Magnetic Solitons due to interfacial chiral interactions

We study solitons in a zig-zag lattice of magnetic dipoles. The lattice comprises two sublattices of parallel chains with magnetic dipoles at their vertices. Due to orthogonal easy planes of rotation for dipoles belonging to different sublattices, the total dipolar energy of this system is separable into a sum of symmetric and chiral long-ranged interactions between the magnets where the last takes the form of Dzyaloshinskii-Moriya coupling. For a specific range of values of the offset between sublattices, the dipoles realize an equilibrium magnetic state in the lattice plane, consisting of one chain settled in an antiferromagnetic parallel configuration and the other in a collinear ferromagnetic fashion. If the offset grows beyond this value, the internal Dzyaloshinskii-Moriya field stabilizes two Bloch domain walls at the edges of the antiferromagnetic chain. The dynamics of these solitons is studied by deriving the long-wavelength lagrangian density for the easy axis antiferromagnet. We find that the chiral couplings between sublattices give rise to an effective magnetic field that stabilizes the solitons in the antiferromagnet. When the chains displace respect to each other, an emergent Lorentz force accelerates the domain walls along the lattice.

Summary of results. Aimed to identify new venues for realizing magnetic solitons in antiferromagnetic materials and find new schemes for their dynamics, we study a zig-zag lattice of magnetic dipoles. The lattice extends along thex axis and comprises two parallel sublattices with an offset alongŷ, each with magnetic dipoles at their vertices. Dipoles in different sublattices have perpendicular easy planes of rotation and couple via the longranged dipolar interaction.
In the first part of the paper we examine the magnetization dynamics of the zig-zag lattice when the offset between sublattices is tuned. For this purpose we solve the set of coupled equations of motion for the angular rotation of each dipole using molecular dynamics simulations. The angular dynamics is caused by the internal torques resulting from the dipolar interactions between the magnets. For a given dipole, the internal torques propelling its rotation can be changed if the offset between sublattices is varied. We find that depending on the offset, the system relaxes into four possible magnetic configurations.
Here we focus our analysis in a range of offsets that allow the lattice to settle in the planar phase: with one sublattice relaxed in a collinear magnetic configuration where dipoles order along the lattice, and the other in a state where dipoles settle in a parallel antiferromagnetic arrangement orthogonal to the lattice. While the lattice is in the planar phase, as the offset is increased, two Bloch domain walls (DW) arise at the edges of the antiferromagnetic chain while the angular positions of the ferromagnetic sublattice remain motionless. As the offset increases further, the two DW move from the edges of the lattice until they meet and annihilate at its center. As the offset is increased even further, the system returns to the planar phase.
In the second part of the paper we examine a continuum model that allows unveiling the internal fields responsible for the DW birth and identifying the forces that promote their translations. We find that solitons are stabilized by an emergent magnetic field product of symmetric and chiral internal fields rooted in the inter-sublattice dipolar interactions. We show that dynamics of such solitons along the zig-zag lattice is due to the symmetric coupling between sublattices and is induced by an internal electromagnetic force that emerges along the lattice when one sublattice moves with respect to the other.
The rest of the paper is organized as follows. Section II describes the model, examines the magnetic equilibrium states of the zig-zag lattice as a function of the offset and identifies the inter and intra-chain symmetric and chiral long-ranged couplings between dipoles. In order to understand the dynamics of the Bloch domain walls arose in the planar phase of the lattice, in section III we write the field theory for the antiferromagnetic chain. By describing solitons in terms of collective coordinates, we identify the electromagnetic force on the soliton in regard to the internal dipolar couplings. This allows the formulation of a new strategy for propelling solitons in antiferromagnetic chains. Section IV is devoted to discussion and concluding remarks. Appendix A and Appendix B show respectively details of the numerical simulations and analytical calculations.

II. MODEL
The magnetic dipolar energy for dipoles in the zig-zag lattice reads k | denotes the unit vector joining dipolem α i at site i and sublattice α and dipolem β k at site k and sublattice β. g = µ 0 m 2 0 4πa 3 x sets an energy scale [14] and contains the physical parameters of the system, such as a x , the lattice constant along the x direction, two nearest neighbor sites of sublattice f as shown in Fig.1 (left). f contains n dipoles at its sites and af has n − 1. Dipoles in f rotate in terms of a polar angle θ in the planex −ẑ, that is respect to a local axis alongŷ, fixed to their center. Dipoles in af, on the other hand, rotate in terms of θ in theŷ −ẑ plane, that is, respect to a local axis alongx (see Fig.1).
All dipoles of the zig-zag lattice have fixed angular positions respect to the azimuthal angle ϕ α : ϕ f = 0 and ϕ af = π 2 . Hereafter the magnetic moments are normalized by m 0 , and dipoles belonging to sublattice α : (f, af ) have unit vectorm α i = (sin θ α i cos ϕ α , sin θ α i sin ϕ α , cos θ α i ). Because sublattices have easy planes mutually perpendicular, the full dipolar energy Eq.1 is separable into symmetric and antisymmetric long-range interactions (see AppendixB for details). They give rise to four energetic contributions to the magnetic energy of the system and are consecutively denoted such that U d = g . They correspond respectively to symmetric intra-sublattice interactions in f : , a symmetric inter-sublattice interaction: U J ik = J ik m f i ·m af k , and an antisymmetric inter-sublattice interaction energy U dm . The associated couplings between dipoles i and k read J 0 3/2 which are respectively symmetric intra-chain and interchain couplings.

A. Equilibrium magnetic states
We have used molecular dynamics simulations to study the magnetic evolution of dipoles increases. Consequently, they spread, though slightly and at a very small rate which causes no big damage to their structure, before the two meet. 2) As η increases, the solitons move from the edges toward the center of the lattice. Once they have met at the middle point, af relaxes back to one of its AF ground states ±ŷ (lower row Fig.2(a)). In what follows, we focus on the af sublattice in the soliton regime, which occur for offsets η ≥ 0.8. In this regime, simulations show that af is in the AF state, and hosts two Bloch DWs for a range of values of η. On the other hand, f remains in the F state, with a constant intrachain energy which from now on we set to zero.
B. Long ranged symmetric and anti-symmetric couplings.

III. DYNAMICS OF SOLITONS
The Bloch domain walls [18,19] found above are localised structures which propagates along af with shape retention. Therefore they constitute solitons [20]. In this section we use a continuum field theory to examine how the internal fields h J (y) and h dm (y) stabilize and accelerate solitons in the af chain.

A. AFM field theory
In the non-dissipative continuum limit, the dynamics of magnetization fields in af is determined by the lagrangian density [9,12,21,22] The term ρ 2 |ṅ| 2 is the kinetic energy of the staggered magnetization and ρ = χ/γ 2 is the density of inertia of n. The second term ργh ef · (n ×ṅ) also known as gyroscopic [23] quantifies the effective geometric phase for the dynamics of n, due to h ef [12]. The two last terms correspond to the potential energy. In the context of the zig-zag chain studied here, the first term is proportional to the exchange strength A = m 0 γ 2 hex > 0 where the exchange field h ex is proportional to the energy of the uniform antiferromagnetic ground state. This term is due to the symmetric intrachain interactions in af which favor a uniform antiferromagnetic state (∂ x n = 0). The potential term ρ 2 |γh ef × n| 2 ∼

B. Electromagnetic force on the soliton
The dynamics of n is obtained from the Euler Lagrange equations from the lagrangian density Eq.3. In the cartesian coordinate system shown in Fig.1, axisx is perpendicular to the DW and axisẑ parallel to it. For the analysis of the solitons in Fig.2, it is convenient to use the language of collective coordinates [25] and parametrize them by the variables q 1 ≡ X and q 2 ≡ Φ, corresponding to their position along the lattice and the azimuthal angle, respectively [26]. Thus, the DW profile can be written as: Eq.5 defines a static DW and minimizes Eq.4, where θ and φ are the polar and azimuthal angles parameterizing the unit vector n [1]. To describe the low energy dynamics of the domain wall, we promote the two collective coordinates to dynamics variables in the DW anzats Eq.5. The variation of n = (sech x−X λ cos φ, tanh x−X λ , sech x−X λ sin φ) in time is mediated by the change of these collective coordinates:ṅ =q i ∂n/∂q i =Ẋ∂n/∂X where repeated indices are implicitly summed over and we used thatΦ = 0. The kinetic energy of the DW becomes M ijqiqj /2, where the inertia tensor is defined as M ij = ρ ( ∂n ∂q i · ∂n ∂q j )dx [9]. In the Lagrangian of the soliton, the gyroscopic term [27] can be written in terms of a gauge field as A iqi where A i (X, Φ) = ργ h ef · ( ∂n ∂q i × n)dx [9]. The gauge potential for the domain wall of Eq.5 has components that depend on y (AppendixB): The curl of A yields an emergent magnetic field [9], This field, the product of the interchain dipolar coupling, decays with y. When B XΦ is time-dependent, it induces an electric field E i = ργḣ ef · ( ∂n ∂q i × n)dx [25]. In the experiment of [14], the sublattices of the zig-zag lattice move apart at speed v = ∂y ∂t . In this case, the effective magnetic field becomes dependent on timeḣ ef = ∂h ef ∂y v and gives rise to an emergent electric field with components E X (y) = ±ργv π 2 sin Φ ∂h J ∂y and E Φ (y) = ±λργv ∂h dm ∂y . The emergent fields satisfy Jacobi identities and hence Maxwell equations ∇ × E +Ḃ = 0 and ∇ · B = 0 [12,25]. The previous numerical analysis indicate that solitons are rigid, thus they have only one continuous degrees of freedom which is the zero mode X associated to the global translational symmetry along an infinite af. The effective field h ef (y) respects that symmetry. Therefore for the solitons of Fig.2 where φ = π/2 we haveφ = 0 and B XΦ = ±ργh dm . The electromagnetic force on the DW along the lattice axis (x), is such that F X = E X = ±ργv π 2 ∂h J ∂y = − ∂A X ∂t and depends only on the emergent electric field. Consequently, the implicit time dependence of h J (y) allows the acceleration of the DW along the zig-zag lattice as long as there is a relative velocity between its sublattices (AppendixB).

IV. SUMMARY AND DISCUSSION
Finding new strategies to propel antiferromagnetic domain walls remains a challenge.
The purpose of this work has been to examine whether and under which circumstances stable antiferromagnetic domain walls could be driven by internal emergent fields arising at magnetic interfaces. To that aim, we modeled an interface of two magnetic lattices by a zig-zag chain of easy plane dipoles interacting via dipolar coupling. We found that such a model realizes a planar magnetic state with one sublattice antiferromagnetic and the other in the ferromagnetic state. The internal Dzyaloshinskii-Moriya field, which arises from the inter-sublattice dipolar interactions, stabilizes chiral solitons at each edge of the antiferromagnetic sublattice. The dynamics of such solitons is studied by deriving the longwavelength lagrangian density of this non-conventional easy axis antiferromagnetic chain in the soliton regime. Using the collective coordinates formalism, we find that aside from conservative forces, the internal fields due to the chiral and symmetric couplings between sublattices generate an effective magnetic field exerting a gyrotropic force on the solitons and an induced electric field when the two chains move apart along the lattice inversion axis.
The emergent electromagnetic fields satisfy Maxwell equations and produce a Lorentz force that can accelerate solitons. As a result, from the edges of af , two Bloch DW migrate to the center of the lattice before they meet and disappear. An important aspect of this work is that, once the magnetization vectors are plugged into the formula for the dipolar energy, it is possible to separate antisymmetric and symmetric interchain dipolar contributions. This step allows to track the role of each of them in the subsequent continuum analysis of the soliton regime: while the antisymmetric contribution gives rise to an internal magnetic field able to produce Bloch DWs at the edge of the lattice, once the solitons have arisen, it is the symmetric contribution to the energy which produces the force that accelerates the solitons along the lattice as long as the gap between sublattices changes with time. This means that a possible strategy to accelerate solitons could involve magnetic fields that point along the lattice but change in time orthogonal to it.
A natural next step could generalize this model into two dimensions. That would emulate a more realistic version of the magnetic interface and allow us to test whether the unconventional dynamics shown here remain when the two chains are extended in the planex −ŷ.
Another exciting venue to be studied is the effect of an electric current passing through the interface. When the spin of electrons encounters a magnetic texture undergoes precession and exchange angular momentum with the soliton [25]. While it has been shown [9] that the adiabatic spin torque cannot propel a DW on its own, it would be interesting to examine its effect in the model presented here. The angular variable φ i is fixed and defines a plane of rotation with normal vector p i = (− sin φ, cos φ, 0). The angular variable θ i is a dynamical variable that changes in time due to the interaction of the magnetic dipole i with its environment. The equation of motion for the polar angles of inertial dipoles located at sites i in sublattice α, interacting through the full long-range dipolar potential with all other dipoles in the lattice, reads: Where I denotes the moment of inertia of the magnets, ξ is the damping for the rotation of dipoles in the lattice. The first term at the right hand side of the previous equation is the magnetic torque due to the action of the internal magnetic field from dipolar interaction between all dipoles in the system We solved this set of coupled equations using a discretized scheme for the integration of the differential equation, given by the recursion where the functions f a , f b and f c corresponds to We performed the simulations for systems with a total of 2n − 1 = 77, 97 and 117 magnetic dipoles, starting with the sublattices close together and separating af from f at constant speed. The fixed angular variable φ that defines the plane of rotation was chosen as 0 for f n and π/2 + δφ i for af. The quantity δφ i corresponds to a Gaussian random noise with zero mean and standard deviation equal to 0.01.
The time interval used to integrate the equations of motion was ∆t = 2 × 10 −6 , and at each time step, sublattice af was moved apart from f along theŷ direction by a distance of δη = 3.5 × 10 −8 , or a y = 1.6 × 10 −6 a x , with a x = 2.2 × 10 −2 the distance between neighbour magnetic dipoles in the same sublattice. This integration time step is smaller than the shortest time scale in the system which can be estimated from the dipolar force between two nearest neighbor dipoles, and scales as τ c ∼ 8π πIL µ 0 ax m 0 ∼ 0.2 s. Therefore, at every time step of the integration, dipoles have relaxed to their equilibrium angular positions.
To examine the soliton regime (η > 0.8) the initial distance between sublattices was set to η = 0.5. Initial conditions for angular positions of dipoles in the the zig-zag lattice were antiferromagnetic for both sublattices, which is the stable magnetic configuration at η ∼ 0.5, and the transition to the solitonic regimen was seen at η ∼ 1. As η increased from 0.5 to 2 we stored the angular variable θ i for each dipole i at every time step.
Stability was also studied, by suddenly stopping the relative movement between the chains in the soliton regime and then waiting a time of ∼ 2s to observe the evolution of the angles.
Solitons were found to be stable for distances near the transition, with small oscillations that did not destroy the profile of the chiral structures.
In Fig.2, we show the magnetization of each dipole in af chain for different distances along the simulation in the soliton regimen. Fig.2(a) Fig.2(b) and Fig.2(c) show the projections of the magnetization along theŷ and z axis respectively. The energy landscape shown in Fig.3 corresponds to the effective torque T that produces chain f over each magnet of chain af projected on the plane perpendicular to the magnet affected by said torque. This is computed as And the dipolar energy becomes: and J ik = 1 η 2 + (i − k + 1 2 ) 2 3/2 are respectively the exchange coupling between dipoles belonging to same and different chains. The third term in the right hand side of Eq. B6 is a Dzyaloshinskii-Moriya (DM) antisymmetric type of exchange perpendicular to the plane of the system,

Internal fields in the soliton regime
Consider the energy from the antisymmetric interchain coupling with h k dm = ± g 2m 0 i∈f D ikŷ , and where we used that in the soliton regimem af = ±(1, 0, 0). Similarly, the energy from the symmetric interchain coupling yields, where h k J = ± g 2m 0 i∈f J ikx

Inertia Tensor
The kinetic energy of the DW is given by M ijqiqj /2, where the inertia tensor is defined as M ij = ρ ( ∂n ∂q i · ∂n ∂q j )dx. In our case, M XΦ = 0 and The gauge potential for the domain wall of Eq.5 is defined as A i (X, Φ) = ργ h ef · ( ∂n ∂q i × n)dx. Its components depend on y,

Emergent magnetic field B XΦ
The curl of the vector potential yields an emergent magnetic field product of the interchain dipolar coupling, where F (L) = arctan sinh L−X λ + sech L−X λ tanh L−X λ

Lorentz Force
The force on the soliton along the x direction is given by the electric field along X and the gyroscopic force, analogous of the Lorentz force [12] F X = E X + B XΦΦ = E X (B16) Using the previous results is easy to find that the time derivative of the gauge potential dA X dt = −E X , and therefore F X = − dA X dt