Magnetic skyrmions under confinement

We present a variational treatment of confined magnetic skyrmions in a minimal micromagnetic model of ultrathin ferromagnetic films with interfacial Dzylashinksii-Moriya interaction (DMI) in competition with the exchange energy, with a possible addition of perpendicular magnetic anisotropy. Under Dirichlet boundary conditions that are motivated by the asymptotic treatment of the stray field energy in the thin film limit we prove existence of topologically non-trivial energy minimizers that concentrate on points in the domain as the DMI strength parameter tends to zero. Furthermore, we derive the leading order non-trivial term in the $\Gamma$-expansion of the energy in the limit of vanishing DMI strength that allows us to completely characterize the limiting magnetization profiles and interpret them as particle-like states whose radius and position are determined by minimizing a renormalized energy functional. In particular, we show that in our setting the skyrmions are strongly repelled from the domain boundaries, which imparts them with stability that is highly desirable for applications. We provide explicit calculations of the renormalized energy for a number of basic domain geometries.

1 Introduction Magnetic skyrmions are particle-like non-collinear spin textures that were predicted to exist in noncentrosymmetric ferromagnets some 30 years ago [10,9,8] and have been recently observed in a number of magnetic systems [35,50,21,41].These coherent spin states are enabled by the non-trivial topological characteristics of their magnetizations [36,19] which endow them with a considerable degree of thermal stability down to nanoscale and permit observation of magnetic skyrmions at room temperature [34,11,49].The latter property makes magnetic skyrmions attractive candidates as information carriers in a new generation of spintronic devices for information technology [19,51].
In ultrathin ferromagnetic films exhibiting skyrmions, the magnetization of the material may be described as a map from a two-dimensional plane to a three-dimensional sphere at the level of the continuum [26,40].As skyrmions are particle-like localized perturbations of the uniform ferromagnetic state, they must belong to a homotopy class of the equivalent (after a stereographic projection) continuous maps from 2 to itself.These classes are characterized by an integer topological degree, and the observed magnetic skyrmion configurations display the degree +1 of the identity map from 2 to 2 [36]. 1 Mathematically, these configurations may be viewed as local minimizers of a suitable micromagnetic energy functional among configurations within the above homotopy class, and their existence was established for several models [32,5,3,4].
In a minimal model relevant to ultrathin ferromagnetic films capped with a layer of a heavy metal [40], the energy consists of a sum of the exchange energy forcing the magnetization to be constant in space, the interfacial Dzyaloshinskii-Moriya interaction (DMI) that promotes rotation of the magnetization vector, as well as the perpendicular magnetic anisotropy that forces the magnetization to align normally to the film plane and/or the Zeeman energy associated with the perpendicular applied magnetic field that has the same effect (for technical details, see section 2).Note that the problem of existence above is closely related to the one studied by Lin and Yang in a two-dimensional Skyrme model [29,30].Bernand-Mantel et al. established the asymptotic behavior of skyrmion solutions in the case of vanishingly small DMI strength and demonstrated that in this limit the magnetization profiles are close to the shrinking Belavin-Polyakov profiles, i.e., the degree +1 harmonic maps from ℝ 2 to 2 [2], which in the minimal model described above are of Néel type [5,3].
In the physics literature, the inability to continuously deform a topologically non-trivial skyrmion configuration into the topologically trivial uniform ferromagnetic state is often referred to as topological protection of magnetic skyrmions [36].We note that this is somewhat of a misnomer, as a topologically non-trivial skyrmion configuration may in fact be deformed discontinuously into the uniform ferromagnetic state via core collapse by crossing a finite energy barrier [6].In contrast, in finite samples such as nanodots or nanostrips that are of particular interest to applications, there is strictly speaking no topological obstruction that prohibits a homotopy between a skyrmion solution and the trivial solution for example by moving the skyrmion "through" the boundary.Nevertheless, these two solutions may still be separated by an energy barrier, and the question of existence of skyrmion solutions becomes more subtle.
In the minimal micromagnetic model that includes the stray field effect only via an effective anisotropy term [48], Rohart and Thiaville numerically constructed the Néel type radiallysymmetric skyrmion solutions in a circular nano-dot [40].It is unclear, however, whether these solutions always represent local energy minimizers, as the exchange energy in such a solution may be continuously lowered by moving the skyrmion towards the domain boundary, breaking the radial symmetry of the solution (see also section 2.5).Numerical studies of the minimal model in confined geometries do indicate the presence of a finite energy barrier towards skyrmion disappearance through the boundary under certain conditions [14,13,39].The solutions in nanodisks were further analyzed numerically within the full micromagnetic model that includes the non-local stray field effects [43,46,1].In particular, the obtained numerical profiles exhibit a strong perpendicular alignment of the magnetization at the domain edges, which can be explained by an additional contribution of the stray field enhancing the perpendicular magnetic anisotropy there (see also the experimental observations in [23]).As was shown in [16], in suitable thin film limits for the considered class of materials the effect of the stray field may be asymptotically accounted for via an effective penalty term forcing the magnetization to align with the normal to the film plane at the domain boundary, similarly to what happens in other ferromagnetic thin film problems [24].In our problem, this should lead to the skyrmion being repelled from the sample edges.
In view of the above arguments, it is physically reasonable to consider the situation in which the magnetization at the film edge is rigidly aligned with a normal to the film plane.This may either be achieved via sending the penalization of the deviations at the boundary to infinity (corresponding to an appropriate choice of the material and geometric parameters [16]), or it could be the result of patterning the substrate of an extended ferromagnetic film with a strongly magnetically anchoring material (for a related approach, see [37]).Using these Dirichlet boundary conditions restores the possibility of topological protection, as continuous maps from a bounded two-dimensional domain to 2 with the boundary values pinned to a single direction can once again be classified by their topological degree.However, it is still not a priori clear whether minimizers would be attained in such a setting, as the possibility of a skyrmion shrinking to a point and collapsing is not excluded.
In this paper, we present a variational treatment of the minimal micromagnetic model of confined magnetic skyrmions in ultrathin ferromagnetic films with interfacial DMI, in which the confinement is provided by the Dirichlet boundary condition that forces the magnetization to take one direction normal to the film plane at the two-dimensional domain edge.We first prove existence of degree +1 minimizers of the energy consisting of the sum of the exchange and the interfacial DMI terms (with a possible addition of the perpendicular magnetic anisotropy term).We then focus on the regime in which the DMI is a perturbation to the exchange energy and develop a Γ-expansion of the energy in the limit of vanishing DMI strength.This leads to the appearance of a renormalized energy which determines asymptotically both the location and the radius of the skyrmion, whose shape is shown to be close to a Néel type degree +1 harmonic map from ℝ 2 to 2 .Lastly, we explicitly construct the minimizers of the renormalized energy in the case of disk and strip domains.In particular, we show that the energy minimizing skyrmions are located in the disk center and on the strip midline, respectively, due to the effective repulsive interaction provided by the excess exchange energy from the tail of the magnetization profile.This confirms the physical expectation based on the numerical simulations that skyrmions can be robust particle-like objects even in finite samples of varying geometry.

Informal discussion of the results
From a mathematical standpoint, the confinement provided by the boundary data in fact simplifies the proof of existence of skyrmions compared to the case of the whole plane, since the translational symmetry of the problem is broken.In order to obtain the parameters describing the asymptotic behavior, we apply the rigidity of degree ±1 harmonic maps from ℝ 2 to 2 obtained by Bernand-Mantel, Muratov and Simon [3] after extending the magnetizations by a constant outside the domain using the Dirichlet boundary condition.This allows us to define the location, radius and rotation angle of the skyrmion.As is common in Γ-convergence arguments, we first obtain qualitative information such as linear scaling of the radius in the DMI constant or the fact that skyrmions are repelled from the boundary via non-optimal estimates, in order to obtain compactness properties of the energies.
A finer analysis requires us to also keep track of the tail correction to the skyrmion profile necessary to enforce the boundary condition.Here, the skyrmion position interacts with the boundary through the solution of the linearization of the harmonic map problem at the constant state given by the boundary condition, i.e., Laplace's equation for the in-plane components.A correction to the skyrmion core is not necessary to first order as the Belavin-Polyakov profiles are the exact degree one minimizers of the Dirichlet energy on the whole plane.The location of the optimal skyrmion is set by minimizing the interaction with the boundary, and the Néel character of the profile arises via minimizing the DMI term among all rotation angles.The radius then optimizes the balance of the two contributions.In simple domains, such as balls and strips, the Laplace's equation determining the tail correction can be explicitly solved by means of complex analysis, thus giving the full solution of the limiting problem in these cases.
Finally, we additionally include the perpendicular anisotropy at an appropriate scaling in the DMI constant.As the radius scales linearly in the DMI constant and the anisotropy energy of an exact Belavin-Polyakov profile is well-known to have a logarithmic divergence in its tail [17,3], we consider effective anisotropies scaled down with the logarithm of the DMI strength.In this regime the anisotropy is essentially a continuous perturbation of our original problem with respect to the topology we determine the Γ-limit in.Furthermore, due to the fact that it is only the tail that contributes to the anisotropy at leading order and that it is of logarithmic character, we obtain that its contribution in the limit is in fact independent of the shape of the domain.
We note that the variational problem considered by us bears several similarities with the one for the classical Ginzburg-Landau model (without the magnetic field) [7,38], in which the boundary data with a non-trivial topological degree force minimizers to form point-like vortices in the domain interior as the small parameter of the model goes to zero.Our problem for the magnetic skyrmion behavior in the limit of vanishing DMI strength thus provides a micromagnetic setting for the celebrated questions of Matano for Ginzburg-Landau vortices [7,45].Towards answering this type of questions, we show that the skyrmion in a disk concentrates at the disk center in the limit and explicitly compute its asymptotic magnetization profile.We point out, however, that the analysis of the limit micromagnetic problem is considerably more delicate, as in contrast to the Ginzburg-Landau problem the energy of a single skyrmion remains finite in the limit, and, therefore the tail contribution of the Dirichlet energy does not decouple from the problem for the skyrmion core.In particular, contrary to the Ginzburg-Landau vortex problem [33,44], the radius of the skyrmion turns out to be affected by the shape of the domain through the solution of the limit problem in the tail.
As in the problem of Ginzburg-Landau vortices [7,38], it is also natural to ask whether multiple skyrmion configurations may be similarly described in the vanishing DMI strength limit.In fact, the micromagnetic energy has been shown numerically to exhibit a multitude of local energy minimizers other than a single magnetic skyrmion [42,25].However, our present analysis does not easily extend to the case of magnetization configurations of degree other than +1.Even at the level of existence we cannot rule out the collapse of minimizing sequences, failing to yield minimizers with a prescribed degree in this case.For the limit behavior of vanishing DMI strength, we also no longer have the quantitative rigidity estimate for the harmonic maps of arbitrary degree, which is the key tool in our analysis of a single skyrmion [3].In fact, such an estimate has been recently shown to be false for degree 2 harmonic maps [15].Similarly, we cannot give a positive answer to the existence of anti-skyrmions, i.e., minimizers among configurations with degree −1, as we do not know whether the basic energy bound in Lemma 3.2 holds in this class.

Outline of the paper
This paper is organized as follows.In section 2, we give the precise definitions of the micromagnetic energy, admissible classes, and the limit processes under consideration and then formulate our main results.In section 3, we prove existence of minimizers in the considered non-trivial topological class of maps with degree +1.In section 4, we derive the first-order term in the Γ-expansion of the energy in the DMI strength beyond the classical topological lower bound at zeroth order.Then, in section 5 we explicitly compute the renormalized energy for a number of geometries.Finally, in section 6 we show how to include the perpendicular magnetic anisotropy as a continuous perturbation to the limit energy.

Definition of the energy
On a bounded domain Ω ⊂ ℝ 2 with Lipschitz boundary, we consider the set of admissible functions where the degree of a function ∈ ̊ 1 (ℝ 2 ; 2 ) is defined as and we extend ∈  to the whole of ℝ 2 by setting = − 3 outside Ω.Here, as usual, we define It is well known that  ( ) ∈ ℤ for any ∈ ̊ 1 (ℝ 2 ; 2 ), see Brezis and Coron [12].For ∈  we wish to minimize the energy where ∈ ℝ is the DMI constant and we use the convention = ( ′ , 3 ), with ′ taking values in ℝ 2 .Passing from to ̃ ∶= (− ′ , 3 ) when minimizing  in the case of < 0, throughout the rest of the paper we may assume that ≥ 0.

Statement of the results
We first make sure that the energy indeed admits minimizers.Due to the Dirichlet boundary conditions, minimizers exist for all > 0 sufficiently small even in the absence of the anisotropy penalizing the out-of-plane component of the magnetization.Note that at the same time the infimum of the energy is not attained for = 0 (see below).
Theorem 2.1.There exists 0 > 0 depending only on Ω such that for all 0 < < 0 there exists a minimizer of  over .
More importantly, we are also able to give a precise description of the minimizers for being small, i.e., the parameter regime in which one does have skyrmions.In particular, we can express their location and radius in terms of an optimization over the tail of the skyrmion.
To make this statement precise, we define the standard Belavin-Polyakov profile for ∈ ℝ 2 , which is the negative of the inverse stereographic projection, and we denote the set of all Belavin-Polyakov profiles by They arise as configurations achieving equality in the sharp topological bound with degree  = 1.In particular, they are precisely the minimizing harmonic maps of degree one, see Belavin and Polyakov [2] and [12,Lemma A.1].It is therefore not surprising and indeed well known [17,3], that minimizers of micromagnetic-type energies augmented with DMI should approach the set  when the Dirichlet energy dominates, i.e., when ≪ 1.We can thus attempt to express the location and the radius of the skyrmions as ∈ ℝ 2 and > 0 of an approached Belavin-Polyakov profile in this regime.Notice that for = 0 an equality in (2.7) is achieved by a sequence of truncated Belavin-Polyakov profiles with vanishing radius, which fails to converge to an element in .This statement remains true also in the presence of an additional out-of-plane anisotropy term (see section 2.4).However, as we expect the radius of the minimizers to shrink compared to the size of the domain as → 0, we can only expect the close-by Belavin-Polyakov profiles to converge after a rescaling.Consequently, we have to find a Belavin-Polyakov profile for each minimizer at positive in a controlled way.An appropriate set of tools for such a purpose has been identified by Bernand-Mantel, Muratov, and Simon in the form of a quantitative rigidity result for degree one harmonic maps: Theorem 2.2 ([3, Theorem 2.4]).For ∈ , let the Dirichlet excess be and the Dirichlet distance to the set of the Belavin-Polyakov profiles be where as usual is extended outside of Ω by − 3 .Then the infimum in the definition of ( ; ) is achieved, i.e., there exists a Belavin-Polyakov profile closest to each ∈ .Moreover, there exists a universal constant > 0 such that for all ∈  we have (2.10) Shorter, alternative proofs of this statement have later been provided by Hirsch and Zemas [22] and Topping [47].
In order to identify the Belavin-Polyakov profiles corresponding to minimizers of  in the limit → 0, we turn to computing the Γ-limit in a suitable topology retaining the location, the radius, the global rotation and the skyrmion tail.To this end, we have to identify the correct higher order Γ-expansion of the energy.By roughly minimizing over the above quantities, we will find in Lemma 3.2 below that there exists a constant > 0 depending only on ( This suggests to seek a Γ-limit of the functional . However, in order to rule out some behaviors of finite energy sequences that minimizers will not exhibit, such as skyrmions shrinking too fast or their centers approaching the boundary of Ω, we will restrict our attention to magnetizations whose energy is sufficiently low, i.e. we restrict the admissible set to Furthermore, in the Γ-limit we will only consider magnetizations ∈  which satisfy Note that this corresponds to a finite energy sequence for the functional
We are now in a position to give the Γ-limit of with respect to the above convergence.
Definition 2.5.For ( 0 , 0 , 0 ) ∈ 0 let where the Dirichlet contribution of the tail correction is (2.20) We furthermore define a restricted admissible set We can then state the Γ-convergence.
Theorem 2.6.The Γ-limit as → 0 of the functionals restricted to  with respect to the BP-convergence is given by  0 restricted to  0 in the sense that we have the following: < 0 there exists a subsequence (not relabeled) and ( 0 , 0 , 0 ) ∈  0 such that BP-converges to ( 0 , 0 , 0 ).
Then we have (iii) For every ( 0 , 0 , 0 ) ∈  0 and every sequence of → 0 there exist Notice that the last term in the definition of  0 is clearly minimized by 0 = id among all 0 satisfying 0 3 = 3 , since this achieves an absolute maximum of the integrand by pointwise Cauchy-Schwarz inequality in view of the fact that Φ ′ is collinear to ∇Φ 3 .Thus from the fact that In particular, this theorem says that as → 0 the appropriately translated and dilated minimizer ( 0 (•) + 0 ) converges to the canonical Belavin-Polyakov profile Φ in Dirichlet distance, up to a subsequence.In the original variables, the energy minimizing profile is, therefore, close to the Belavin-Polyakov profile of Néel type centered at 0 and with the small radius = 0 .

Explicit minimizers for specific domains
We next give several examples of geometries, in which an explicit minimizer of the limit problem may be obtained.We use the standard identification of the complex plane with ℝ 2 and write ∈ ℂ to denote a vector in the plane.The symbol ̄ denotes the complex conjugate of .We also introduce the Wirtinger derivatives = 1 2 ( − ) and ̄ = 1 2 ( + ), acting on ∶ ℂ → ℂ.Clearly, the infimum in (2.20) is attained by the unique harmonic extension of from one can then write the limit energy associated with a skyrmion centered at 0 ∈ ℂ as The following proposition allows us to reduce the computation of ( 0 ) to evaluating a derivative of 0 ( ) at = 0 for the considered geometries.Proposition 2.9.Let Ω ⊂ ℂ be a simply connected bounded domain with a boundary of class 1, , for some ∈ (0, 1).Then we have We note that due to the continuous dependence of the boundary values of on 0 , the function ( 0 ) is continuous for all 0 ∈ Ω.Moreover, since |∇ ( )| behaves like | − 0 | −2 for all ∈ ℝ 2 ⧵Ω and all competitors from equation (2.20), which is not 2 -integrable on ℝ 2 ⧵ Ω if 0 = 0 ∈ Ω, we have ( 0 ) → +∞ as 0 approaches Ω.Therefore, ( 0 ) always attains its minimum for some 0 ∈ Ω.

Disks
For the special choice Ω = (0) with > 0 we can fully solve the above minimization problem, obtaining that the skyrmion will be located in the disk's center.Proposition 2.10.For Ω = (0) and 0 ∈ Ω, the map achieving ( 0 ) is given by Its energy is given by which is minimized by 0 = 0 with (0) = 16  2 .The rescaled skyrmion radius is 0 = 2 4 and the corresponding limiting energy is Note that the minimizer achieving ( 0 ) has the special property of being a holomorphic function in Ω.In the next example of strips we will see that this does not necessarily have to be the case.

Strips
We can also consider the energy (2.4) on strips Ω = ℝ × (− ∕2, ∕2) for > 0. Technically, the previous statements do not apply as Ω is not bounded.However, the arguments can be adjusted straightforwardly as strips support Poincaré inequalities.We will give the modifications in section 5.2 below.
The only change in the resulting statement is that in the BP-convergence we will, due to the translational invariance of Ω in the first component, only track the second component of the skyrmion center, so that the limiting set is Furthermore, the limiting energy is given by where Also this problem can be solved explicitly.
The formula for 0 above was obtained by computing the harmonic extension of the boundary data in Fourier space, but to verify its validity we only need to check that it satisfies the conditions defining 0 .

Half-plane
For the half-space Ω = ℝ × (−∞, 0) our rigorous arguments cannot be salvaged, and indeed in this case the energy can be easily seen to be unbounded from below.However, we may still consider the problem as arising from a limiting procedure where the distance of the skyrmion center to the boundary of growing, smooth domains is fixed.Then we obtain the problem defined on and where where the skyrmion is located at 0 = 0 with 0 < 0 in the limit.
The straightforward solution then gives information about how the energy of the skyrmion behaves as it approaches the boundary.Of course, the repelling effect of the boundary can also be seen from our rigorous analysis.However, in this situation, the estimate is especially transparent.Proposition 2.12.For Ω = ℝ × (−∞, 0), and 0 ∈ (−∞, 0), the map achieving ( 0 ) is given by Its energy is given by ( 0 ) = 4 2 0 , the corresponding rescaled skyrmion radius is 0 = 2 0 and the limiting energy is  0 id, 2 0 , 0 = −4 2 0 .

Adding anisotropy
We may also consider the case where we augment our energy by an anisotropy term.In order not to significantly change the behavior of the Γ-limit, we choose the quality factor in dependence of such that, after the renormalization in the thin film limit by the local contribution of the stray field term, the resulting term is essentially a compact perturbation of our above results.As it is well known that the anisotropy contribution of a skyrmion with radius > 0 behaves like 2 | log |, see for example [17,3], and as in our case ∼ , the appropriate scaling is − 1 = | log | −1 for some > 0. Consequently, we obtain the modified energy Notice that the statement of Theorem 2.1 remains valid for minimizers of  , .
The following proposition then implies that the Γ-limit with respect to the BP-convergence at order 2 is given by (2.46) In particular, the above result shows that the addition of anisotropy does not affect the center of the skyrmion in the limit → 0 in the considered regime.As before, the limit of  0 is achieved by the Néel profile, 0 = id, the center 0 = argmin 0 ∈Ω ( 0 ) and The corresponding minimal energy is . (2.48)

A note on free boundary conditions
We wish to also mention the difference between the behavior of the energy for BP-converging sequences for the Dirichlet problem associated with the admissible class  and that of the analogous free problem in which the Dirichlet boundary condition at Ω is absent.We point out that in this case the BP-limit does not give rise to a well-behaved energy whose minimization would yield the position of the skyrmion in Ω as → 0. Indeed, in the latter case the restriction to Ω of the Néeltype Belavin-Polyakov profile = Φ( −1 (• − )) with = id, ∕ → 0 and → 0 ∈ Ω in Definition 2.3 is an example of a BP-convergent sequence, and, therefore, we have an upper bound on the Dirichlet energy excess for a sequence of BP-converging to (id, 0 , 0 ) by

.49)
A straightforward computation shows that as → 0 we have where is the outward unit normal to Ω, and in the last line we carried out an integration by parts.This is a negative contribution that goes to negative infinity as 0 approaches Ω.
For example, if, as in section 2.3.3,we take Ω = ℝ × (−∞, 0) and 0 = (0, 0 ) with some 0 ∈ (−∞, 0), then by (2.50) we have explicitly for the renormalized energy: where the infimum is over sequences of that BP-converge to (id, 0 , 0 ).This energy clearly does not have a minimum in 0 , suggesting that the skyrmion is not able to stabilize its radius at a fixed distance towards the boundary.Similarly, at fixed radius the skyrmion is attracted towards the boundary.Dynamically this would give rise to the disappearance of a skyrmion from Ω via escape towards the boundary, with zero energy barrier.This is in contrast with the case of the Dirichlet boundary conditions considered in section 2.3.3, in which the exchange contribution has the opposite sign.
Finally, notice that an addition of a sufficiently strong anisotropy as in section 2.4 may restore existence of local minimizers.To get some sense for this, consider again a skyrmion in the halfplane as in the previous paragraph.With the addition of anisotropy we would then get and it is clear that the skyrmion should experience a repulsive interaction and have a well-defined optimal radius far enough from Ω, while it would still be attracted towards the boundary close enough to Ω.

Notation and presentation
Throughout the rest of the paper, we extend ∈  to ℝ 2 by − 3 .Furthermore, unless explicitly stated otherwise, the letters , ′ denote generic, positive constants only depending on Ω and which may change from line to line.Each subsection first lists its statements and provides a description for their proof and use throughout the rest of the paper.The actual proofs are collected at the end of the respective subsections.

Existence of minimizers
We first provide a simple lower bound for the energy that controls the 2 -norm of ∇ for sufficiently small .
Lemma 3.1.For all > 0 and ∈ 1 (Ω; 2 ) satisfying = − 3 on Ω we have for some > 0 depending only on Ω.Furthermore, if also ∈  and < 1∕(2 ) we have Next, we show by a construction that the infimum energy is strictly below the topological lower bound for the case of the pure Dirichlet energy.In particular, the restricted admissible sets  , see definition (2.12), are non-empty.Furthermore, there exist constants > 0 and 0 > 0 depending only on Ω such that for all ∈ (0, 0 ), we have Proof of Lemma 3.1.For ∈ 1 (Ω; 2 ) satisfying = − 3 on Ω we have by Cauchy-Schwarz and Poincaré inequalities

Proof of Lemma 3.2. Step 1: Truncation of the Belavin-Polyakov profile
We truncate the standard Belavin-Polyakov profile by choosing > 1 and setting for > 0 and (3.7) for ∈ ℝ 2 .One may then compute, see [3, equation (A.66)], that so that and for all ∈ 2 (0) ⧵ (0).Consequently, we have Furthermore, as can be seen by a direct computation we have Step 2: Construction of competitors Let > 0 be the in-radius.Without loss of generality, we may assume that the corresponding in-circle is centered at the origin, so that (0) ⊂ Ω.Let now > 0 and > 1 be such that 2 ≤ .Then the function , ( ) ∶= Φ ( −1 ) satisfies , ∈ .For < 1 we compute To minimize the preceding expression, we need to choose as big as possible, i.e., = 2 .This yields In particular, choosing big enough, we obtain that which yields (3.3).Furthermore, optimizing in gives = ∕( ) for some suitably chosen > 0 depending only on Ω, so that > 1 for < ∕ and which completes the proof.
Arguing as in [12,32], we complete the squares to get for all ∈ 1 (Ω; 2 ) that As a result, by the lower semicontinuity of the right-hand side in (3.18) and the continuity of the DMI term we have Therefore, for small enough we get with the help of Lemmas 3.1 and 3.2: from which  ( ∞ ) = 1 immediately follows.
4 The next-order Γ-limit

Preliminaries
Before we turn to the actual proof of Theorem 2.6, we establish a number of preliminary statements designed to provide compactness in BP-convergence.First, we prove that in fact the skyrmion center for 0 < ≪ 1 satisfies ∈ Ω, as well as a lower bound for the Dirichlet excess of a minimizer in terms of its radius and dist( , ℝ 2 ⧵ Ω).The idea is that for achieving the Dirichlet distance from to  we have ∇( − )( ) = −∇ ( ) for all ∈ ℝ 2 ⧵ Ω, so that the control of Theorem 2.2 can be translated into a control over the radius and the center.

Lemma 4.1.
There exist 0 > 0 and , ′ > 0 depending only on Ω such that for all 0 < < 0 the following statement holds: Let ∈  and let ( ) = Φ( −1 ( − )) with ∈ (3), > 0, ∈ ℝ 2 achieve the Dirichlet distance from  to .Then we have ∈ Ω and We now record some basic estimates for the skyrmion tail.As we wish to apply this result also in the construction of a recovery sequence, we take care to only assume ∈ , not ∈  .

Lemma 4.2.
There exists a constant > 0 only depending on Ω such that we have the following statement: Let furthermore ∈ .For ∶= + 3 − − 3 we have With these bounds we can give an estimate for the DMI term.Again, we only assume ∈  to keep the statement applicable for the construction of the recovery sequence.(3), ∈ 0, 1 2 , ∈ Ω.Then there exist 1 > 0 universal and 2 = 2 (Ω, ) > 0 such that the following holds: If additionally ∈  and achieves the Dirichlet distance of to , then there exists 0 > 0 and 3 > 0 depending only on Ω such that for all ∈ (0, 0 ) the estimate takes the form . (4.7) Proof of Lemma 4.1.
Step 3: ∈ Ω, provided is small enough.Towards a contradiction, let us assume that ∉ Ω.We claim that since Ω is a Lipschitz domain, there exist > 0 and ̃ > 0 depending only on Ω such that  ( ) ∩ ̃ ( ) ⊂ ℝ 2 ⧵ Ω, where  ( ) is a cone with vertex at and the opening angle .Indeed, we may assume that is sufficiently close to Ω, and near the set Ω is locally a subgraph of a Lipschitz function.Translating the point vertically down towards Ω, we obtain a cone  ( ̃ ) pointing up with ̃ ∈ Ω that lies above Ω in ( ̃ ) for some > 0 and > 0 depending only on Ω. Hence the claim follows by translating the cone  ( ̃ ) vertically upward until its vertex coincides with .We now compute (again, as in [3, Equation (A.67)], and using (2.10) and (3.2)) By step 2, the left-hand side is uniformly bounded from below, giving a contradiction for small enough.
Step 4: We have estimate (4.1).Once again, since Ω is a Lipschitz domain there exist > 0 and ̃ > 0 depending only on Ω such that for any ̂ ∈ Ω we have We next fix > 0 small enough (depending only on ).If dist( , ℝ 2 ⧵ Ω) ≥ ̃ then by the estimate (4.9) shown in Step 2 we have with ′ depending only on Ω. If, on the other hand, dist( , ℝ 2 ⧵ Ω) < ̃ (meaning | ̂ − | is very small comparing to ̃ ) then using basic geometry arguments we deduce that there exist a cone  ( ) with the opening angle > 0, and a constant > 0 (both depending only on and ) such that Similarly to the previous calculations (see (4.11)), we obtain We further calculate (4.13) By step 2, we know that 2 ≤ ( ) and, therefore, we have Taking small enough and recalling ( ) ≤ , we deduce that (4.16) Therefore, we have (4.17) Using the fact that 2 + | − | 2 ≥ 2 | − | and the fact that ∈ (3), we obtain (4.2).Taking the gradient of both parts of (4.17) we arrive at (4.3) in a similar way.
Proof of Lemma 4.3.Letting ∶= + 3 − − 3 , we compute The first term gives Similarly, we get the estimate (4.5).In total, we obtain We treat the second term by using Young's inequality to get Therefore, we get For the third term, we find by Hölder's inequality that for > 2 and ′ = −1 ∈ (1, 2) we have Noticing that |∇Φ| ′ decays sufficiently fast to be integrable, we furthermore compute By the Sobolev embedding, for some > 0 depending only on Ω and we have Therefore, together with the estimates (4.4) and (4.27) we see that Applying Young's inequality to For the last term, we have by Young's inequality and estimate (4.4) that Combining the estimates ( , we get the desired estimate (4.6).Furthermore, the only dependence of the constant on in (4.6) is through estimate (4.23), and for achieving the Dirichlet distance we can uniformly absorb this term into ( ) using Lemma 4.1.This gives us estimate (4.7).

Γ-convergence
With the preliminary statements above, we can now argue for all the relevant compactness properties.Essentially, the centers cannot approach the boundary since otherwise the Dirichlet excess will be too large by Lemma 4.1.Estimates for the radii and the Dirichlet excess ( ) easily follow.To control pinning of the rotation, we refer to the Moser-Trudinger-type inequality [3, Lemma 2.5] to side-step the fact that in two dimensions 1 does not embed into ∞ .Recall that  0 and 0 are defined in (2.21) and (2.14), respectively.there exists a subsequence (not relabeled) and ( 0 , 0 , 0 ) ∈ 0 such that BP-converges to ( 0 , 0 , 0 ).Note that the above lemma does not yet yield a limit in  0 .
We now proceed to formulate the Γ-convergence result by first giving the lower bound statement.The convergence of the DMI term will follow from Lemma 4.3, and therefore we only need to deal with the Dirichlet excess.In particular, we have to prove that deviations from the Belavin-Polyakov profile are only energetically favorable in the tail of the skyrmion.To this end, we split the Dirichlet excess into a part localized in the core and a tail contribution.The localized part turns out to be given by the Hessian of the Dirichlet energy after using a new parametrization by mapping ℝ 2 to the sphere, with the Belavin-Polyakov profile closest to .Since Belavin-Polyakov profiles are minimizers of the Dirichlet energy, the Hessian is non-negative and thus the contribution of any possible core correction is non-negative.Then we have and, in particular, ( 0 , 0 , 0 ) ∈  0 .
We next turn to the construction of a recovery sequence.Essentially, we take the Belavin-Polyakov profile determined by the limit problem and modify the tail according to the harmonic function arising as the minimizer of ( 0 ), see (2.20).The DMI term has again been treated in Lemma 4.3, so that after an appropriate construction only the Dirichlet term remains to be analyzed.To ensure that the tail correction does not affect the skyrmion core, we modify to satisfy = 0 in a small neighborhood of 0 , which is possible since points have zero capacity in 1 (ℝ 2 ).Proposition 4.6.For every ( 0 , 0 , 0 ) ∈  0 and all sequences of → 0 there exists a sequence ∈  BP-converging to ( 0 , 0 , 0 ) such that Throughout the proofs of these statements, we will omit the index from the notation by abuse of notation.
Step 1: Estimate dist( , ℝ 2 ⧵ Ω) and ( ), and prove By our assumption (4.33), for > 0 small enough there exists a subsequence (not relabeled) and 1 > 0, such that we have Completing the square on the left-hand side, we get Since the left-hand must be non-negative and constants , 1 , 2 are positive, we obtain the estimate so that after passing to a further subsequence we have → 0 with 0 ∈ Ω. Continuing from (4.42), we also have Consequently, we obtain lim sup →0 < ∞.Since we also obtain lim inf →0 > 0. Extracting a further subsequence, if necessary, we find ∈ (0, ∞) such that → .By the estimate (4.40), Theorem 2.2 and taking into account lim sup →0 < ∞, we also get Step 2: Prove 0 3 = 3 .
Proof of Proposition 4.6.In contrast to the rest of the paper, in this proof the constant may depend on 0 and 0 .We fix ( 0 , 0 , 0 ) ∈  0 and take ∈ ̊ 1 (ℝ In particular, is harmonic in Ω, unique, and by the maximum principle satisfies ∈ ∞ (ℝ 2 ; ℝ 2 ).We start by constructing two auxiliary functions which will be useful in the construction of the recovery sequence.
Step 3: Definition of the recovery sequence.
Step 4: Computation of the energy.By Lemma 4.   Proof of Theorem 2.6.After we established the compactness of sequences obeying (4.33) with respect to the BP-convergence in Lemma 4.4, the statement of Theorem 2.6 follows by combining Propositions 4.5 and 4.6, and noting that by Proposition 4.5 the limit of the sequence in Lemma 4.4 belongs to  0 .

Disks
Proof of Proposition 2.10.Without loss of generality, we may assume = 1.Let 0 be defined by (2.33).We recall that for ∈ ℂ∖ 1 (0) we have 0 ( ) = 2 ̄ − ̄ 0 . (5.12) In particular, up to complex conjugation, 0 is invariant under the Kelvin transform, i.e., for all ∈ 1 (0) we have 0 which is holomorphic and, therefore, harmonic in 1 (0).Furthermore, the function 0 is continuous across 1 (0).By the uniqueness of boundary value problems for harmonic functions and continuity at the boundary, 0 is indeed the function achieving ( 0 ).By Proposition 2.9, we obtain (5.14) Clearly, this expression is minimized for 0 = 0.The rest of the statement is obtained by a direct substitution.

Anisotropy as a continuous perturbation
Proof of Proposition 2.13.Due to the properties of the BP-convergence, there exist sequences ∈ (3), > 0 and ∈ Ω such that with ( ) ∶= Φ( −1 ( − )) for ∈ ℝ 2 the estimates of Definition 2.3 hold.Again, by Friedrichs' inequality [31,Corollary 6.11.2] we have which allows to control the 2 -distance between and the Belavin-Polyakov profile that approximates it.In particular, by Lemma 4.2 and the properties of the BP-convergence, the two error terms in the right-hand side of (6.1) are of order 2 for all large enough.Therefore, additionally reparametrizing the integral by the factor −1 in the second step and using the assumption lim →∞ = 0 , we have that where we recalled that ∕ → 0 as → ∞ by the BP-convergence.
At the same time, as lim →∞ = 0 ∈ Ω, there exists an ̃ > 0 such that ̃ ( ) ⊂ Ω for all ∈ ℕ.By the estimate (6.4) the expression on the right-hand side of estimate (6.3) is bounded, so that we can pass to the limit in the rotation lim →∞ = 0 and exploit 0 3 = 3 , to get lim (6.5) The statement then follows from combining the estimates (6.4) and (6.5).
, if ∶= represents the unit tangent to Ω in the counter-clockwise direction, the tangential derivative = ′ on Ω.Therefore, using (5.7) we can write At the same time, by continuity of 0 across Ω we have on Ω, and the integral in (5.6) can be rewritten as a Cauchy type contour integral( 0 ) = −4 ∮ Ω ′