Goldstone Mode of Skyrmion Crystal

We discuss the Goldstone mode of skyrmion crystal in a model of two-dimensional ferromagnet with the Dzyaloshinskii–Moriya interaction in magnetic field. We use stereographic projection approach to construct skyrmion crystal and consider skyrmion displacement field. The small overlap of the individual skyrmion images restricts the potential energy to the interaction of nearest neighboring displacements. The closed form of the Goldstone mode dispersion is found and its dependence on the magnetic field is studied. We use semiclassical quantization to define the Green’s function and show that the propagation of displacements through the crystal changes its tensorial form from anisotropic to isotropic one at large times.

Introduction.Magnetic skyrmions are topologically nontrivial whirls of local magnetization.They may serve as building blocks for novel racetrack memory devices 1 or programmable logic devices 2 , thanks to topological protection, small size and ability to be manipulated by spin torques.
One can consider a magnetic skyrmion 3 as an extremely small magnetic bubble 4 (cylindrical domain wall 5 ).In terms of domain walls, skyrmion's radius is comparable with its width and defined by Dzyaloshinskii-Moriya interaction (DMI) constant 3,6 .
The dynamics of a local magnetization is usually described by the Landau-Lifshitz-Gilbert(LLG) equation.LLG equation can be further transformed into the Thiele 7 equation in case of domain wall steady motion.Thiele equation and its generalizations 8 are the main tool for skyrmions motion analysis 9 .Thiele equation allows one to take into account the spin current impact on skyrmions and discuss the situation of skyrmions on a track, see 9 In non-centrosymmetric magnets with DMI magnetic skyrmions are often arranged into regular lattices [10][11][12] .Such lattices (called also skyrmion crytals (SkX) 13 ) are preferable to uniform, helix or cone configurations in case when a single skyrmion configuration becomes energetically more favorable 14 .It was shown that the densely packed skyrmion configurations is characterized by both pairwise repulsive and triple attractive interaction between skyrmions 15 .Hence, the motion of individual skyrmions in a lattice depends on its neighbours, and the SkX dynamics cannot generally be reduced to the motion of solitary skyrmion in a potential well.
Excitations of SkX have are described by the complicated band structure [16][17][18][19] .The lattice excitations of different angular symmetry correspond to different distortions of individual skyrmions, among them elliptical deformation, breathing mode, clockwise and counter-clockwise motion etc.The modes with certain symmetries 20 show up in magnetic resonance experiments 21 .
The soft Goldstone mode of SkX, also called gyrotropic mode, is associated with displacement of skyrmions in SkX and was predicted in Ref. 22 .SkX was represented there as a sum of three magnetic helices with the corresponding phase shifts, and it was shown that the topological term in the Lagrangian leads to quadratic dispersion of the soft mode (it was recently verified numerically in Ref. 23 ).The gyrotropic mode does not manifest itself in magnetic resonance experiments 24 , but somehow appears in inelastic neutron scattering 25 .
In this work we consider a simplest model of noncentrosymmetric ferromagnet with DMI in external magnetic field, whose ground state is SkX in a certain range of parameters.Staying in a framework of stereographic projection approach and regarding skyrmions in quasiparticles paradigm, we consider a displacement field of skyrmions positions in the lattice.We numerically show the nearest-neighbour character of displacements' coupling and obtain closed form of dispersion of the Goldstone mode.The dependence of force constants on external magnetic field is also numerically found.The dynamical Green's function of displacements is isotripic at small distances, while showing anisotropic tensor structure at larger distances. Model.
We consider planar model of noncentrosymmetric ferromagnet with DMI in uniform external magnetic field perpendicular to the plane.The energy density is given by: where C is an exchange parameter, D is DMI constant, and B is an external magnetic field magnitude.There is a convenient way to choose measurement units in the model (1): we will measure length in the units of l = C/D, and energy density in the units of CS 2 l −2 = S 2 D 2 /C.Then the energy of the model depends only on the dimensionless parameter b = BC/SD 2 .We consider low temperature limit, when local magnetization is saturated, and its magnitude doesn't change from point to point S = Sn, with |n| = 1.The above planar model is applicable also to thin films, whose thickness is less or comparable to l.We ignore the magnetic dipolar interaction here, because it can be reduced to uniaxial anisotropy for ultrathin films.Small anisotropy leads only to minor changes of SkX parameters.The stereographic projection representation of the vector n reads as with f a complex-valued function, and f its complex conjugate.A single skyrmion's stereographic function is conveniently represented by where κ is a smooth real profile function, z 0 is a skyrmion size parameter.The ansatz (3) is more convenient for the description of SkX case, while in case of one skyrmion it reproduces the profile obtained by usual bubble domain ansatz.It was shown previously that the multiskyrmion configurations can be built as a sum of stereographic functions of individual skyrmions 15 .Particularly, the regularly arranged SkX corresponds to the following stereographic function: where a 1 = (0, a), a 2 = (− √ 3a/2, a/2), and a is a cell parameter of SkX.The static properties of this ansatz (3)-( 4) was discussed to some detail in previous works 14,15 .It was shown that the proposed SkX configuration has lower energy than helix or uniform configuration at magnetic fields, 0.25 ≲ b ≲ 0.8.
The dynamics of local magnetization follows from the Lagrangian, L = T − E, with the kinetic term here φ and θ define the magnetization direction n = (cos φ sin θ, sin φ sin θ, cos θ), and γ 0 gyromagnetic ratio.
This form of the Lagrangian leads to the well known Landau-Lifshitz equation.The expression (5) may be rewritten in terms of f as and the factor S/γ 0 was included into the time scale.The exact equation of motion for f (t) is highly nonlinear and cannot generally be solved.In previous works 19,20 we have discussed the normal modes of infinitesimal fluctuations of the function f .In this work we develop a special approach for consideration of the gyrotropic mode of SkX.
Displacement field and dispersion.In our previous papers we associated the above configuration f SkX , Eq.( 4), with the equilibrium spin configuration, f 0 , and considered small fluctuations, δf , around it, writing with the dynamics of ψ is discussed at length in 19 .
In this study we assume that the skyrmion lattice is imperfect, in the sense that the stereographic image is still given by the sum of individual images of skyrmions, and the shape of each image is unchanged, but the only imperfection is the position of the center of skyrmions.It is similar to the description of ions' displacements in crystals in the theory of phonons.
where r (0) l = na 1 +ma 2 with integer n, m and a 1,2 lattice vectors.For infinitesimal displacements u l we can write In order to make use of our previously found formulas in 19 , we define the quantity ψ(r) as also and similarly for complex conjugated ψ.As a result, we obtain For the uniform shift u j = u, using the property j (f j , fj ) = (f 0 , f0 ), we restore the previously discussed zero modes (Eqs.(29) in 19 ) : The quadratic in displacements part of the Lagrangian takes the form with the explicit form of U , V and A is given in 19 . 26et us discuss a few general properties.(iv) It can be shown that j Klj = πσ 3 , it corresponds to the value of topological charge per unit cell of skyrmion crystal.
(v) For triangular lattice with six NN we have d = (a cos ϕ d , a sin ϕ d ) with ϕ d = π 3 (n − 1/2) and n = 0, . . . 5. Individual skyrmions are characterized by certain chirality, f 1 (r) ∝ e iϕ /r.Considering the symmetry of the potentials U, V and matrix O j under the rotation ϕ → ϕ + ϕ d , we notice that the phase ϕ d doubles in the off-diagonal components and is absent in diagonal components of Ĥlj .As a result we have a structure Dependence of the hopping constants, h1, h2, and the stiffness, A, on magnetic field b.
with h 1,2 depending only on the distance, d.For the onsite term l = j, the off-diagonal components are absent, h 2 = 0. Numerically, we find that h 1,2 < 0 for l ̸ = j.
(vi) The Thiele equation for the motion of l'th skyrmion is obtained by putting u ± j = 0 in ( 13) for all j ̸ = l.In this case the Lagrangian becomes 2 ), cf. 27.Notice that even if u ± j = 0 initially, the collective character of Eq. ( 13) leads to eventual propagation of perturbation around the initial displacement.We discuss it in more detail below.
Using the above properties, we represent the quadratic part of the Lagrangian as where we defined u x j ± iu y j = q e iqrj u ± q , and the sums over six NN are with the property γ s,d (0) = 0.The zero value of diagonal components of Ĥq at q = 0 is the explicit use of the above property (ii), and we will return to it below.The dispersion law ω = ϵ q in the time dependence u ± q (t) = e iωt u ± q is given by solving the equation det(ω Kq − Ĥq ) = 0, which yields We obtain numerically that h 1 ≃ 0.84 h 2 in the whole range of relevant fields, b ∈ (0.3, 0.8).The dependence of h 1,2 on b is shown in Fig. 2. We see here that both coefficients vanish simultaneously at the critical value of the field, b c ≃ 0.8.In terms of elastic theory, discussed in 22 , it corresponds to Lamé coefficients λ, µ decreasing and vanishing at b = b c , whereas λ ≃ 1.94µ, see also Eq. ( 19) below.The dependence of ϵ q along the symmetry lines of the Brillouin zone is shown in Fig. 3.
The values of ϵ q are 9|h 1 |/π and 2 4h 2 1 − h 2 2 /π at q = K and q = M , respectively, and we have ϵ q=M /ϵ q=K ≃ 0.71 for all values of b.The diminishing of the gyrotropic bandwidth with b while maintaining the above ratio 0.71 is consistent with the results reported in 18 .The amplitude k 1 for the hopping contribution in the kinetic term Kq is negative and small, |k 1 | < 2 • 10 −2 , and we can safely ignore it in a qualitative discussion below.
In the limit of small wave vectors, |q| ≪ 1, we have The dependence of the stiffness coefficient, A, on b is also shown in Fig. 2 ; there is no simple proportionality between h 1,2 and A, since d also depends on b, see 19 .It is worthwhile to compare the value of A with the dispersion law of uniform ferromagnet.The latter case corresponds to f 0 ≡ 0, and further to U = b, V = 0 and A = 0 in (13), see Eqs. ( 16)- (18) in 19 .This results to ϵ q = q 2 + b, i.e. to A = 1 and the gapped character of the spectrum.We see that compared to the uniform ferromagnetic case, the lowest-lying part of the spectrum of SkX is gapless and its stiffness decreases with the field.One may ask, how this gapless character becomes gapful one at b c , when the SkX dissolves.The answer is that the applicability of ( 16) shrinks to q = 0 in this case, as a → ∞ at b c , cf. 28 .Let us now discuss the above property (ii), which we incorporated into the definition of γ s (q).Our numerical computation with the use of our trial function shows that the property Ĥq = 0 at q = 0 is satisfied to the accuracy of 5•10 −3 in the range of the fields, 0.35 < b < 0.75.This shows rather good quality of our trial function, in accordance with our previous paper 19 .At the same time, since our trial function does not provide the exact extremum of the action, its first variation is not identically zero.In this case additional terms should be added to the effective Hamiltonian, Ĥlj in (13), as is explained below.
We expand the variation of our function to second order in displacements, This expression should be multiplied by δL/δf , which is now assumed not being identically zero.The first order term here, ∝ u α l , when integrated over r, produces the force applied to lth skyrmion out of equilibrium position.The second-order term in (17), when multiplied by δL/δf and integrated over r, adds to expression (13) in a following way.
First we note that the kinetic part of δL/δf , being symmetric in indices α, β, leads to terms proportional to full time derivatives, ∝ d dt (u α l ) 2 , which can be discarded.The potential part of the Lagrangian can be integrated by parts and the boundary term vanishes due to the rapid decrease of f 1 (r) with distance.The rest can be represented, after some calculation, in a form which means that we should replace Ĥlj in (13) by Hlj = Ĥlj − δ lj m Ĥlm .Clearly, if the property j Ĥlj = 0 is not fulfilled, due to imprecise character of our trial function, then this property is ultimately restored for the corrected form, Hlj , with j Hlj = 0. Using the symmetry of (17) in indices α, β, one can also show that l Hlj = 0 as well.It justifies the above general property (ii) and the subtraction of number 6 in our definition of γ s (q).Green's function in small q limit.
Let us discuss now the propagation of displacements through the SkX.To simplify our discussion, we perform two subsequent rotations of our Lagrangian.First, we return to the Cartesian basis u ± q = u x q ± iu y q , which corresponds to rotation U = 1, −i 1, i .Second, we bring Ĥq to principal axes, parallel and perpendicular to q = q(cos ϕ q , sin ϕ q ), by writing (u x q , u y q ) = (u ∥ q , u ⊥ q ) • U † 1 with with U 1 = cos ϕ q , − sin ϕ q sin ϕ q , cos ϕ q .As a result we reduce the Green's functions, G ∥ (t, r) and G ⊥ (t, r), plotted for t = 6A −1 a 2 , describing pairwise correlation for displacements, u ∥ and u ⊥ , parallel and perpendicular to r.

FIG. 1 .
FIG. 1.A sketch of SkX with one displaced skyrmion.Red arrows and a hexagon illustrate lattice vectors and the primitive cell of SkX, the orange circle shows a typical value of z0 parameter, green arrow indicates a displacement of skyrmion in the right top corner.The Brillouin zone with symmetry points is depicted in the bottom right corner.
(i) Due to translation invariance, the quantities Klj and Ĥlj depend only on the difference d = r The zero mode corresponds to summation over j, and we should have j Ĥlj = l Ĥlj = 0, see below.(iii)The image of a single skyrmion f 1 (r) decreases exponentially with distance.It follows that Klj , Ĥlj decrease rapidly with |r For practical reasons it suffices to consider only on-site term, l = j, and the nearest neighbors (NN).