Lattice Solutions in a Ginzburg–Landau Model for a Chiral Magnet

We examine micromagnetic pattern formation in chiral magnets, driven by the competition of Heisenberg exchange, Dzyaloshinskii–Moriya interaction, easy-plane anisotropy and thermodynamic Landau potentials. Based on equivariant bifurcation theory, we prove existence of lattice solutions branching off the zero magnetization state and investigate their stability. We observe in particular the stabilization of quadratic vortex–antivortex lattice configurations and instability of hexagonal skyrmion lattice configurations, and we illustrate our findings by numerical studies.

tensor ∇m × m and is therefore sensitive with respect to reflections and independent rotation in the domain and the target space, respectively. It is well-known that DMI gives rise to modulated phases. The basic phenomenon is that the energy of the homogeneous magnetization state m = const. can be lowered by means of spiralization in the form of periodic domain wall arrays, the helical phase. A prominent form of doubly periodic lattice state arises from the stabilization of topological structures, so-called chiral skyrmions, in two-dimensional chiral ferromagnets. Chiral skyrmions are localized structures which are topologically characterized by a unit S 2 degree and a well-defined helicity depending on the specific form of DMI. In the presence of sufficiently strong perpendicular anisotropy and/or Zeeman field interaction, chiral skyrmions occur as local energy minimizers in form of isolated topological solitons. Zeeman field interaction enables the possibility of an intermediate regime where chiral skyrmion embedded into a hexagonal lattice is expected to be globally energy minimizing. Micromagnetic theories including DMI have been proposed in Bogdanov and Yablonskii (1989) with the idea that skyrmion lattice configurations represent a magnetic analog of the mixed state in type-II superconductors. Corresponding phase diagrams and stability questions have been examined analytically and numerically in the seminal work Hubert 1994, 1999), see Leonov et al. (2016) for a recent review. A fully rigorous functional analytic theory on the existence, stability, asymptotics, internal structure and exact solvability of isolated chiral skyrmions has recently started to emerge (Melcher 2014;Döring and Melcher 2017;Li and Melcher 2018;Komineas et al. 2019Komineas et al. , 2020Barton-Singer et al. 2020;Bernand-Mantel et al. 2019).
Lattice states in two-dimensional Ginzburg-Landau models for chiral magnets including thermodynamic effects have been proposed theoretically in Rößler et al. (2006), see also Mühlbauer et al. (2009) and Yu et al. (2018). Mathematically, one may expect a close analogy with Abrikosov's vortex lattice solutions in Ginzburg-Landau models for superconductors (Abrikosov 1957) with a well-established theory in mathematical analysis. The occurrence of Abrikosov lattices in the framework of gauge-periodic solutions of Ginzburg-Landau equations and the optimality of hexagonal lattices has been thoroughly investigated by means of variational methods and bifurcation theory (Barany et al. 1992;Odeh 1967;Tzaneteas and Sigal 2009;Aydi and Sandier 2009;Aftalion et al. 2006;Sandier and Serfaty 2012).
Ginzburg-Landau models in micromagnetics are, in contrast to superconductivity, directly formulated in terms of the physically observable magnetization field. A class of such models has been proposed and examined computationally in Rößler et al. (2006) and Mühlbauer et al. (2009). Compared to the purely ferromagnetic case |m| = const., Ginzburg-Landau models offer a larger variety of patterns, including vortex and halfskyrmion arrays of opposite in-plane winding on square lattices, and skyrmions on hexagonal lattices, see Fig. 1.
We shall examine the occurrence of periodic solutions near the paramagnetic state of zero magnetization, i.e., in a high temperature regime. Our starting point are magnetization fields m : R 2 → R 3 governed by energy densities of the form The Dirichlet term with A > 0 is referred to as Heisenberg exchange interaction. The helicity term with D = 0 is a prototypical form of DMI arising in the context of noncentrosymmetric cubic crystals. Note that for m = (m, m 3 ) defined on a twodimensional domain A key analytical feature induced by DMI is the loss of independent rotational symmetry in magnetization space R 3 and the domain R 2 . Finally, the Landau term f is an even polynomial in the modulus |m| a(T − T C )|m| 2 + b|m| 4 + c|m| 6 + d|m| 8 + · · · where T − T C is the deviation from the Curie temperature. We shall focus on a minimal model with For stability reasons, we also include the easy-plane anisotropy K (m·ê 3 ) 2 with K > 0, which typically emerges as a reduced form of magnetostatic stray-field interaction in thin-film geometries, see, e.g., Gioia and James (1997). We are interested in magnetization fields m which are periodic with respect to a two-dimensional lattice r (r > 0) with where τ is a complex number in the fundamental domain of the modular group, referred to as the lattice shape parameter, see Sect. 2.1. Rescaling space we may assume r = 1. The rescaled energy density reads with dimensionless constants Since a sign reversal of the DM density can be achieved by reflections such as m 3 → −m 3 , we may assume w.l.o.g that κ > 0.
Euclidean Symmetry The planar model to be examined arises from dimensional reduction. It is instructive to return to the original setting and consider the energy density on fields m from R 3 . In the case β = κ = 0, we have invariance with respect to the following action of the Euclidean group in R 3 In the case β = 0 but κ = 0, the reflection symmetry is broken and invariance is restricting to the special Euclidean group with R ∈ SO(3), see Lemma 6. Including anisotropy β = 0 and κ = 0 amounts to a further restriction of the rotation group to elements of the form defining an embedding O(2) → SO(3). Restricting to only horizontal translations t = (t, 0) with t ∈ R 2 amounts to invariance of the two-dimensional model with respect to the action of the Euclidean group in R 2 where R ∈ SO(3) is given by (3). We shall investigate the occurrence and stability of non-trivial -periodic critical points m of the average energy over a primitive cell i.e., of non-trivial -periodic solutions m to the Euler-Lagrange equation We first discuss energy minimizing solutions.
(i) If λ > κ 2 , then m ≡ 0 is the unique energy minimizer on every lattice.
(ii) If λ < κ 2 and β = 0, then the helix (see Fig. 2) is up to a joint rotation the unique energy minimizer on suitable lattices.
In the isotropic case β = 0, Theorem 1 indicates the existence of only two phases, paramagnetic or helical, while the picture in the anisotropic case β > 0 is incomplete. The occurrence of helical phases is common to other mathematically related theories for condensed matter such as the Oseen-Frank model for chiral liquid crystals (see, e.g., Virga 1995) or the Gross-Pitaevskii model for spin-orbit coupled Bose-Einstein condensates (see, e.g., Aftalion and Rodiac 2019). Helical structures in chiral ferromagnets are also discussed in Davoli and Di Fratta (2020) and Muratov and Slastikov (2017).
Here, we are interested in doubly periodic solutions. Given a lattice , we aim to find λ and a non-trivial -periodic solution m of (6) at λ. We call such pairs (m, λ) -lattice solutions and will prove the following: Theorem 2 Suppose α > 0, β ≥ 0, κ > 0, and = 2π Im τ (Z ⊕ τ Z). Then, (6) has a branch of -lattice solution (m s , λ s ), analytically parameterized by a real parameter s near 0, in a neighborhood of m 0 ≡ 0 and provided λ 0 satisfies the non-resonances condition that for any ω ∈ * \S 1 where * denotes the dual lattice, see Sect. 2.3. The branch (m s , λ s ) has the form as s → 0 with ν 2 < 0 and ϕ 1 explicitly determined. Furthermore, we have and Journal of Nonlinear Science (2020) 30:3389-3420 where · denote the average over a primitive cell .
The morphology of bifurcation solutions is related to symmetry properties of the underlying lattice. Depending on this, the first-order bifurcation solution ϕ 1 , arising from the first critical wave number, indicates a threefold pattern formation: 1. Helical pattern exists on all lattices, the first-order bifurcation solution (29) is given by a single helical mode (see Reducing the domain of the bifurcation parameter s if necessary we shall prove the following stability properties of quadratic vortex-antivortex and hexagonal skyrmion bifurcation solutions obtained in Theorem 2: and where γ is the second critical wave number, i.e., depending on the lattice shape τ = |τ |e iθ . κ, see Li (2020).

Corollary 1 The quadratic vortex-antivortex lattice configuration exists and is stable if
The admissible set of (κ, β) is not empty, see Fig. 3.
The existence and stability results are only an initial step towards understanding the stabilization of two-dimensional lattice solutions in chiral magnets. In particular, stability of lattice solutions is only examined under the simplest perturbations which preserve lattice periodicity. A more general stability result in the style of Sigal and Tzaneteas (2018) is beyond the scope of this work and requires a different approach.
The mathematical framework for our construction of lattice solutions is the equivariant branching lemma (Chossat and Lauterbach 2000;Golubitsky et al. 2012), a concept of symmetry-breaking bifurcation based on a particular type of (axial) symmetry group. More precisely, letting where P ⊂ O(2) is the point group (or holohedry) of and T = R 2 / is the torus of translations modulo , the Euclidean symmetry (4) induces an action of on spaces of -periodic fields m. For each lattice, we identify all isotropy subgroups ⊂ (up to conjugacy) so that the fixed subspace of in the kernel of linearized operator D m F(m 0 , λ 0 ) is one-dimensional. By means of an equivariant Lyapunov-Schmidt procedure, (6) reduces to a one-dimensional bifurcation equation. The implicit function theorem provides a solutions to the bifurcation equation in the one-dimensional fixed subspace of , from which a solution to (6) can be reconstructed. This solution is the bifurcation solution and inherits the symmetries featured by . Fig. 3 The admissible set of (κ, β) for a stable quadratic vortex-antivortex lattice (indicated by the grey shaded area) In Sect. 2, we shall briefly recall the representation of lattices in the plane with an emphasis on symmetry and Fourier series which are key to our bifurcation argument. In Sect. 3, we shall derive energy bounds proving Theorem 1. Solving the linearized version of Eq. (6) explicitly by Fourier methods is the key ingredient to the proof of Theorem 2 in Sect. 4 and provides insight about the morphology and topology of bifurcation solutions. In Sect. 5, we investigate the stability of bifurcation solutions under -periodic perturbations proving Theorem 3 . Finally, in Sect. 6 we validate our analytical results by a series of numerical simulations of gradient flows using a modified Crank-Nicolson scheme.

Representation of Lattices
Recall that a planar lattice is the integer span of two linearly independent vectors t 1 , t 2 ∈ R 2 , i.e., Given x ∈ R 2 , a primitive cell of is a set of the form = {x + a 1 t 1 + a 2 t 2 , a 1 , a 2 ∈ [0, 1]}. The lattice basis {t 1 , t 2 } is clearly non-unique. Identifying R 2 ∼ = C, however, the complex ratio τ ∈ C of two basis vectors of contained in the fundamental domain parametrizes the lattice shape uniquely, see, e.g., Ahlfors (1953). Writing τ = |τ |e iθ , the range of |τ | and θ corresponding to fundamental domain (16) is A lattice is called equilateral if |τ | = 1, where the borderline cases θ = π/2 and θ = π/3 are referred to as square and hexagonal lattice, respectively; other equilateral lattices are called rhombic. There are two distinct types of symmetries preserving the lattice: the lattice translations and the holohedry group P , which is a finite subgroup of O(2). Non-equilateral lattices have holohedry Z 2 (oblique) or D 2 (rectangular); rhombic lattices have holohedry D 2 ; square lattices have holohedry D 4 ; hexagonal lattices have holohedry D 6 , where D k is the dihedral group generated by rotation through 2π/k and a reflection, see, e.g., Chossat and Lauterbach (2000).

Function Spaces on Lattices
As the governing energy densities and the Euler-Lagrange equations are invariant under translation and joint rotation, we can fix one basis vector of the lattice as 2πrê 1 so that = 2πr (Z ⊕ τ Z) is uniquely characterized by r and τ . Upon rescaling, we can arrange = 2π Im τ (Z ⊕ τ Z) spanned by { 2π Im τê 1 , 2π Im τ τ } and consider the rescaled density (1) containing only dimensionless parameters, see Fig. 4. For -periodic functions or fields f , g on R 2 , we denote the average by the L 2 scalar product by and the L 2 norm by f := √ f , f once existent. Accordingly, we define and for k ∈ N the Sobolev spaces respectively. Thanks to Sobolev embedding, the average energy (5) defines an analytic functional on H 1 . Critical points m ∈ H 2 of E satisfy loc is the nonlinear operator given by

Dual Lattice and Fourier Series
Fourier expansion on requires the notion of dual lattice given by In particular, * = A −T Z 2 for = 2π AZ 2 where in our setting In the equilateral case |τ | = 1, dual lattices remain square if θ = π/2 and hexagonal if θ = π/3. For f ∈ L 2 and v ∈ * , Fourier coefficients are defined as and the following Fourier expansion holds true in the L 2 sense along with Parseval's identity

Equivariance and Lattice Symmetry
The action of an element γ = (R, t) of the group = P T , the semi-direct product of the holohedry of and translations modulo , on a field m : R 2 → R 3 given by where the corresponding R ∈ SO(3) is determined by (3), is an isometry on H k for every k ∈ N 0 , and the operator (18) is -equivariant in the sense that for all γ ∈ , m ∈ H 2 and λ ∈ R, see Lemma 6 The symmetry of a field φ ∈ H 2 is given in terms of the isotropy subgroup i.e., the largest subgroup of which fixes φ. Given a subspace X ⊂ H 2 , the fixed subspace associated with a subgroup ⊆ in X is L 2 orthogonal projections on such invariant subspaces of H 2 are equivariant. Equivariance therefore propagates to the Lyapunov-Schmidt decomposition enabling a reduction in the bifurcation equation to Fix X 0 ( ) where X 0 is the kernel of the linearization of F at a bifurcation point (0, λ 0 ), see, e.g., Chossat and Lauterbach (2000). Bifurcation solutions arising from the equivariant branching lemma turn out to have full symmetry and are unique in this class. Anticipating the results in Sect. 4, we introduce a set of isotropy subgroups i ⊆ on different lattice types, which play a central role in our bifurcation argument. In Proposition 2, we shall prove that the i are indeed axial, i.e., have one-dimensional fixed-point subspace in the kernel X 0 .
On non-equilateral lattices (|τ | > 1), we consider the symmetry group where T 1 are the translations in x 1 -direction and Z 2 = {I , R} with associated SO(3) elements The corresponding bifurcation solutions feature a 1 -invariant pattern of helices propagating in the x 2 direction (Fig. 2). Equilateral lattices (|τ | = 1 and π 3 ≤ θ ≤ π 2 ) have an additional symmetry given by reflections across the diagonals of the lattice cell. Therefore, in this case, we consider, in addition to 1 , the symmetry group where the associated SO(3) elements are Bifurcation solutions corresponding to 2 are doubly periodic array of vortices and antivortices (Fig. 1a).

Energy Bounds on Lattices
For a lattice and e(m) given by (1), we examine ansatz-free lower bounds We start by expressing the total exchange density as a sum of sign definite terms and a null Lagrangian also know as Frank's formula in the theory of liquid crystals, see, e.g., Virga (1995) Chapter 3.
Lemma 1 For m ∈ H 1 loc (R 3 ; R 3 ), the following holds in the sense of distributions. In particular for m ∈ H 1 with equality if and only if ∇ × m + κ m = 0 and β = 0.
From the lemma, we obtain immediately claim (i) in Theorem 1, and moreover: Proposition 1 If λ < κ 2 , the energy admits a lower bound which for β = 0 is precisely attained for m of constant modulus |m| = κ 2 − λ α and such that ∇ × m + κ m = 0.
The unimodular Beltrami fields being parallel to their curl have been classified by Ericksen within the variational theory of liquid crystals, see, e.g., Virga (1995) and references therein. For the present case of constant κ, those are helices of pitch 2π/|κ|, i.e., (7). For the convenience of the reader, we present this fundamental result in Appendix A Lemma 7, which yields claim (ii) in Theorem 1. Thus, in order to realize the lower energy bound, the underlying lattice is required to accommodate such a helix. In this case, the zero state loses its linear stability at λ = κ 2 . In fact, the Hessian By the preceding arguments, it satisfies and has a helical instability at λ = κ 2 .

Bifurcation on Lattices
In this section, we prove Theorem 2 based on the equivariant branching lemma. The requisite assumptions are summarized in Proposition 2. Bifurcation points λ 0 arising from the first critical wave number are identified by means of a Fourier expansion in Lemma 2 in combination with Lemma 3. We examine the linearization of F at m = 0 for arbitrary λ given by We need to find non-trivial -periodic solutions φ of the equation for λ = λ 0 depending on κ and β.
Proof We expand φ ∈ H 2 in Fourier series the linearized equation (24) is equivalent to the system and We focus on the wave vectors v ∈ * \{0} of shortest length which are characterized by minimizing problem A straightforward analysis yields (see Fig. 5
Proof The normalized Fourier coefficients (26) obtained in the proof of Lemma 2 are The corresponding real-space solutions of (24) to the wave vector v are We need to find all solutions of L 0 φ = 0 for a given wave number |v|. According to Lemma 3, we consider following cases separately.
Proof of Theorem 2 Clearly, F(0, λ) = 0 for all λ ∈ R. According to Proposition 2, the fixed subspace of the symmetry group = i , i = 1, 2, 3 is one-dimensional in all three cases Invoking the equivariant branching lemma (see Chossat and Lauterbach 2000;Golubitsky et al. 2012), we conclude the existence of bifurcation solutions in the form of  (6) and (5), we obtain (10)-(12). Explicit calculations are carried out in Appendix B.

Topology of Bifurcation Solutions
On hexagonal lattices, the bifurcation solution corresponding to the isotropy subgroup 3 is nowhere vanishing and features in every primitive cell a skyrmion, i.e., a vortex-like structure with the magnetizations pointing upwards at the core and downwards at the perimeter, see Fig. 6a.
On equilateral lattices, the horizontal component of ϕ (2) 1 has a finite number of isolated zeros in a primitive cell and forms a vortex or an antivortex around each zero. The antivortices are half-skyrmions (sometimes referred to as merons, see, e.g., Yu et al. 2018) and have magnetizations pointing upwards or downwards at the core; while the center of vortices are singularity points (m = 0) due to the continuity and the 2 -invariance of bifurcation solutions, see Fig. 6b.

Linear Stability of the Bifurcation Solutions
In this section, we discuss the stability of bifurcation solutions following perturbation methods as, e.g., in Kielhöfer (2011). Suppose (m s , λ s ) is a bifurcation solution as in Theorem 2 with We focus on the larger root as positivity turns out to be necessary for the stability of bifurcation solutions. The linearization of (6) at (m s , λ s ) We first investigate the spectrum of L 0 . (13) and (14) hold. Otherwise, L 0 has negative eigenvalues.
From now on, we focus on the case λ 0 > 0. It follows from Lemma 4 and the standard perturbation theory of eigenvalue (Kato 2013) that the spectrum of L s consists of eigenvalues of the same multiplicities in an neighborhood of the eigenvalues of L 0 . Thus, the stability of (m s , λ s ) depends on the perturbation of the critical eigenvalue 0. It follows from Proposition 2 that there exist the following topological decompositions We first consider the perturbation of the zero eigenvalue corresponding to the eigenvector ϕ 1 spanning Fix X 0 ( ).
Proof We introduce the smooth operator Since L s φ ∈ Fix X 0 ( )⊕ran L 0 for φ ∈ Fix X 0 ( )⊕ X 1 , this operator is well-defined. As G(0, λ 0 , 0, 0) = 0 and the differential given by is invertible, the implicit function theorem provides a smooth map It remains to examine the properties of μ(s). Differentiating (32) with respect to s in s = 0 yieldṡ Calculating the second derivative of μ(s) at s = 0, we obtain
The helical solution on non-equilateral lattices is stable.

Proposition 3 For β ≥ 0 and every nonzero s ∈ (−δ, δ), the bifurcation solution on non-equilateral lattices is linearly stable in the sense that
L s ≥ 0 with ker L s = span{∂ 2 m s }.
Proof Recall that on non-equilateral lattices where both ϕ (1) 1 = φ 1,v (1) and φ 2,v (1) depend only on the spatial variable x 2 and ∂ 2 ϕ (1) is, for small s, a non-trivial element of ker L s , and the claim follows with Lemma 5.
The quadratic vortex-antivortex lattice is stable under large enough anisotropy.
The same argument proves that the helical bifurcation solution on square lattice is stable if β < 4 √ 3 κ and unstable for β > 4 √ 3 κ, see Li (2020). The hexagonal skyrmion lattice is unstable independently of any additional easyplane anisotropy: for example, for any nonzero s ∈ (−δ, δ) 1,v (5) . Similarly, it can be shown that the vortex-antivortex bifurcation solution on hexagonal lattice is unstable and the helical bifurcation solution is linearly stable under any easy-plane anisotropy, for details see Li (2020).

Numerical Scheme
To examine critical points, we consider the L 2 -gradient flow equation for the energy functional E We aim to find equilibria of the energy functional E by solving (35) numerically on a primitive cell induced by lattice spanned by {2πê 1 , 2πτ }. Equation (35) is discretized by a modified Crank-Nicolson approximation for the time variable and a Fourier collocation method for the space variable. We denote m N the trigonometric interpolation function of m on the discretized grid by N 2 collocation points for N N = {0, . . . , N − 1}, N ∈ N and odd. For continuous fields u, v on , we define the discrete L 2 scalar product the associated norm · N , defined by u 2 N := u, u N and the discrete energy Our numerical scheme at time iteration n + 1 reads: find m n+1 such that where I N denotes the trigonometric interpolation operator and At each time iteration, in order to find the solution m n+1 N , we use a fixed point iteration for some time step t. Well-posedness and a-priori error bounds of this numerical scheme follow analogously as in Condette et al. (2011). Given initial data m 0 N , the corresponding sequence (m n N ) n∈N 0 satisfies the following energy law We have implemented this numerical scheme in MATLAB. At each time-step, the iteration process stops if a certain norm of the difference of two successive iterations becomes smaller than a chosen stopping tolerance. In our case, we choose the L ∞norm and set the stopping tolerance to 10 −8 . The discrete energy is evaluated in each time step, and the terminal time is controlled through a smallness condition for the discrete energy gradient, i.e., After the termination of the scheme, an equilibrium configuration is reached approximately.

Numerical Experiments
We have implemented the method on a lattice of 275 × 275 grid points and with a time increment t = 0.1 for different parameters and a randomly distributed initial field with modulus between 0 and 0.1 as initial condition. (13) First, we implemented the simulations on a square lattice for different parameters κ and β near the bifurcation point by setting λ = λ 0 + δν 2 , where δ = 0.01, λ 0 and ν 2 are calculated according to (13) and (11), respectively. For κ ∈ (0, 0.5), the bifurcation point λ 0 is negative for any β ≥ 0. In this case, the stability condition (13) is not fulfilled we obtained an almost homogeneous field.

Parameter Study and Assessment of the Stability Condition
When the value of κ was increased over 0.5, vortex-antivortex lattice configurations were observed for β ≥ 0 small enough so that λ 0 > 0. For β large enough so that λ 0 < 0, the vortex-antivortex lattice configuration decayed to an almost homogenous field, as shown in Fig. 7.
Other patterns emerged for κ > 1.2 and small β ≥ 0. For example, at κ = 1.4 and small β ≥ 0 the solution converged to a stripe pattern, i.e., helices with a pitch smaller than 2π . Vortex-antivortex lattice configurations were observed for β in the admissible region, see Fig. 3, i.e., β larger than the stability threshold (about 2.3κ) and λ 0 > 0. Increasing β further so that λ 0 < 0, we obtained the almost homogeneous field again, as shown in Fig. 8.

Stability of Vortex-Antivortex Solution Under the Perturbation of Lattices
We proved the stability of quadratic vortex-antivortex solutions in certain parameter region under -periodic perturbations (see Sect. 5). Complementarily, we examine the persistence of quadratic vortex-antivortex solutions under the perturbations of lattice shape by implementing the simulations on different lattices with the same parameters. For small perturbations of τ = e iπ/2 , we obtained the vortex-antivortex lattice configuration, while for |τ | large enough only the helix state was observable, as shown in Fig. 9. material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Appendix A: Helical State
In this section, we consider vector fields m defined on a domain in R 3 . Given R ∈ O (3) Proof Upon rescaling, one may assume M = 1 and κ = 1. Taking the divergence, it follows that ∇ · m = 0 and hence m + m = 0 in the sense of distributions by taking the curl. So m is smooth by virtue of standard elliptic regularity theory. In particular, it is enough to prove the claim locally. Denoting the componentwise gradient by (∇m) jk := (∂ j m k ), we claim that rank(∇m) = 1 and (∇m) 2 = 0.
Fixing a point x, we may assume after rotation m(x) =ê 3 , so that by (38) the matrix ∇m(x) is given by a 2 × 2 matrix A such that tr A = 0 and tr A 2 = 0 by (39). It is easy to see that A 2 = 0 which implies det A = 0. According to (36), there exist local smooth unit vector fields X and Y and a function λ such that ∇m = λX ⊗ Y and m = X × Y . Now, ∇ × m = λ X × Y = λ m, hence λ = −1 and ∇m = −X ⊗ Y .
Assuming X is constant so that after rotation X =ê 1 , it follows that m = m(x 1 ) and m 1 = 0. Now, the equation implies for the remaining components m 2 = −m 3 and m 3 = m 2 , so that after a rotation around theê 1 axis, m = h.
To show that X = const., one may invoke the spectral theorem. In fact, symmetry of ∇ X follows from the symmetry of ∇ 2 m = (∇ ⊗ ∇)m since −∇ 2 m = ∇ X ⊗ Y + X ⊗ ∇Y so that − ∇ 2 m · Y = ∇ X.