density as follows
where . In the above Lagrangian density, the auxiliary field (no Kinetic term) is a Lagrange multiplier carrying the constraint relation . The is a potential energy density which play an important role for describing 1d kinks with finite size. Notice also that the variation gives precisely the constraint while the gives the spacetime dynamics of described by the spacetime equation . By substituting , we obtain . If setting , we end up with the free field equation that expands like ; it is invariant under spacetime translations with conserved current symmetry with standing for the energy momentum tensor given by the 22 symmetric matrix . The energy density is given by and the momentum density reads as . Focussing on , the conserved energy reads then as follows
with minimum corresponding to constant field (). Notice that general solutions of are given by arbitrary functions ; 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 ) by the following expression
where is a positive parameter representing the width where the soliton acquires a significant variation. Notice that for a given , the field varies from to regardless the value of . These limits are related to each other by a period .
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 going beyond the spacetime translations generated by the energy momentum tensor . The conserved spacetime current of this exotic symmetry can be introduced in two different, but equivalent, manners; either by using the free degree of freedom ; or by working with the constrained field In the first way, we think of the charge density like and of the current density as . This conserved current is a topological spacetime vector that is manifestly conserved; this feature follows from the relation between and the antisymmetric as follows ,
Because of the ; the continuity relation vanishes identically due to the antisymmetry property of . The particularity of the above conserved is its topological nature; it is due to the constraint without recourse to the solution . Indeed, Eq. (13) can be derived by computing the Jacobian of the mapping from the 2d spacetime coordinates to the target space fields (). Recall that the spacetime area can be written in terms of like and, similarly, the target space area can be expressed in terms of as follows . The Jacobian is precisely given by (9); and can be presented into a covariant form like . This expression of the Jacobian captures important informations; in particular the three following ones. It can be expressed as a total divergence like with spacetime vector
and where 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 ]. Because of the constraint following from , the Jacobian vanishes identically; thus leading to the conservation law ; i.e. and then . The conserved charge associated with the topological current is given by ; it is time independent despite the apparent t- variable in the integral (). By using (13), this charge reads also as and after integration leads to
Moreover, seen that is an angular variable parameterising ; it may be subject to a boundary condition like for instance the periodic with N an integer; this leads to an integral topological charge interpreted as the winding number of the circle. In this regards, notice that: the winding interpretation can be justified by observing that under compactification of the space variable , the infinite space line gets mapped into a circle with angular coordinate ; so, the integral gets replaced by ; and then the mapping is a mapping between two circles namely ; the field then describes a soliton (one space extended object) wrapping the circle N times; this propery is captured by , a homotopy group property . The charge is independent of the Lagrangian of the system as it follows completely from the field constraint without any reference to the field action. Under a scale transformation with a scaling parameter , 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 ; 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 . This can done by turning on an appropriate potential energy density in Eq. (10). An example of such potential is the one given by , with positive , breaking ; by using the constraint , it can be put . In terms of the angular field , it reads as leading to the well known sine-Gordon Eq. [64, 65] namely with the symmetry property . So, the solitonic solution is periodic with period ; that is the quarter of the old 2period of the free field case. For static field , the sine Gordon equation reduces to ; its solution for is given by representing a sine- Gordon field evolving from to and describing a kink with topological charge For , the soliton is an anti-kink evolving from to with charge Time dependent solutions can be obtained by help of boost transformations .
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 so that the skyrmion field is a real 3-vector with three components constrained as in Eqs. (4) and (5); see Figure 2. Second, here we have ; that is a 3-component field living in the space time with Lorentzian metric and coordinates . This means that ; explicitly .
3.2.1 Dzyaloshinskii-Moriya potential
The field action describing the space time dynamics of has the same structure as Eq. (10); except that here the Lagrangian involves two space variable like and the density ; this is a function of the constrained 3-vector and its space time gradient ; it reads in term of the Hamiltonian density as follows.
In this expression, the is the continuous limit of a lattice Hamiltonian Hinvolving, amongst others, the Heisenberg term, the Dyaloshinskii- Moriya (DM) interaction and the Zeeman coupling. The in the first expression of 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 is an auxiliary 3D spacetime field; it is a Lagrange multiplier that carries the constraint which plays the same role as in subSection 3.1. By variating this action with respect to the fields and ; we get from precisely the field constraint ; and from the following Euler–Lagrange equation 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 where is an operator acting on 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 . To that purpose, we start by noticing that there are two manners to deal with the field constraint ; either by using the Lagrange multiplier ; or by solving it in terms of two angular field variables as given by Eq. (5). In the second case, we have the triad and
but now and with and Notice also that the variation of the filed constraint leads to teaching us interesting informations, in particular the two following useful ones. the movement of in the target space is a rotational movement; and so can be expressed like
where the 1-form is the rotation vector to be derived below. By substituting (20) back into we obtain which vanishes identically due to the property . Having two degrees of freedom and , we can expand the differential like with the two vector fields and as given above. Notice that the three unit fields plays an important role in this study; they form a vector basis of the field space; they obey the usual cross products namely and its homologue which given by cyclic permutations; for example,
Putting these Eq. (21) back into the expansion of in terms of ; and comparing with Eq. (20), we end up with the explicit expression of the 1-form angular “speed” vector ; it reads as follows . Notice that by using the space time coordinates , we can also express Eq. (20) like with given by From this expression, we can compute the Laplacian ; which, by using the above relations; is equal to reading explicitly as or equivalently like with operator . Notice that the above operator has an interesting geometric interpretation; by factorising , we can put it in the form where appears as a gauge covariant derivative with a non trivial gauge potential given by Comparing with (18) with , we obtain and then a scalar potential energy given by . The second term in this relation vanishes identically because ; thus reducing to
containing as a sub-term. In the end of this analysis, let us compare this sub-term with the with giving the general structure of the DM coupling (see end of subSection 2.1). For D spacetime, the general structure of DM interaction reads ; by setting and as well as , one brings it to the form 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 D topological current discussed in subSection 2.1, which reads as , one can wonder the structure of the D topological current that is associated with the 2d Skyrmion described by the 3-vector field . It is given by
where is as before and where is the completely antisymmetric Levi-Civita tensor in the D spacetime. The divergence of the above spacetime vector vanishes identically; it has two remarkable properties that we want to comment before proceeding. The is nothing but the determinant of the Jacobian matrix relating the three field variables to the three spacetime coordinates ; this Jacobian is generally given by it maps the spacetime volume into the target space volume . In this regards, recall that these two 3D volumes can be expressed in covariant manners by using the completely antisymmetric tensors and introduced earlier; and as noticed before play a central role in topology. The target space volume can be expressed like ; and a similar relation can be also written down for the spacetime volume . Notice also that by substituting the differentials by their expansions ; and putting back into we obtain the relation where is precisely the Jacobian . The conservation law has a geometric origin; it follows from the field constraint relation degenerating the volume of the 3D target space down to a surface. This constraint relation describes a unit 2-sphere ; and so a vanishing volume thus leading to and then to the above continuity equation. Having the explicit expression (23) of the topological current in terms of the magnetic texture field , we turn to determine the associated topological charge with charge density given by . Substituting by , we have . Moreover using the differentials , we can calculate the area in terms of the angles and we find where we have used So, the topological charge reads as which is equal to 1. In fact this value is just the unit charge; the general value is an integer with N being the winding number see below. Notice that can be also presented like
Replacing by their expression in terms of the angles , we can bring the above charge density into two equivalent relations; first into the form like ; and second as which is nothing but the Jacobian of the transformation from the space to the unit 2-sphere with angular variables . The explicit expression of in terms of the space variables is given by
but this is nothing but the stereographic projection of the 2-sphere on the real plane. So, the field defines a mapping between towards with topological charge given by the winding number around ; this corresponds just to the homotopy property .
4. Three dimensional magnetic skyrmions
In this section, we study the dynamics of the 3d skyrmion and its topological properties both in target space (with euclidian metric ) and in 4D spacetime parameterised by (with Lorentzian metric ). The spacetime dynamics of the 3d skyrmion is described by a four component field obeying a constraint relation ; here the is given by the quadratic form invariant under SOtransformations isomorphic to SUSU. The structure of the topological current of the 3d skyrmion is encoded in two types of Levi-Civita tensors namely the target space and the spacetime 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 is described by a field action with Lagrangian realized as the space integral . Generally, the Lagrangian density is a function of the soliton which is a real 4-component field  constrained like For self ineracting field, the typical field expression of is given by where is a scalar potential; and where the auxiliary field is a Lagrange multiplier carrying the field constraint. This density reads in terms of the Hamiltonian as Below, we consider a 4-component skyrmionic field constrained as ; and focuss on a simple Lagrangian density to describe the degrees of freedom of . Being a unit 4-component vector, we can solve the constraint in terms of three angular angles by setting
where is a unit 3-vector parameterising the unit sphere . Putting this field realisation back into , we obtain Notice that by restricting the 4D spacetime to the 3D hyperplane z=const; and by fixing the component field to , the above Lagrangian density reduces to the one describing the spacetime dynamics of the 2d skyrmion. Notice also that we can expand the differential in terms of ; we find the following
For convenience, we sometimes refer to the three collectively like so we have with
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 . Then, we turn to study the induced topological properties of the 3d skyrmion viewed from the side of the 4D space time .
4.2.1 Topological current in target space
The 3d skyrmion field is described by a real four component vector subject to the constraint relation ; so the soliton has SOsymmetry leaving invariant the condition that reads explicitly as The algebraic condition induces in turns the constraint equation leading to and showing that and orthogonal 4-vectors in . From this constraint, we can construct which is a 44 antisymmetric matrix generating the SOrotations; this contains 3+3 degrees of freedom generating the two and making SO; the first three degrres are given by with ; and the other three concern Notice also that, from the view of the target space, the algebraic relation describes a unit 3-sphere sitting in ; as such its volume 4-form , which reads as vanishes identically when restricted to the 3-sphere; i.e.: . This vanishing property of on is a key ingredient in the derivation of the topological current of the 3D skyrmion and its conservation . Indeed, because of the property (where we have hidden the wedge product ), it follows that can be expressed as with the 3-form given by
This 3-form describes precisely the topological current in the target space; this is because on , the 4-form vanishes; and then vanishes. By solving, the skyrmion field constraint in terms of three angles as given by Eq. (26); with these angular coordinates, we have mapping with . By expanding the differentials like with ; then the conserved current on the 3-sphere reads as follows
where we have substituted the 3-form on the 3-sphere by the volume 3-form . In this regards, recall that the volume of the 3-sphere is .
4.2.2 Topological symmetry in spacetime
In the spacetime with coordinates , the 3d skyrmion is described by a four component field and is subject to the local constraint relation . 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 and thought of as follows
with giving the radius of and associated with the circle fibered over ; the value corresponds to and to . Notice that in Eq. (30) has a spherical symmetry as it is a function only of (no angles ). Moreover, as this configuration obeys and ; we assume and . Putting these relations back into (26), we obtain the following configuration
describing a compactification of the space into which is homotopic to . From this view, the is then a mapping from into with topological charge given by the winding number characterising the wrapping on ; and for which we have the property . In this regards, recall that 3-spheres have a Hopf fibration given by a circle sitting over (for short ); this non trivial fibration can be viewed from the relation and the factorisation with the coset identified with ; and with . Applying this fibration to and , it follows that and the same thing for the bases and for the fibers . Returning to the topological current and the conserved topological charge , notice that in space time the differential expands like ; then using the duality relation , we find, up to a normalisation by the volume of the 3-sphere , the expression of the topological current in terms of the 3D skyrmion field
In terms of the angular variables, this current reads like with . From this current expression, we can determine the associated topological charge by space integration over the charge density
Because of its spherical symmetry, the space volume can be substituted by ; then the charge reads as the integral whose integration leads to the sum of two terms coming from the integration of . The integral first reads as ; by substituting , it contributes like . The integral of the second tem gives ; 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 action describing the time evolution of a coherent spin vector modeled by a rotating magnetic moment with velocity ; and make some accommodations. For that, recall that the Lagrangian has the structure where is the Berry term having the form with geometric (Berry) potential ; and where is the Hamiltonian of the magnetic moment obeying the constraint . This magnetisation constraint is solved by two free angles they appear in the Berry term Below, we think of the above magnetisation as a ferromagnetic background filling the spatial region of with coordinates ; and of the skymion as a massive point- like particle moving in this background.
5.1.1 Rigid skyrmion
We begin by introducing the variables describing the skyrmion in the magnetic background field . We denote by the mass of the skyrmion, and by and its space position and its velocity. For concreteness, we restrict the investigation to the spacetime and refer to by the components and to by the components . Because of the Euclidean metric ; we often we use both notations and without referring to . Furthermore; we limit the discussion to the interesting case where the only source of displacements in is due to the skyrmion (rigid skyrmion). In this picture, the description of the skyrmion dissolved in the background magnet is given by
In this representation, the velocity of the skyrmion dissolved in the background magnet can be expressed into manners; either like ; or as ; this is because . With this parametrisation, the dynamics of the skyrmion is described by an action with Lagrangian given by a space integral and spacetime density as follows
In this relation, the density where now the angular variables are spacetime fields and . Similarly, the density is the Hamiltonian density with arguments as and magnetic as in Eq. (35). In this field action , the prefactor ais required by the continuum limit of lattice Hamiltonians living on a square lattice with spacing parameter a. Recall that for these ‘s, one generally has discrete sums like , and so on; in the limit where a is too small, these sums turn into 2D space integrals like a. To fix ideas, we illustrate this limit on the typical hamiltonian , it describes the Heisenberg model on the lattice augmented by the Dzyaloshinskii-Moriya and the Zeeman interactions [66, 67]
with ; that is where is the square lattice parameter. So, the continuum limit of this lattice Hamiltonian involves the target space metric and the topological Levi-Civita tensor of the target space ; it involves as well the metric and the completely antisymmetry of the space time . In terms of and tensors, the continuous hamiltonian density reads as follows
with . Below, we set J=1 and, to factorise out the normalisation factor , scale the parameters of the model like and . For simplicity, we sometimes set as well a
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 . General arguments indicate that the effective equation of the skyrmion with topological charge 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 () of a moving electron with in an external electromagnetic field the corresponding Lagrangian is . This similarity between the skyrmion and the electron in background fields is because the skyrmion has a topological charge that can be put in correspondence with ; and, in the same way, the background field magnet can be also put in correspondence with the electromagnetic field . To rigourously derive the spacetime Eqs. (39) and (40), we need to perform some manipulations relying on computing the effective expression of and its time variation . However, as has two terms like , the calculations can be split in two stages; the first stage concerns the block with which is a function of the magnetic texture (35); that is . The second stage regards the determination of the integral . The computation of the first term is straightforwardly identified; by performing a space shift , the Hamiltonian density becomes with and where the dependence in becomes explicit; thus allowing to think of the integral as nothing but the scalar energy potential So, we have
Concerning the calculation of the , the situation is somehow subtle; we do it in two steps; first we calculate the because we know the variation which is equal to . Then, we turn backward to determine by integration. To that purpose, recall also that the Berry term is given by ; and its variation is equal to . To determine the time variation , we first expand it like ; and use to put it into the form -. Then, substituting by its expression with expanded like -, we end up with
Next, using the relation , the first factor becomes gives precisely the skyrmion topological charge So, the resulting reduces to that reads also like
This variation is very remarkable because it is contained in the variation of the effective coupling which can be presented like where we have set ; this vector can be interpreted as the vector potential of an effective magnetic field . By adding the kinetic term , we end up with an effective Lagrangian associated with the Berry term; it reads as follows So, the effective Lagrangian describing the rigid 2d skyrmion in a ferromagnet is
From this Lagrangian, we learn the equation of the motion of the rigid skyrmion namely ; for the limit , it reduces to .
5.2 Implementing dissipation
So far we have considered magnetic moment obeying the constraint with time evolution given by the LLequation where the force . Using this equation, we deduce the typical properties from which we learn that the time variation of the Hamiltonian, which reads as , vanishes identically as explicitly exhibited below,
In presence of dissipation, we loose energy; and so one expects that ; 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 acting on the rigid skyrmion has two terms, the old conservative and an extra force linearly dependent in magnetisation velocity . Due to this extra force , the LL equation gets modified; its deformed expression is obtained by shifting the old force like with 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 . From this generalised relation, we still have (ensuring ); however as it is equal to the Gilbert term namely Notice that Eq. (46) still describe a rotating magnetic moment in the target space (); but with a different angular velocity which, in addition to , depends moreover on the Gilbert parameter and the magnetisation . By factorising Eq. (46) like we find
Notice that in presence of dissipation (), the variation of the hamiltonian given by (45) is no longer non vanishing; by first replacing and putting back in it, we get
then, substituting ; we find that is given by indicating that ; and consequently a decreasing energy (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 ; and compute the expression of the skyrmion velocity in terms of the conservative force and the parameter . To that purpose we start from Eq. (46) and multiply both equation sides by while assuming (the conservative force transverse to magnetisation), we get . Then, multiply scalarly by which corresponds to a projection along the magnetisation, we obtain
Substituting and by their expansions and then multiplying by ; we end up with a relation involving (which reads as ); so we have
where we have set defining the magnetization density, and where we have replaced by . By integrating over the 2d space while using and setting , we arrive at the relation
with . The remaining step is to replace the conservative force by and proceeds in performing the integral over . Because of the explicit dependence into , the can be expressed like ; the explicit derivation term has been added because the Hamiltonian density has an explicit dependence Recall that is given by which is equal to Notice also that the term can be also expressed like . Therefore, the integral has two contributions namely the which, being a total derivative, vanishes identically; and the term that gives Putting this value back into (51), we end up with
Implementing the kinetic term of the skyrmion, we obtain the equation with dissipation where the constant 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) . The electron-skyrmion interaction is given by Hund coupling which leads to emergent gauge potential that mediate the interaction between the spin texture and the two spin states 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 and mobile carriers represented by a two spin component field ; the components of the fields and are respectively given by with a=1,2,3; and by with . Using the electronic vector density , the interaction between localised and itinerant electrons of the magnetic atom are bound by the Hund coupling reading as with Hund parameter promoting alignment of and . So, the dynamics of the interacting electron with the backround is given by the Lagrangian density expanding as follows
where and .
6.1.1 Emergent gauge potential
Because of the ferromagnetic Hund coupling (), the spin observable of the conduction electron tends to align with the orientation of the magnetisation — with angle —this alinement is accompanied by a local phase change of the electronic wave function which becomes where is a unitary transformation mapping into ; that is . For later use, we refer to the new two components of the electronic field like (for short with label ) such that the gauge transformation reads as ; that is . This local rotation of the electronic spin wave induces a non abelian gauge potential with components mediating the interaction between the electron and the magnetic texture. Indeed, putting the unitary change into , we end up with an equivalent Lagrangian density; but now with new field variables as follows
Here, the vector potential matrix is valued in the Lie algebra generated by the Pauli matrices ; so it can be expanded as with components . Notice that in going from the old to the new , the spin texture has disappeared; but not completely as it is manifested by an emergent non abelian gauge potential ; so everything is as if we have an electron interacting with an external field . To get the explicit relation between the gauge potential and the magnetisation, we use the isomorphism and the Hopf fibration to write the unitary matrix as follows
where the factor describes and where, for later use, we have set and . So, a specific realisation of the gauge transformation is given by fixing (say )it corresponds to restricting down to and SUreduces down to SU/U. In this parametrisation, we can also express the unitary matrix U like with magnetic vector obeying the property the same constraint as before. By putting back into , and using some algebraic relations like we obtain . Then, substituting by its expression , we end up with the desired direction appearing in Eq. (54). On the other hand, by putting back into , we obtain an explicit relation between the gauge potential and the magnetic texture namely . From this expression, we learn the entries of the potential matrix of Eq. (55); the relation with the texture is given in what follows seen that .
6.1.2 Large Hund coupling limit
We start by noticing that the non abelian gauge potential obtained above can be expressed in a condensed form like (for short ; so it is normal to ; and then it can be expanded as follows
where we have used the local basis vectors and . 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 and involving angle instead of . 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 which is equal to with ; and the third component has the remarkable form whose structure recalls the geometric Berry term (7). Below, we set as in Eq. (55); it contains the temporal component and the three spatial ones — denoted in Section 2 respectively as and —.
In the large Hund coupling (), the spin of the electron is quasi- aligned with the magnetisation ; so the electronic dynamics is mainly described by the chiral wave function denoted below as . 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 . By setting into Eq. (54) and using and as well as replacing by , the Lagrangian (54) reduces to the polarised given by
where define the four components of the emergent abelian gauge prepotential associated with the Pauli matrix ; their explicit expressions are given by and ; their variation with respect to the magnetic texture are related to the magnetisation field like .
6.2 Skyrmion with spin transfer torque
Here, we investigate the full dynamics of the electron/skyrmion system described by the Lagrangian density containing the parts the electronic Lagrangian is given by Eq. (54). The Lagrangian , describing the skyrmion dynamics, is as in eqs (5)–(7) namely with By setting , the full Lagrangian density with can be then presented like like
with expanding as . The equations of motion of and are obtained as usual by computing the extremisation of this Lagrangian density with respect to the corresponding field variables. In general, we have which vanishes for and .
6.2.1 Modified Landau-Lifshitz equation
Regarding the spin texture , the associated field equation of motion is given by ; the contributions to this equation of motion come from the variations and with respect to namely
The variation depends on the structure of the skyrmion Hamiltonian density ; its contribution to the equation of motion can be presented like with some factor . However, the variation describes skyrmion-electron interaction; and can be done explicitly into two steps; the first step concerns the calculation of the time like component ; it gives it is normal to and to velocity and involves the eletron spin density .
The second step deals with the calculation of the space like component ; the factor gives with a 3-component current vector density reading as follows
This vector two remarkable properties: it is given by the sum of two contributions as it it reads like with
These vectors are respectively interpreted as two spin polarised currents; the is associated with the wave function as it points in the same direction as ; the is however associated with pointing in the opposite direction of . Each one of the two and are in turn given by the sum of two contributions as they can be respectively split like and with vector density standing for the usual current vector . The contribution is proportional to the emergent gauge field ; it defines a spin torque transfert to the vector current density .
Regarding the factor , it gives ; by substituting, the total contribution of leads to that reads in a condensed form like . Putting back into Eq. (60), we end up with the following modified LL equation
To compare this equation with the usual LL equation (in absence of Hund coupling (which corresponds to putting to zero), we multiply Eq. (63) by in order to bring it to a comparable relation with LL equation. By setting , describing the electronic spin density ; we find
where, due to , the space gradient is normal to ; and so it can be set as with . 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 , this equation reduces to showing that the vector rotates with . By turning on , we have indicating that the LL rotation is drifted by coming from two sources: the term which deforms LL vector drifted by the ; and the electronic spin density ; this term adds to the density of the magnetic texture per unit volume; it involves the number with standing for the filling factor of polarized conduction electrons. Moreover, if assuming with a uniform , then the drift velocity and . Putting back into the modified LLG equation, we end up with the following relation between the and velocities where we have set and .
6.2.2 Rigid skyrmion under spin transfer torque
Here, we investigate the dynamics of a 2D rigid skyrmion [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 from the computation space integral of and Eq. (36). To begin, recall that in absence of the STT effect, the Lagrangian of the 2D skyrmion’s point- particle, with position and velocity is given by with effective scalar energy potential and a constant Under STT induced by Hund coupling, the Lagrangian gets deformed into , that is
To determine , we start from with Lagrangian density as with given by Eq. (59). For convenience, we set and set
The deviation with respect to in (65) comes from those terms in Eq. (66). Notice that this expression involves the wave function coupled to the emergent gauge potential field ; that is and . Thus, to obtain , we first calculate the variation and put . Once, we have the explicit expression of this variation, we turn backward to deduce the value of . To that purpose, we proceed in two steps as follows: We calculate the temporal contribution ; and we compute the spatial Using the variation , the contribution of the first term can be put as follows
where we have set and Notice that the right hand side in above relation can be also put into the form indicating that must contain the term which reads as well like . Regarding the spatial part , we have quite similar calculations allowing to put it in the following form
where we have set with given by Eq. (61). Here also notice that the right hand of above equation can be put as well like indicating that contains in addition to , the term which reads also as with two component vector . 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 spacetime dimensions with ; while emphasizing the analysis of their topological properties and their interaction with the environment. After having introduced the quantum SUspins, their coherent vector representation with standing for the highest weight spin state; and their link with the magnetic moments , 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 SUgauge fields. It is found that the magnetic skyrmions, existing in a ferromagnetic (FM) medium, show interesting behaviors such as emergent electrodynamics  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 . Such condition supports the topological symmetry of magnetic solitons which is found to be characterised by integral topological charges that are interpreted in terms of magnetic skyrmions and antiskyrmion; these topological states can be imagined as (winding) quasiparticle excitations with and 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  and MnRhIrSn . It is then deduced that stabilized antiskyrmions can be observed in materials exhibiting Dsymmetry 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 . 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 . 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 . At ambient temperature, these skyrmions move at a speed of with a reduced skyrmion Hall angle of . 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 Dsymmetry. 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).
- 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.