Homogenization of chiral magnetic materials-A mathematical evidence of Dzyaloshinskii’s predictions on helical structures

In this paper we investigate the in uence of the bulk Dzyaloshinskii-Moriya interaction on the magnetic properties of composite ferromagnetic materials with highly oscillating heterogeneities, in the framework of ¡-convergence and 2-scale convergence. The homogeneous energy functional resulting from our analysis provides an e ective description of most of the magnetic composites of interest nowadays. Although our study covers more general scenarios than the micromagnetic one, it builds on the phenomenological considerations of Dzyaloshinskii on the existence of helicoidal textures, as the result of possible instabilities of ferromagnetic structures under small relativistic spinlattice or spin-spin interactions. In particular we provide the rst quantitative counterpart to Dzyaloshinskii's predictions on helical structures.


Introduction
Composite ferromagnetic materials are the subject of growing interest, as they often display unusual properties which turn out to be strikingly dierent from the corresponding ones of their constituents. For this reason, it is possible to engineer ferromagnetic composites exhibiting physical and chemical behaviors which rarely, if ever, emerge in bulk materials [30].
A systematic study of composite materials, and more generally of media with microstructures, is the primary source of inspiration for the mathematical theory of homogenization. The theory aims at a description of composite materials with highly oscillating heterogeneities, through a simplied homogeneous model whose material-dependent properties are now related to specic averages of the physical and geometrical parameters of the constituents (cf., e.g., [8,7]). The origins of homogenization in micromagnetics date back to 1824, when Poisson, in his Mémoire sur la théorie du magnétisme [34], laid the foundations of the theory of induced magnetism, proposing a model in which a ferromagnet is composed of conducting spheres embedded in a nonconducting material.
The homogenization analysis performed in our paper is motivated by recent technological advances in the eld of spintronics; rst and foremost, by the observation, in magnetic systems lacking inversion symmetry, of chiral spin textures known as magnetic skyrmions [19], whose origin is ascribed to the Dzyaloshinskii-Moriya interaction (DMI) [18,23]. More precisely, our 1 work builds on Dzyaloshinskii's observations in [16,17] where, based on the Landau theory of second-order phase transitions, the emergence of helicoidal structures is predicted. According to Dzyaloshinskii, the appearance of these textures is the result of possible instabilities of the ferromagnetic structure created by relativistic spin-lattice or spin-spin forces, or by a sharp anisotropy in the exchange interaction. The results of our paper, based on the continuum theory of micromagnetics, make Dzyaloshinskii predictions quantitative (cf. Theorem 3).
From the mathematical point of view, magnetic skyrmions emerge as topological defects in the magnetization texture that carry a specic topological charge, also referred to as the skyrmion (winding) number . If H is a compact smooth hypersurface of R n+1 and m: H! S n is a suciently smooth vector eld on H, the skyrmion number of m is dened by the Kronecker integral [31] According to Hadamard [33], N sk (m) is always an integer number and coincides with the topological degree of m. By Hopf's theorem [29], skyrmions with dierent topological charges belong to dierent homotopy classes and, therefore, from the physical point of view, skyrmions are expected to be topologically protected against external perturbations and thermal uctuations [12]. Since their discovery, magnetic skyrmions have been the object of intense research work in condensed matter physics. Their stability, the reduced size, and the small current densities sucient to control them, make skyrmions extremely attractive for applications in modern spintronics [20,21,25].
In this paper, in the framework of ¡-convergence and 2-scale convergence, we investigate the inuence of the bulk Dzyaloshinskii-Moriya interaction [18,23] on the magnetic properties of composite ferromagnetic materials with highly oscillating heterogeneities. The homogeneous energy functional resulting from our analysis provides an eective description of most of the magnetic composites of interest nowadays. Indeed, although the homogenized coecients of the limiting energy functional involve the solution of a system of PDEs, chiral multilayers are essentially one-dimensional structures, and this allows us for a complete characterization of the minimal congurations and their topological degree, at least under some simplied hypotheses on the distribution of the constituents. Precisely, we show that depending on the eective DMI constant of the homogeneous model, two Bloch-type chiral skyrmions with opposite topological charges can arise. Our results provide a solid ground to the experimental observations that ground states with a non-trivial topological degree do exist here in a stable state [40,22,11] (see Remark 13 below).
In order to describe our main contributions, we rst collect below some preliminary notation and results.
1.1. The micromagnetic theory of (single crystal) chiral magnets. In the continuum theory of micromagnetism [10,24], which dates back to the seminal work of LandauLifshitz [26] on ne ferromagnetic particles, the observable states of a rigid ferromagnetic body, lling a region W R 3 , are described by the magnetization M : W ! R 3 , a vector eld subject to the fundamental constraint of micromagnetism: the existence of a material-dependent constant M s such that jM j = M s in W. For single-crystal ferromagnets (cf. [1,4]), the saturation magnetization M s := M s (T ) depends only on the temperature T and vanishes above a critical value T c , characteristic of each crystal type, known as the Curie temperature. When the specimen is at a xed temperature well below T c , the function M s is constant in W and the magnetization takes the form M := M s m, where m: W ! S 2 is a vector eld with values in the unit sphere of R 3 (cf. [10,24]).
Although the length of m is constant in space, this is, in general, not the case for its direction, and the observable states of the magnetization result as the local minimizers of the micromagnetic energy functional, which, for non-centrosymmetric (chiral) magnets, reads as where m W denotes the extension by zero of m to the whole space. The exchange energy E W penalizes spatial variations of the magnetization. The quantity a ex > 0 represents a phenomenological (material-dependent) constant that summarizes the eect of short-range exchange interactions.
The second term, K W (m), represents the bulk Dzyaloshinskii-Moriya interaction (DMI), and accounts for possible lacks of inversion symmetry in the crystal structure of the magnetic material. The material-dependent constant 2 R is the bulk DMI constant; its sign aects the chirality of the ferromagnetic system [39,37].
The third term, W , is the magnetostatic self-energy, that is, the energy due to the demagnetizing (or stray) eld h d generated by m. The stray eld ¡h d [M s m W ] is characterized as the projection of m W 2 L 2 (R 3 ; R 3 ) on the closed subspace of gradient vector elds The physical constant 0 denotes the vacuum permeability. The competition among the contributions in (3) explains most of the striking pictures of the magnetization observable in ferromagnetic materials [24]; in particular, the emergence of chiral spin textures with a non-trivial topological degree, i.e., magnetic skyrmions [21,22].
We note that, usually, the micromagnetic energy includes two additional energy contributions: the magnetocrystalline anisotropy energy A W and the Zeeman energy Z W : The energy density ' an : S 2 ! R + accounts for the existence of preferred directions of the magnetization: it vanishes on a nite set of directions, called easy axes, that depend on the crystallographic structure of the material. Instead, Z W models the tendency of a specimen to have the magnetization aligned with the external applied eld h a 2 L 2 (W; R 3 ), assumed to be unaected by variations of m. Although both A W and Z W are of fundamental importance in ferromagnetism, in a homogenization setting they behave like ¡-continuous perturbations, and their analysis has already been performed in [4]. Therefore, to shorten notation, they will be neglected in our investigation.

The micromagnetic theory of periodic chiral magnets.
When considering a ferromagnetic body composed of several magnetic materials, the material-dependent parameters a ex ; ; M s are no longer constant in the region W occupied by the ferromagnet. Moreover, one has to describe the local interactions of two grains with dierent magnetic properties at their touching interface [1]. There are dierent ways to take into accounts interfacial eects, and we will follow the approach of [4,3]: we will assume a strong coupling condition, meaning that the direction m of the magnetization does not jump through an interface, and only the magnitude M s is allowed to be discontinuous. This assumption allows for the analysis of the homogenized problem under the standard requirement that the magnetization direction m is in H 1 (W; S 2 ), i.e., that m belongs to the topological subspace of H 1 (W; R 3 ) consisting of vector-valued functions taking values on S 2 . The previous considerations lead to consider, for every " > 0, the family of energy functionals where the exchange constant a ex , the DMI constant , and the saturation magnetization M s are now replaced by Q-periodic functions in R 3 of period Q := (0; 1) 3 , and where a " (x) : Note that, a " , " , and M " are "-periodic functions that describe the oscillations of the material-dependent parameters of the composite. The main object of this paper is the asymptotic ¡-convergence analysis of the family of functionals (F " ) "2R + in the highly oscillating regime, i.e., when " ! 0.

State of the art.
Although the periodic homogenization of Dirichlet-type energies has been the focus of several studies (see, e.g., [38,28,2]), it is only recently that the analysis has been extended to the case of manifold-valued Sobolev spaces by means of ¡-convergence techniques [13,5]. In [5], a general result is proven for Caratheodory integrands of the type f (x/"; rm), with f being Q-periodic in the rst variable, and subject to classical growth conditions, and where m 2 H 1 (W; M) is constrained to take values in a connected smooth submanifold M of R n . Under these assumptions, it is shown that the behavior of f (x/"; rm) as " ! 0, can be described by a suitable tangentially homogenized energy density dened on the tangent bundle of M.
However, the analysis in [5] being purely local, does not cover long-range interactions such as the magnetostatic ones; this motivated the work in [4] (recently generalized to the stochastic setting in [3]). Two main novelties were introduced therein: i. The identication of the ¡-limit of the family of magnetostatic self-energies W W " , and the proof that it constitutes a ¡-continuous perturbation of the micromagnetic energy functional.
ii. While the analysis of the exchange energy density was already covered by the general results in [5], the treatment of the manifold-valued constraint in [4], via 2-scale convergence, allowed to obtain the result in a more concise and direct way, however under a bothering convexity assumption on M that in this paper we are going to remove.
For what concerns the statement in i. we recall the following result which we state here (without proof) in the slightly more general setting of a bounded, C 2 orientable hypersurface M of R 3 .

Proposition 1. (Prop. 4.4 in [4])
The family of magnetostatic self-energies (W W " ) "2R + ¡continuously converges in R + L 2 (W; M) to the functional where for almost every In particular, this guarantees that W W " can be treated as a continuous perturbation (cf. [14,Prop. 6.20,p. 62]). Namely, ¡-lim "!0 F " = ¡-lim "!0 (E W " + K W " ) + W 0 . For this reason, in the sequel, our analysis will be focused on the family Regarding point ii. we observe that the energy densities in (E W " + K W " ) "2R + explicitly depend on m and cannot be expressed in the form f (x/"; rm) for some Caratheodory integrand f tting the analysis in [5].

Contribution of the present work.
Moving beyond [5] and departing from the observations in [4], our analysis tackles the more general setting of periodic chiral magnets, that is composite chiral magnets in which the heterogeneities are evenly distributed inside the media.
The contribution of the present work is threefold. First, we provide a characterization of the asymptotic behavior of the energy functionals (G " ) "2R + in terms of ¡-convergence in the weak H 1 (W; M)-topology. Our homogenization result reads as follows (see Proposition 5 and Theorem 7).

Theorem 2.
Let M be a bounded, C 2 orientable hypersurface of R 3 that admits a tubular neighborhood of uniform thickness. Then, the family (G " ) "2R + ¡-converges with respect to the weak topology in H 1 (W; M), to the energy functional is the unique solution of the cell-problem described in Proposition 5.
We point out that the range of surfaces included in our study is quite broad. Indeed, any compact and smooth surface is orientable and admits a tubular neighborhood (of uniform thickness), cf. [15,Prop. 1,p. 113]. In particular, our analysis covers the class of bounded surfaces that are dieomorphic to an open subset of a compact surface (e.g., a nite cylinder, or the graph of a C 2 function). The proof strategy relies on a characterization of the twoscale asymptotic behavior of sequences in H 1 (W; M) (see Proposition 8), on an application of the theory of two-scale convergence (see [2,32,27]), and on a careful projection argument guaranteeing the optimality of G 0 as a lower bound for the energies G " , as " converges to zero.
Our second main result concerns the case in which M = S 2 and W R 3 has a laminated structure (see Figure 1 below). In this micromagnetic setting of chiral multilayers we provide an explicit identication of minimizers of the functional G 0 . Our theorem reads as follows (see Theorem 12).
a(y) dy: The arising of helical magnetic structures in composite alloys had been originally theorized by Dzyaloshinskii in [16,17] (see also [36]), as the result of possible instabilities of ferromagnetic structures with respect to small relativistic spin-lattice or spin-spin interactions. A concrete realization of Dzyaloshinskii's conjecture has been shown in [6], where the authors exhibited long-period structures in MnSi and FeGe alloys stemming from the phenomenon described in [16,17]. Despite the growing interest in chiral skyrmions (see, e.g., [35] for a review of the main properties) a rigorous theoretical justication for the presence of helical magnetic structures in alloys was, so far, still missing. To the authors' knowledge, Theorem 3 provides thus the rst mathematical evidence of Dzyaloshinskii's conjectures in multilayers when long-range eects are neglected (see Remark 15).
Our third main contribution consists of an extension of the characterization in Theorem 2 to the higher-dimensional setting. To be precise, we consider the family of energy functionals where for every i = 1; :::; n the maps function, taking values in the set of n n-matrices. Additionally, we assume that each map A i is uniformly positive denite, namely that for every i = 1; :::; n there exists c i > 0 such that A i (y) > c i jj for every 2 R n and for all y 2 Q: As highlighted in Remark 16 the class of energies densities as above includes the setting in which both the exchange energy coecient and the material-dependent DMI constant are anisotropic. In Theorem 18 we prove the following.
The paper is organized as follows: In Section 2 we introduce the setting of the problem and prove some rst preliminary results. In Section 2.1 we characterize the two-scale limits of H 1 (W; M)-maps. Sections 2.2 and 2.3 are devoted to the proof of Theorem 2. The study of chiral multilayers and the higher-dimensional setting are the subject of Section 3 and Section 4, respectively.

The three-dimensional setting
In what follows, W will be an open bounded domain of R 3 . Our analysis will focus on vectorvalued functions taking values on surfaces M in R 3 . We will always assume that M is a bounded, C 2 orientable hypersurface of R 3 that admits a tubular neighborhood of uniform thickness.
The normal eld associated with the choice of an orientation for M will be denoted by n: M ! S 2 . For every m 2 M and every 2 R we denote by`(m) := fm + tn(m): ¡ < t < g the normal segment to M having radius and centered at m. We recall that if M admits a tubular neighborhood (of uniform thickness) then there exists a 2 R + such that the following properties hold (cf. [15, p. 112 In what follows, Q will be the unit cube in R 3 . We will denote by H ] 1 (Q) the set of corresponding periodic H 1 -maps, namely the collection of functions u 2 H 1 (R 3 ) such that u(x + ke i ) = u(x) for every k 2 N, and for almost every x 2 R 3 , i = 1; 2; 3. With a slight abuse of notation, we will identify H ] Throughout the paper, the symbol will denote weak two-scale convergence. The symbol D(W) will represent the class of smooth functions having compact support in W. Also, to shorten notation, for every map 2 L 1 (Q) we will denote by h i Q the average of on Q.
We consider the energy density with : s 2 M 7 ! (e 1 s; e 2 s; e 3 s) T 2 R 33 . We aim at identifying a homogenized functional capturing the limiting behavior of minimizers of G " as " ! 0, that is, as the period over which the heterogeneities are evenly distributed inside the media shrinks to zero. Before stating our main result, we introduce the so-called tangentially homogenized energy density T hom : (s; T ) 2 TM ! R, dened by the minimization problem for every s 2 M and T 2 T s M. We rst show an explicit characterization of solutions to (14), guaranteeing, as a by-product, the measurability of the map x! T hom (m(x); rm(x)) for every Here the operator div acts on columns. Then, the unique solution [s; (14) is given by Proof. We rst observe that for every 2 H ] where we set (s) := ( 1 (s); 2 (s)) 2 R 32 and (y) = ( 1 (y); 2 (y)), for every s 2 M and for almost every y 2 Q. Additionally, we have ( j (s) + r j (y)) (s j ) = ( (s) + r (y)):(s); with ( which in turn can be rephrased as the combination of two scalar inmization procedures. Indeed, for every ; ! 2 R 3 we consider the minimization problem Accordingly, the unique solution of the main minimization problem (14) reads as for every s 2 M, and almost every y 2 Q. The expression of [s; T ] can be further simplied. Indeed, by (15) and (16) we have ¡div (a r(' a )) = ¡ div(ar' a ) = ra and ¡div (a r(' !)) = ¡ r !. Therefore [; !] = ' a + ' !, and we deduce the following identities: for every s 2 M and almost every y 2 Q. This completes the proof of (17).
To prove (18)  Our main result is to show that T hom represents the eective energy density associated to our homogenization problem. Proof. The proof of Theorem 7 is subdivided into two main steps: the compactness of sequences with equibounded energies and the liminf inequality are the subject of Theorem 10; the optimality of the upper bound follows from Theorem 11. The second equality in (7) is a direct consequence of Proposition 5.

Two-scale limits of elds in H 1 (W; M).
In this section, we characterize the two-scale asymptotic behavior of sequences in H 1 (W; M).
Proof. Since M is a C 2 orientable hypersurface, there exist an open tubular neighborhood U R 3 of M; and a C 2 function : U ! R, which has zero as a regular value, and is such that M = ¡1 (0). In view of (28), we have, up to the extraction of a (not relabeled) subsequence, 0 = (u " (x)) ! (u 0 (x)) = 0 for almost every x 2 W; therefore u 0 (x) 2 M for almost every x 2 W. Additionally, for every " > 0 there holds, 0 = r( u " ) = ru " r(u " ): By (28) it follows that r(u " ) ! r(u 0 ) strongly in L 2 (W; R 3 ). Thus, by (29) we obtain, for In particular, considering a test function independent of y, since R Q r y u 1 (x; y)dy = 0 for almost every x 2 W, we conclude that ru 0 r(u 0 ) = 0 in W. In particular, from (30) we infer that Z W Q r y (u 1 (x; y) r(u 0 (x))) (x; y) dxdy = 0 8 2 D(W; C ] 1 (Q; R N )): Hence, u 1 (x; y) r(u 0 (x)) = c(x) for almost every (x; y) 2 W Q, for some function c 2 L 2 (W). As R Q u 1 (x; y)dy = 0 for almost every x 2 W, it follows that c 0. Thus, u 1 r(u 0 ) 0: The thesis follows by observing that the vector r(u 0 (x)) is orthogonal to T u 0 (x) M for almost every x 2 W.

Remark 9.
The proposition holds if we assume, more generally, that M is the inverse image of a regular value of a C 2 function : U R N ! R M with M < N . Indeed, in this case, there exist M linearly independent normal vector elds, (n j ) j 2N M , which at every point p 2 M span the orthogonal complement of T p M. Repeating the same argument, one then nds that u 1 n j (u 0 ) = 0 for every j 2 N M , and therefore u 1 (x; y) 2 T u 0 (x) M.

Compactness and ¡-Liminf
rm " rm 0 + r y m 1 weakly in L 2 (W Q; R 33 ): Additionally, Proof. The compactness result is a direct consequence of Proposition 8, the assumptions on a and , and the boundedness of M. First, we observe that the energy density (12) can be rearranged as follows: with : s 2 M 7 ! (e 1 s; e 2 s; e 3 s) T 2 R 33 . Therefore where a " (x) (m " (x)) 2 dx; and jm " (x)j 2 dx: Next, we point out that for every 2 D(W; H ] 1 (Q; R 3 )) there holds because the dierence of the integrand on the left-hand side with the integrand on the righthand side is a perfect square. Thus, by standard properties of two-scale convergence, and owing to the regularity of , we obtain the inequality : ). Therefore, the previous inequality yields Since " 2 /a " * h 2 /ai Q weakly in L 1 (Q); we conclude that lim "!0 By combining (35), (36), and (37) we obtain (33).

The Limsup inequality in the 3d-setting.
In this section we show that the lower bound identied in Theorem 10 is optimal. To be precise, we prove the following result. and Proof. Denote by U the tubular neighborhood of size around M, and let M : U ! M be the pointwise projection operator. Note that, since M is C 2 , the projection satises M 2 C 1 (U ; M). Clearly, we can choose small enough so that M 2 C 1 (U ; M). For convenience of the reader we subdivide the proof into two steps.
Step 1. Given 2 C 1 (W ; W ] 1;1 (Q; R 3 )), for every " > 0 and almost every x 2 W we set We note that, Given the regularity of , for " small enough there holds m " 2 U for almost every x 2 W. By the regularity of M , there exists a constant c M > 0 depending only on M, such that jrm " j 6 c M jrm " j a.e. in W: By (40) and the boundedness of (rm " ) in L 2 (W; R 33 ), we deduce that, up to the extraction of a not relabelled subsequence, Moreover, by Proposition 8, we infer that, up to the extraction of a not relabelled subsequence, there holds for almost every x 2 W. By (43), and by the regularity of and M , it follows that "r x ¡ x; x " ! 0 strongly in L 1 (W; R 33 ); (48) r y ¡ x; x " r y strongly two-scale in L 2 (W Q; R 33 ): By combining the above convergences and (46), we conclude that Therefore, in view of (45), almost everywhere in W Q. Since both and have null average in Q, this implies that r M [m 0 ]rm 0 = rm 0 for almost every x 2 W. In particular, by (50) we infer that To see that the previous convergence is actually stronger, we observe that and Now, the rst term in the right-hand side of (53) converges to zero in L 2 (W; R 33 ) owing to (47), the regularity of M, and the Dominated Convergence Theorem. The second term converges to zero strongly two-scale in L 2 (W Q; R 33 ) due to (41) and the regularity of M. Finally, the last term in the right-hand side of (53) converges to zero strongly in L 2 (W; R 33 ) because of (47), the boundedness of (rm " ) in L 2 (W; R 33 ), and Lebesgue's dominated convergence theorem. This proves that In view of (54) and of the regularity of we directly obtain that By the arbitrariness of , the limsup inequality follows then from classical properties of ¡convergence (see Section 1.2 in [9]).

The micromagnetic setting: applications to multilayers
In this section, we specify the characterization of the eective energy to the micromagnetic setting of chiral multilayers. In this case, M = S 2 and W R 3 has a laminated structure as in Figure 1. We have the following result.
Theorem 12. Assume that W := ! I with ! R 2 , j!j = 1, > 0; I = (0; ), and that the material-dependent functions a; 2 L ] 1 (Q; R) depend only on the third coordinate: a(y) = a(y e 3 ) and (y) = (y e 3 ). Then, the homogenized energy functional (7) is given, for every where the eective parameters are dened by with 0 2 R arbitrary, and the minimum value of the energy is G 0 (m ) = ¡ 2 h 2 /ai Q : Remark 13. It is interesting to note that the DMI layers contribute to an increase in the crystal anisotropy of the magnetic system. When hi Q = 0, the minimizers of G 0 are always planar and constant on each layer. Moreover, depending on the eective DMI constant 0 of the homogeneous model, two Bloch-type chiral skyrmions with opposite (and possibly nonintegral) topological charges can arise. Indeed, the sign of 0 , equivalently the sign of h/ai Q , controls the chirality of the system: the minimizers describe right-handed helices when 0 > 0 and left-handed helices when 0 < 0 (cf. Figure 2).
one can interpret m as a map from S 1 to S 1 , whose skyrmion number (cf. (2)) coincides with the winding number of m around the origin: In other words, for the values of specied in (63), the sign of h /ai Q determines that of the topological degree of m .

Remark 14.
Note that, the shape of the minimizers does not depend on the height of the multilayer. Instead, the minimum value of the energy scales linearly in .
Remark 15. Our analysis in Theorem 12 does not take into account long-range eects such as the ones originating from magnetostatic interaction, as well as magnetocrystalline eects. As already pointed out these contributions can be superimposed to our energy functional because, from the variational point of view, they play the role of a continuous perturbation. For example, to include magnetostatic interaction, one has to consider the augmented energy functional (cf. Proposition 1) G 0 + W 0 ,with W 0 given by (6). In this case, although we expect similar qualitative considerations, an explicit characterization of the minimizing proles of G 0 + W 0 will be hardly achievable.
Testing (64) against vector elds 2 H ] 1 (Q; R 3 ) that do not depend on t we get that, for a.e. t 2 I , the distribution ' (; t) is harmonic: where we set ! ' (; t): We conclude that ' (y) depends only on the t-variable. Therefore, we set ' (y) = ' (y e 3 ) : = ' (t). In view of (64) it follows that ' (t) solves the ordinary dierential equation Integrating in [0; t] yields the equation where we set a 0 := a(0); 0 := (0). Integrating again in I and imposing periodicity we deduce that The previous relation implies that ' 0 (0) e 1 = 0, ' 0 (0) e 2 = 0, and Therefore, for every t 2 I Noting that ' a solves an analogous dierential equation as ' , in which is replaced by ¡a, we conclude that ' a 0 (0) e 1 = 0, ' a 0 (0) e 2 = 0, and ' a 0 (0) e 3 = ¡ 1 + (a 0 ha ¡1 i I ) ¡1 . Thus, for every t 2 I, We stress that K(t) and a(t) in (66) and (67) have been introduced because they are the only quantity of interest for the tangentially homogenized energy density T hom .
We recall in fact that T hom reads as (cf. (18) In the micromagnetic setting, in which the vector eld m take values in M := S 2 ; an explicit identication of jrj 2 is available. Indeed, we nd that 1 (s) = 2 (s) and 2 (s) = ¡ 1 (s) for every s 2 S 2 . Hence, recalling that ' a and ' k depend only on the t-variable, for almost every y 2 Q, s 2 M and T 2 T s M, we deduce with K(t) and a(t) given by (66) and (67). A direct computation shows that Thus, for almost every y 2 Q, s 2 M, and T 2 T s M, we have jr(y)j 2 = ja(y) T e 3 + K(y)e 3 sj 2 : Substituting the previous expression in (18) gives Z Q a(y)ja(y) T e 3 + K(y)e 3 sj 2 dy with 0 := haa 2 i Q , 0 := haK 2 i Q , and 0 := haaKi Q explicitly given by and This yields (60). To complete the proof of the theorem it remains to show that when hi Q = 0 the energy minimizers depend on y e 3 only and can be fully characterized. We proceed in two steps:

1.
We assume that any minimizer m of G 0 is of the form m (x) = u(x e 3 ) for some planar one-dimensional prole u: I ! S 1 f0g, and we characterize the minimizers in this class.

2.
We prove that every minimizer of G 0 satises the assumptions in step 1.
Step 1. We start noting that under the symmetry assumptions in 1. the minimization problem for the micromagnetic energy functional reduces to the minimization in H 1 (I ; S 2 ) of the functional Since u(t) 2 S 1 f0g for almost every t 2 I, we have that u(t) u _(t) = 0 and therefore, for almost every t 2 I there holds u _(t) (e 3 u(t)) = e 3 (u(t) u _(t)) = ju _(t)j (u(t); u _(t)); with (u(t); u _(t)) = 1 if the couple (u; u _) induces a positively-oriented basis of R 2 , and (u(t); u _(t)) = ¡1 otherwise. In particular, if u + is a critical point of the energy for 0 > 0, then for every R 2 SO(3), the prole Ru + is again a critical point. By contrast, if R ¡ 2 O(3) and det R ¡ = ¡1, then u ¡ := R ¡ u + is a critical point of the energy for 0 = ¡j 0 j. Therefore, without loss of generality, we can assume that 0 > 0. The Euler-Lagrange equations associated with (70) read as In particular, parameterizing u in polar coordinates, we obtain that the general solution of (71) is given by with 0 := (0); := () arbitrary real numbers. The corresponding values of the energy depends on ¡ 0 , and on the height of the multilayer. Precisely, evaluating G ? on the family (72) we nd that Minimizing with respect to ( ¡ 0 ), and taking into account (68) and (69), we deduce that the corresponding minimum value of G ? is achieved when ( ¡ 0 ) = 0 / 0 = h/ai Q , it is strictly negative, and it is given by Summarizing, the energy G ? is minimized by all proles of the form u(t) = cos (t)e 1 + sin (t)e 2 ; (t) := 0 + h/ai Q t: As we were expecting, depending on the sign of 0 , equivalently on the sign of h/ai Q , the rotation is clockwise or counter-clockwise (cf. Figure 2). Note that the structure of optimal proles does not depend on the height of the multilayer. The length aects only the minimal energy value, which is a decreasing function of .
Step 2. We decompose each element m 2 H 1 (W; S 2 ) as m = u + m 3 e 3; with u: W ! R 3 being the projection of m on R 2 f0g. We expand the energy G 0 as and we set I 1 (u) := for almost every x 2 R n , and for all (s; T ) 2 TM, where we have used the symmetry of A i and the skew-symmetry of K i . We point out that for almost every x 2 W: For this reason, the analysis of energy densities f ṽ as in (79)  ; m(x); rm(x) dx The main result of this section is the proof that the eective functional for every s 2 M and T 2 T s M. We rst provide the counterpart to Proposition 5 in the higher dimensional setting.

Proposition 17.
For every (s; T ) 2 TM, the minimization problem (84) has a unique solution. Specically, let ' J ; ' À 2 H ] ¡1 (Q; R nn );`= 1; :::; n be the unique solutions to the cell elliptic systems and X i=1 n @ y i (A i (y)@ y i ' À (y)) = @ y`( A`(y));`= 1; :::; n: : (@ i m 0 (x) + @ y i (x; y) ¡ J i (y)m 0 (x))dy dx; owing to the fact that the dierence of the integrand on the left-hand side with the integrand on the right-hand side is a perfect square. Thus, by standard properties of two-scale convergence, and owing to the regularity of , we deduce liminf "!0 : (@ i m 0 (x) + @ y i (x; y) ¡ J i (y)m 0 (x))dy dx :(@ i m 0 (x) + @ y i (x; y) ¡ J i (y)m 0 (x)) dy dx: By density, there exists ( n ) n2N such that n ! m 1 in L 2 (W; H ] 1 (Q; T m 0 M)). Therefore, the previous inequality yields liminf "!0 Finally, since K " i * R Q K i (y)dy weakly in L 1 (Q; R nn ); we get lim "!0 Combining (92) and (93) we deduce (89). The optimality of the lower bound is a straightforward adaptation of the arguments in Theorem 11.
programme when part of the work on this paper was undertaken. G. Di F. acknowledges support of the Vienna Science and Technology Fund (WWTF) through the research project Thermally controlled magnetization dynamics (Grant MA14-44).