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Research article

Variational analysis in one and two dimensions of a frustrated spin system: chirality and magnetic anisotropy transitions

  • Received: 18 July 2022 Revised: 03 June 2023 Accepted: 04 June 2023 Published: 14 June 2023
  • We study the energy of a ferromagnetic/antiferromagnetic frustrated spin system where the spin takes values on two disjoint circles of the 2-dimensional unit sphere. This analysis will be carried out first on a one-dimensional lattice and then on a two-dimensional lattice. The energy consists of the sum of a term that depends on nearest and next-to-nearest interactions and a penalizing term related to the spins' magnetic anisotropy transitions. We analyze the asymptotic behaviour of the energy, that is when the system is close to the helimagnet/ferromagnet transition point as the number of particles diverges. In the one-dimensional setting we compute the Γ-limit of scalings of the energy at first and second order. As a result, it is shown how much energy the system spends for any magnetic anistropy transition and chirality transition. In the two-dimensional setting, by computing the Γ-limit of a scaling of the energy, we study the geometric rigidity of chirality transitions.

    Citation: Andrea Kubin, Lorenzo Lamberti. Variational analysis in one and two dimensions of a frustrated spin system: chirality and magnetic anisotropy transitions[J]. Mathematics in Engineering, 2023, 5(6): 1-37. doi: 10.3934/mine.2023094

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  • We study the energy of a ferromagnetic/antiferromagnetic frustrated spin system where the spin takes values on two disjoint circles of the 2-dimensional unit sphere. This analysis will be carried out first on a one-dimensional lattice and then on a two-dimensional lattice. The energy consists of the sum of a term that depends on nearest and next-to-nearest interactions and a penalizing term related to the spins' magnetic anisotropy transitions. We analyze the asymptotic behaviour of the energy, that is when the system is close to the helimagnet/ferromagnet transition point as the number of particles diverges. In the one-dimensional setting we compute the Γ-limit of scalings of the energy at first and second order. As a result, it is shown how much energy the system spends for any magnetic anistropy transition and chirality transition. In the two-dimensional setting, by computing the Γ-limit of a scaling of the energy, we study the geometric rigidity of chirality transitions.



    Lattice systems are discrete variational models, whose energy depends on a spin field defined in a lattice. In frustrated lattice systems, spins cannot find an orientation that simultaneously minimizes the nearest-neighbor (NN) and the next-nearest-neighbor (NNN) interactions. Such interactions are said to be ferromagnetic or antiferromagnetic if they favour alignment or anti-alignment (we address the reader to [13] for a complete dissertation).

    Three-dimensional frustrated magnets generally exist in the magnetic diamond and pyrochlore lattices (see [14]) and edge-sharing chains of cuprates provide a natural example of frustrated lattice systems (see [16]). Furthermore, jarosites are the prototype for a spin-frustrated magnetic structure, because these materials are composed exclusively of kagomé layers (see [20]).

    A different frustration mechanism can also be caused by magnetic anisotropy, as it is common in spin ices (see [17]). Magnetic anisotropy refers to the dependence of the magnetization of a material on the direction of the applied magnetic field, which acts as a potential barrier (we address the reader to [23] for a comprehensive overview of magnetism, including a chapter on magnetic anisotropy and the energy barrier). The interplay between the two frustration mechanisms may result in very complicated Hamiltonians (see [22]). Most recently, the physics community attempts to find new fundamental effects such as the magnetization plateaus and the magnetization jumps which represent a genuine macroscopic quantum effect. For example, kagomé staircases have been of particular interest because of the concurrent presence of both highly frustrated lattice and strong quantum fluctuations (see [24]).

    In this paper we study a frustrated lattice spin system whose spins take values on the unit sphere of R3. More precisely, a spin of the system u is a vectorial function whose codomain is the union of two fixed disjoint circles, S1 and S2, of the unit sphere, which have the same radius R and are identified by two versors, v1 and v2, Figure 1. We set the problem in one and two dimensions: in the one-dimensional case (Section 3) spin fields are parametrized over the points of the discrete set [0,1]λnZ and satisfy a periodic boundary condition; in the two-dimensional case (Section 4) they are parametrized over the points of the discrete set ΩλnZ2, where ΩR2 is an open bounded regular domain. In both cases {λn}nN is a vanishing sequence of lattice spacings. In the first setting, the energy of a given spin of the system u:λni[0,1]λnZuiS1S2 is

    En(u)=En(u)+Pn(u),
    Figure 1.  S1 and S2 circles of anisotropy transitions.

    with

    En(u)=i[0,1]λnZλn[αuiui+1+uiui+2]andPn(u)=λnkn|DA(u)|(I),

    where α(0,+) is the frustration parameter of the system that rules the NN and NNN interactions and {kn}nN is a divergent sequence of positive numbers. The term A(u) indicates the spins' magnetization direction (the so-called magnetic anisotropy) in the two circles. If the number of magnetic anisotropy transitions, i.e., the number of the jumps between the two circles, is finite, A(u) is a BV function and |DA(u)|(I) counts them. According to physical considerations, we require that the energy Pn gives a penalizing contribution to the total energy.

    It is easy to see that while the first term of the energy En is ferromagnetic and favors the alignment of neighboring spins, the second one, being antiferromagnetic, frustrates it as it favors antipodal next-to-nearest neighboring spins. A more refined analysis, contained in Proposition 3.5 and Remark 3.6, shows that, for n sufficiently large, the ground states of the system take values on one of the two circles and for α4 are ferromagnetic (the spins are made up of aligned vectors), while for 0<α4 they are helimagnetic (the spins consists in rotating vectors with a constant angle ϕ=±arccos(α/4)). The property of the latter case is known in literature as chirality simmetry: the two possible choices for the angle correspond to either clockwise or counterclockwise spin rotations, or in other words to a positive or a negative chirality.

    In this paper, we address a system close to the ferromagnet/helimagnet transition point (see [15]), that is when α is close to 4 from below. We also require that λnkn is close to some positive value (that can be also infinite). This assumption is reasonable, since from a physical point of view the change of the spin's polarization involves a larger amount of energy. Our aim is to provide a careful description of the admissible states and compute their associated energy. In particular, we find the correct scalings to detect the following two phenomena: the spins' magnetic anistropy transitions and chirality transitions that break the rigid simmetry of minimal configurations.

    In [12], the authors studied a one-dimensional ferromagnetic/antiferromagnetic frustrated spin system with nearest and next-to-nearest interactions close to the helimagnet/ferromagnet transition point as the number of particles diverges. In that case, spin fields take values in the unit circle. The proposed model is different from that one, where no anisotropy functional Pn was introduced. In [12] the presence of a periodic boundary condition allowed manipulating En in such a way that it can be recast as a discrete version of a Modica-Mortola type energy, whose Γ-convergence is well-known in literature (see [18] and [19]). Indeed, expanding the functional at the first order, under a suitable scaling, spin fields can make a chirality transition on a scale of order λnδn, when λnδn approaches to a finite nonnegative value, as n+ (otherwise no chirality transitions emerge).

    To set up our problem, we let the ferromagnetic interaction parameter α depend on n and be close to 4 from below, that is, we substitute α by αn=4(1δn) for some positive vanishing sequence {δn}nN. As in [12], the Γ-limit of En (with respect to the weak convergence in L) does not provide a detailed description of the phenomena (as a consequence of Theorem 3.12) and suggests that, in order to get further information on the ground states of the system, we need to consider higher order Γ-limits (see [6] and [7]).

    The two phenomena can be detected at different orders. At the first order we are led to normalize the energy En of the system and study the asymptotic behavior of (a rescaling of) the new functional Gn defined by

    Gn=EnminEn.

    Rescaling Gn by λn, we prove that magnetic anisotropy transitions can be captured when λnkn is close to any positive finite value, for n large enough (see Theorem 3.16). At the scale value λn, the energy spent for spin's magnetic anisotropy transitions is equal to the minimal energetic value corresponding to the sum of all the interactions in proximity of the transition points. In Figure 2 it can be seen an occurrence of the phenomenon that we are analyzing.

    Figure 2.  Magnetic anisotropy transitions.

    Chirality transitions can be detected at the next order by means of a technical decomposition of the energy Gn. The idea behind the construction in Subsection 3.6 is to split the problem set in the sphere into finitely many problems set in one of the two circles each. We associate each spin field u with a unique and finite partition of [0,1] containing intervals Ij such that u|Ij takes values only in one circle. We note that the intervals Ij depend on n because u is defined on the lattice [0,1]λnZ. We modify such restrictions u|Ij in such a way that they still satisfy a similar periodic boundary condition on Ij, denoting them as ˜uIj. In Lemma 3.13 we decompose the functional Gn as follows:

    Gn(u)=jMMn(˜uIj)+j(Rn)j(u)+(Rn)M(u)(u)+Rn(u).

    The energy MMn is of discrete Modica-Mortola type and collects the pairwise interactions of spins' vectors pointing to the same circle; the functionals (Rn)j and (Rn)M(u) gather the interactions between consecutive spins' vectors that point to different circles. Rn is a correction addend. The first sum and the other addend in the right-hand side of the previous formula need to be rescaled in different ways, the first sum being a higher order term. Thus, at the second order we deal with the energy

    Gn(u)=Gn(u)j(Rn)j(u)(Rn)M(u)(u)Rn(u)=jMMn(˜uIj).

    In Theorem 3.18 we apply the Γ-convergence result contained in [12] to each functional MMn, rescaled by λnδ3/2n. It turns out that different scenarios may occur, depending on the value of limnλn/δn:=l[0,+]. If l=+, chirality transitions are forbidden. Otherwise a spin field can make a chirality transition on a lenght-scale λn/δn. In particular, if l>0, it may have diffuse and regular macroscopic (on an order one scale) chirality transitions in each Sj whose limit energy is finite on H1(Ij) (provided some boundary conditions are taken into account); if l=0, chirality transitions on a mesoscopic scale are allowed. In this case, the continuum limit energy is finite on BV(Ij) and counts the number of jumps of the chirality of the spin field.

    Systems defined in planar structures are much more difficult to study, due to the higher dimensional setting (see [1,4,5,10,11]). We address here the two-dimensional analogue of the frustrated spin chain studied in the first part of the paper. The energy of a given spin of the system u:(i,j)ΩλnZ2ui,jS1S2 is

    En(u;Ω)=En(u;Ω)+Pn(u;Ω),

    where

    En(u;Ω)=α(i,j)λ2n(ui,jui+1,j+ui,jui,j+1)+(i,j)λ2n(ui,jui+2,j+ui,jui,j+2)

    and

    Pn(u;Ω):=λnkn|DA(u)|(Ω).

    We assume that the functional Pn(;Ω) is bounded. The number α>0 is the frustration parameter of the system and {kn}nN is a divergent sequence of positive numbers. The term |DA(u)|(Ω) is related to magnetic anistropy transitions. In the two-dimensional setting, they occur on the edges of the lattice ΩλnZ2 and the natural number |DA(u)|(Ω)λn|v1v2| is an upper bound on the spins' transitions from a circle to the other in Ω.

    Motivated by the variational analysis of the one-dimensional problem, we assume that the frustation parameter depend on n and is close to the helimagnet/ferromagnet transition point as the number of particles diverges, i.e., αn4. In view of detecting spins' chirality transitions, which cannot be captured by means of the Γ-limit of the energy at the zero order, we are interested in the functional defined by

    Hn(u;Ω):=12λnδ32n12λ2n(i,j)[|ui+2,jαn2ui+1,j+ui,j|2+|ui,j+2αn2ui,j+1+ui,j|2],

    which is the two-dimensional analogue of Gn, up to additive constants.

    In [10] the authors studied a similar frustrated spin chain whose spin fields take values in the unit circle of R2. In [10,Theorem 2.1] they proved the emergence of spins' chirality transitions by means of the Γ-convergence of the functional Hn with respect to the local L1-convergence of two chirality parameters.

    In view of applying their result in our setting, we employ an idea that recalls the construction carried out in the one-dimensional problem. We restrict every spin u to connected open sets Cs that partition Ω in such a way that u|Cs takes values only in one circle. In order to avoid more complicated notation, we do not impose boundary conditions on Ω and we state the result by means of a local convergence. We note that the sets Cs depend on n because u is defined on the lattice ΩλnZ2.

    We decompose

    Hn(u;Ω)=s[Hn(u;Cs)+(Rn)Cs(u)],

    where Hn collects the interactions of spins' vectors pointing to the same circle and (Rn)Cs gathers the interactions between spins' vectors that point to different circles.

    While in the one-dimensional setting the partition associated with a spin contains intervals, which guaranty the compactness results stated, in this case the sets Cs could be very wild, as the spacing of the lattice shrinks. Therefore, we require as additional regularity condition for the components Cs, that is the BVG regularity. Its definition can be found in [21] and is recalled in Definition 4.1.

    With this regularity assumption, we can apply the Γ-convergence result proved in [9] to each addend of the functional

    Gn(h;Ω)=Hn(h,Ω)s(Rn)Cs(h)=sHn(h;Cs),

    as it is shown in Theorem 4.5, that is the main result of Section 4. It turns out that chirality transitions are possible and they can take place both in the vertical and horizontal slices of Cs.

    Given xR, we denote by x the integer part of x. For a set K we denote by co(K) the convex hull of K, by #K the number of its elements and χK its characteristic function. We write vw for the Euclidean scalar product of the vectors v,wR3 and by S2 the unit sphere of R3. For all vR3 we denote by πv the Euclidean projection on v and by πv the projection on the orthogonal complement of v. If A is a subset of the Euclidean space we denote by ¯A its closure respect the Euclidean topology. We denote by C a generic constant that may vary from line to line in the same formula and between formulas. Relevant dependencies on parameters and special constants will be suitably emphasized using parentheses or subscripts.

    If IR is an interval and all wBV(I;R3), we denote by DwMb(I;R3) the distributional differential of w, and by |Dw|Mb(I) the total variation measure of Dw. We say that a sequence {un}nN converges weakly in BV(I;R3) to a function uBV(I;R3) if and only if

    unuin L1(I;R3)  and supnN|Dun|(I)<+,

    (see [3,Definition 3.11 and Proposition 3.13]). We denote it by unBVu.

    Fixing v1,v2S2 and R(0,1), we define the set

    Si:={wS2:|πvi(w)|=R,w,vi>0},for i{1,2}. (2.1)

    It is easy to observe that the set Si is a circle centered in vi1R2 and it can be easily verified that for R<RMax:=1v1v22 the sets S1 and S2 are disjoint. Throughout the paper we assume that R(0,RMax).

    If S is an open set of RN and C is a collection of open subsets of S, we say that C is an open partition of S if C does not contain empty sets and

    ¯S=CC¯C,C1C2=,C1,C2C.

    Given two vectors w=(w1,w2),¯w=(¯w1,¯w2) of R2, we define the function

    χ[w,¯w]:=sign(w1¯w2w2¯w1),

    with the convention that sign(0)=1.

    We let I=(0,1) and we consider a sequence {λn}nNR+ that vanishes as n+. It represents a sequence of spacings of the lattice ¯IλnZ.

    We introduce the class of functions valued in S1S2 which are piecewise constant on the edges of the lattice ¯IλnZ and satisfy a periodic boundary condition:

    PCλn:={u:¯IS1S2:u(t)=u(λni) for tλn[i+[0,1)]¯I and λni¯IλnZ,u0u1=u1λn1u1λn}. (3.1)

    We will identify a piecewise constant function u:¯IS1S2 with the function defined on the points of the lattice given by λni¯IλnZui:=u(λni). Conversely, given values uiS1S2 for λni¯IλnZ, we define u:¯IS1S2 by u(t):=ui for tλn[i+[0,1)].

    There exists a natural projection map A:L(I;S1S2)L(I;{v1,v2}) defined as follows:

    A(u)(t)={v1if u(t)S1,v2if u(t)S2,tI. (3.2)

    For each spin u, the function A(u) indicates the spins' magnetization direction and its jumps correspond the the spins' magnetic anisotropy transitions. In general, A can be defined analogously on L(I;K1K2), if K1 and K2 are two disjoint subsets of R3 containing, respectively, S1 and S2. In this case, we remark that if a spin field uL(I;K1K2) switches from K1 to K2 a finite number of times, i.e., A(u)BV(I;{v1,v2}) and so |DA(u)|(I)<+, the interval I can be partitioned in finitely many regions where the function u takes values only in one of the two sets K1 and K2. In other words, there exist M(u)N and a collection of open intervals, {Ij}j{1,,M(u)}, such that

    {Ij}j{1,,M(u)} is an open partition of I, (3.3)
    either u(Ij)K1 or u(Ij)K2, for any j{1,,M(u)}, (3.4)
    u(Ij)×u(Ij+1)(K1×K2)(K2×K1),for any j{1,,M(u)1}. (3.5)

    The last two properties imply that this partition is unique. We observe that, if uL(I;S1S2) and A(u)BV(I;{v1,v2}) (or, in particular, if uPCλn), then

    M(u)=|DA(u)|(I)|v1v2|+1.

    The following definition will be useful throughout the section.

    Definition 3.1. Let uL(I;S1S2) be such that A(u)BV(I;{v1,v2}). We say that Cn(u)={Ij|j{1,,M(u)}} is the open partition associated with u if M(u)=|DA(u)|(I)|v1v2|+1 and the collection of open intervals {Ij}j{1,,M(u)} satisfies (3.3), (3.4) and (3.5).

    In this subsection we recall some classical properties of the Lebesgue space L(I;K), where KR3 is a compact set. The statements and the proofs are fully analogous if the setting is a N-dimensional Euclidean space.

    Proposition 3.2. Let {fn}nNL(I;K). Then, up to subsequences, fnfL(I;co(K)) in the weak topology of L(I;R3). Moreover, for all uL(I;co(K)) there exists a sequence {un}nNL(I;K) of piecewise constant functions such that unu.

    Proof. Since the set K is bounded then, up to a subsequence, there exists fL(I;R3) such that fnf. Now we prove that f(t)co(K) for almost every tI. For every ξco(K) there exist an affine function hξ:RNR and α<0 such that

    hξ(ξ)>0>α>hξ(x),xco(K).

    By the weak convergence of {fn}nN we have that for any measurable set AI

    Ahξ(f(t))dt=limn+Ahξ(fn(t))dt|A|α<0.

    Hence, by the arbitrariness of A, we obtain

    hξ(f(t))<0, for a.e. tI. (3.6)

    Recalling that

    co(K)=jN{yR3:hξj(y)<0,ξjQNco(K)},

    by formula (3.6) we obtain

    f(t)co(K), for a.e. tI.

    Now we prove the second statement of the proposition. Let uL(I;co(K)). There exists a sequence {un}nNL(I;co(K)) such that un=mj=1ajχIj, where ajco(K) and IjI is an interval, for any j{1,,m}, and un converges to u in L1(I;R3). Hence, unu. Therefore, without loss of generality, we may prove the statement for u=aco(K).

    We define the following function:

    h(t):={a1if t(0,λ),a2if t[λ,1),

    where a=λa1+(1λ)a2 with a1,a2K and for some λ[0,1]. Then the sequence un(t):=h(nt) converges to u in the weak topology of L by Riemann-Lebesgue's lemma.

    Corollary 3.3. The closure of the set L(I;K) with respect to the weak topology of L(I;R3) is the set L(I;co(K)).

    Proof. Since the space L1(I;R3) is separable, every bounded subset of L(I;R3) is metrizable with respect to the weak topology of L(I;R3). Hence the set L(I;K) is metrizable. Therefore, by Proposition 3.2, we have that the set L(I;co(K)) is the weak closure of the set L(I;K).

    In this subsection we cite an abstract Γ-convergence result proved in [2] that will be applied in Subsection 3.5. For this purpose, we introduce the following notation. Let KRN be a compact set and for all ξZ let fξ:R2NR be a function such that

    (H1) fξ(x,y)=fξ(y,x),

    (H2) for all ξZ, fξ(x,y)=+ if (x,y)K2,

    (H3) for all ξZ there exists Cξ0 such that

    sup(x,y)K2|fξ(x,y)|CξandξZCξ<+.

    For any nN we define the functional space

    Dn(I;RN):={u:RRN:u is constant in λn(i+[0,1)) for all λni¯IλnZ}.

    With the notation already used, we denote the value of the function u in the interval λn(i+[0,1)) by ui for all λni¯IλnZ. We introduce the sequence of functionals Fn:L(I;RN)(,+] defined as follows:

    Fn(u):={ξZiRξn(I)λnfξ(ui,ui+ξ) for uDn(I;RN),+ for uL(I;RN)Dn(I;RN),

    where Rξn(I):={λni¯IλnZ:λni+ξ¯IλnZ}, for ξZ. For any open and bounded set AR and for every u:ZRN, we define the discrete average of u in A as

    (u)1,A:=1#(ZA)iZAui.

    Theorem 3.4 (See [2]). Let {fξ}ξZ be a family of functions that satisfies H1, H2, H3. Then the sequence Fn Γ-converges, as n+, with respect to the weak topology of L(I;RN), to

    F(u):={Ifhom(u(t))dtforuL(I;co(K)),+foruL(I;RN)L(I;co(K)),

    where fhom:RNR is given by the following homogenization formula

    fhom(z)=limρ0limk+1kinf{ξZβRξ1((0,k))fξ(u(β),u(β+ξ))|us.t.(u)1,(0,k)¯B(z,ρ)}.

    This subsection is devoted to the mathematical formulation of the model and the characterization of its ground states.

    Let α>0 be a fixed parameter and let {kn}nNR+ be a divergent sequence of positive numbers. Denoting

    In(I):=(¯IλnZ){λn(1λn1),λn1λn},

    and in general if J=(a,b)I we define

    In(J):=(¯JλnZ){λn(bλn1),λnbλn}.

    We define the energy of the system as the sum of two addends. The first addend is a bulk scaled energy of a frustrated F-AF spin chain, En:PCλn(,+), having the following form:

    En(u):=λniIn(I)[αuiui+1+uiui+2].

    The second addend of the energy, Pn:PCλn[0,+), is a term of confinement in S1S2 and is defined as follows:

    Pn(u):=λnkn|DA(u)|(I),

    where A is the function defined in formula (3.2). We consider the family of energies En:PCλn(,+) defined by

    En(u)=En(u)+Pn(u).

    Furthermore, we define the functional Hn:PCλn[0,+) by

    Hn(u):=12λniIn(I)|ui+2α2ui+1+ui|2.

    If uPCλn, since |ui|=1 for all λni¯IλnZ, thanks to the boundary condition contained in the definition of PCλn (see (3.1)), we compute:

    Hn(u)=12λniIn(I)(2+α24+2uiui+2αuiui+1αui+1ui+2)=12iIn(I)λn[α(uiui+1+ui+1ui+2)+2uiui+2]+λn(1+α28)#In(I)=En(u)+λn(1+α28)#In(I). (3.7)

    Thus we gain a new expression for En:

    En=Hn+Pnλn(1+α28)#In(I). (3.8)

    Thanks to this decomposition, we characterize the ground states of En.

    Proposition 3.5 (Characterization of the ground states of En). Let 0<α4. Then, for nN sufficiently large, it holds

    minuPCλnEn(u)=λn#In(I)[R2(1+α28)+(α1)(1R2)].

    Furthermore, a minimizer un of En over PCλn takes values only in one circle S, with {1,2}, and satisfies

    πvuinπvui+1n=R2α4andπvuinπvui+2n=R2(α281),iIn(I).

    Proof. Let us postpone the proof of the following equality:

    minuPCλnu(I)S1oru(I)S2En(u)=λn#In(I)[R2(1+α28)+(α1)(1R2)] (3.9)

    after the next claim. We claim that, for n sufficiently large, if uPCλn is a minimizer of En, then u(I)S1 or u(I)S2. We may assume that the open partition associated with u is {I1,I2}, i.e., M=2, and u(I1)S1, u(I2)S2. The general case MN can be proved similarly. We have that

    En(u)=En(u)+λnknc=λniIn(I1)In(I2)[αuiui+1+uiui+2]+λniIn(I)(In(I1)In(I2))][αuiui+1+uiui+2]+λnknc, (3.10)

    where c:=|v1v2|. We observe that #[In(I)(In(I1)In(I2))]2. We define

    L:=min{αuv+uwαvw+vz:u,vS1,w,zS2}

    and we observe that

    LiIn(I)(In(I1)In(I2))[αuiui+1+uiui+2].

    Therefore by the formula (3.10) we have that

    minuPCλnu(I1)S1En(u)+minuPCλnu(I2)S2En(u)+λn(L+knc)En(u). (3.11)

    In order to prove the claim we are left to show that, for nN sufficiently large,

    minuPCλnu(I)S1oru(I)S2En(u)<minuPCλnu(I1)S1En(u)+minuPCλnu(I2)S2En(u)+λn(L+knc), (3.12)

    which is equivalent to prove that

    λn#In(I)[R2(1+α28)+(α1)(1R2)]<λn(#In(I1)+#In(I2))[R2(1+α28)+(α1)(1R2)]+λn(L+knc),

    where we used formula (3.9). Since #In(I)#In(I1)#In(I2)2, R1 and α4, we have that, for n sufficiently large,

    λn(#In(I)#In(I1)#In(I2))[R2(1+α28)+(α1)(1R2)]12λn<λn(L+knc),

    because kn+. We have proved the validity of (3.12). Thus, combining (3.12) and (3.11), we get

    minuPCλnEn(u)=minuPCλnu(I)S1oru(I)S2En(u).

    We prove that

    minuPCλnu(I)S1oru(I)S2En(u)=λn#In(I)[R2(1+α28)+(α1)(1R2)].

    We fix {1,2} and consider uPCλn such that u(I)S. By geometric and trigonometric identities we deduce that

    uiui+1=1R2+πuiπui+1,

    where πui:=πvui. Of course an analogous statement holds for uiui+2. Thus

    En(u)=iIn(I)λn[απuiπui+1+πuiπui+2](α1)(1R2)λn#In(I)=˜En(u)(α1)(1R2)λn#In(I), (3.13)

    where we have defined

    ˜En(u):=iIn(I)λn[απuiπui+1+πuiπui+2].

    Now we are led to minimize ˜En. We find its minimum by following the same argument in [12]. With an easy computation similar to the one in (3.7), we remark that

    ˜En(u)=12λniIn(I)|πui+2α2πui+1+πui|2R2(1+α28)λn#In(I)=˜Hn(u)R2(1+α28)λn#In(I), (3.14)

    where

    ˜Hn(u):=12λniIn(I)|πui+2α2πui+1+πui|2.

    We fix ϕ[π2,π2] so that cosϕ=α4. We may assume for simplicity of notation that v=en. Let

    ui:=(Rcos(ϕi),Rsin(ϕi),1R2)S,i¯IλnZ,

    so that πui=(Rcos(ϕi),Rsin(ϕi),0). By trigonometric identities, we have that

    πui+πui+2=2cos(ϕ)πui+1=α2πui+1,iIn(I).

    Remarking that ˜Hn(u)=0, we combine the previous identity with (3.14) to get that

    minuPCλnu(I)S1oru(I)S2˜En(u)=R2(1+α28)λn#In(I).

    The computation of the minimum follows from (3.13).

    Now we consider a minimizer uPCλn of En. For n sufficently large, it must hold that u(I)S, for some {1,2}, and

    ˜En(u)=R2(1+α28)λn#In(I),

    thus implying that ˜Hn(u)=0. It follows that

    πui+1=2α(πui+πui+2),iIn(I).

    Squaring the modulus of both sides in the previous equality, we infer

    πuiπui+2=R2(α281).

    Hence

    πuiπui+1=2α(πuiπui+πuiπui+2)=2α(R2+πuiπui+2)=R2α4,

    which concludes the proof.

    From now on we assume that n is sufficiently large to satisfy the thesis of the above proposition.

    Remark 3.6. The case α>4 is trivial and the ground states of En are all ferromagnetic, i.e., ui=¯u, for all i¯IλnZ and for some ¯uS1S2. Indeed, denoting by E(α=4)n the energy of formula (3.8) for α=4, we have that

    En(u)=E(α=4)n(u)λn(α4)iIn(I)uiui+1,

    for all uPCλn. By the above proposition, the energy E(α=4)n is minimized on ferromagnetic states, which trivially also holds true for the second term in the above sum. The minimal value of En is

    minuPCλnEn(u)=λn(α1)#In(I).

    In this subsection we study the Γ-convergence of En at the zero order. With a slight abuse of notation, we extend the energies En, Pn, En and Hn to the space L(I;co(S1)co(S2)), setting their value as + in L(I;co(S1)co(S2))PCλn. With a slight abuse of notation, we extend the projection map A to the space L(I;co(S1)co(S2)) by setting

    A(u)(t)={v1if u(t)co(S1),v2if u(t)co(S2),

    for uL(I;co(S1)co(S2)). Furthermore we define

    D:={uL(I;co(S1)co(S2)):A(u)BV(I;co(S1)co(S2))}=A1(BV(I;co(S1)co(S2))). (3.15)

    It is natural to extend Definition 3.1 to any spin field uD. The following notion of convergence will be used.

    Definition 3.7. Let {un}nNL(I;co(S1)co(S2)) and uD. We say that un D-converges to u (we write unDu) if and only if unu in the weak topology of L(I;R3) and A(un) converges to A(u) weakly in BV(I;{v1,v2}).

    Remark 3.8. We observe that the notion of convergence introduced in the previous definition is induced by the smallest topology on D containing the set

    {A:A is an open set of the weak topology of L(I;co(S1)co(S2))or A=A1(U), where U is an open set of the weak topology of BV(I;co(S1)co(S2))}.

    For further details about the weak topology of a BV space we address the reader to [3,Remark 3.12].

    We prove the following proposition, which relies on the properties contained in Subsection 3.2 and will be useful in this subsection.

    Proposition 3.9. Let {un}nNL(I;S1S2) be such that A(un)BV(I;{v1,v2}), for any nN, and let Cn(un)={Inj|j{1,,M(un)}} be the open partition associated with un. We assume that

    supnNM(un)<+. (3.16)

    Then there exists uD such that, up to subsequences, unDu.

    Proof. By Proposition 3.2 it follows that, up to a subsequence, unuL(I;co(S1S2)). Thanks to (3.16), up to the extraction of a subsequence, M=M(un) is independent of nN. Up to subsequence, InjIj in the Hausdorff sense, for some intervals Ij and for any j{1,,M}. Note that some Ij could be empty. Let us fix j{1,,M}. For all ε>0 there exists n0N such that

    (Ij)ε={tIj:dist(t,Ij)>ε}Injnn0.

    We define the following two sets:

    A1={nn0:un(t)S1 for a.e. t(Ij)ε},A2={nn0:un(t)S2 for a.e. t(Ij)ε}.

    One of the following three alternatives may occur:

    1.#A1=,#A2<;2.#A1<,#A2=;3.#A1=,#A2=.

    In the first case we have that unL((Ij)ε;S1) for all nn0, up to finitely many indices of the sequence. Thus, by Proposition 3.2, unuL((Ij)ε;co(S1)) and hence, by the arbitrariness of ε>0, we obtain that uL(Ij;co(S1)). The second case is fully analogous to the first case. If we repeat the above argument for all j{1,,M}, we deduce that unu.

    Finally, we get the thesis by remarking that

    limn+I|A(un)A(u)|dt=limn+Mj=1Ij|A(un)A(u)|dt=0.

    The third alternative leads to a contradiction. Indeed, if it holds true, we can find two subsequences {n(1)k}kN and {n(2)k}kN such that un(1)kL((Ij)ε;S1) and un(2)kL((Ij)ε;S2), for all kN. By Proposition 3.2, there exist u1L((Ij)ε;co(S1)) and u2L((Ij)ε;co(S2)) such that un(1)ku1 and un(2)ku2. On the other hand, applying again Proposition 3.2, we infer that unuL(I;co(S1S2)). Then, by the uniqueness of the limit in the weak topology, we infer that u1(t)=u2(t)=u(t) for almost every t(Ij)ε, which is a contradiction since co(S1)co(S2)=.

    Firstly, we study the Γ-convergence of En. The following theorem relies on a straightforward application of Theorem 3.4.

    Theorem 3.10. The sequence En Γ-converges to the functional

    E(u):={Ifhom(u(t))dtifuL(I;co(S1S2)),+otherwise,

    with respect to the weak topology of L(I;R3), where fhom:co(S1S2)R is defined by

    fhom(z)=limρ0limk+1kinf{k2i=1[αuiui+1+uiui+2]|us.t.(u)1,(0,k)¯B(z,ρ)}. (3.17)

    Proof. The result immediately follows by applying Theorem 3.4 to

    fξ(u,v)={α2uvif |ξ|=1,12uvif |ξ|=2,0otherwise,

    where u,vK:=S1S2, extended to + outside K.

    Remark 3.11. The function fhom defined in (3.17) does not depend on the parameter λn. Therefore, in the theorem above the Γ-limit does not depend on the choice of λn.

    Furthermore, an analogous statement of Theorem 3.10 above can be obtained if the functional En is defined only in L(I;S) for some {1,2} (see [12,Theorem 3.4]). Its Γ-limit has the same form and it is finite on L(I;co(S)).

    The following theorem is the main result of this subsection.

    Theorem 3.12 (Zero order Γ-convergence of En). Assume that there exists limn+λnkn=:η(0,+]. Then the following Γ-convergence and compactness results hold true.

    (i) If η(0,+), then En Γ-converges to the functional

    E(u)={Ifhom(u(t))dt+η|DA(u)|(I)ifuD,+ifuL(I;co(S1)co(S2))D,

    with respect to the D-convergence of Definition 3.7, where fhom and D are defined in (3.17) and (3.15) respectively. Moreover if {un}nNL(I;co(S1)co(S2)) satisfies

    supnNEn(un)<+,

    then, up to a subsequence, unDuD.

    (ii) If η=+, then En Γ-converges to the functional

    E(u):={Ifhom(u(t))dtif uL(I;co(S1))L(I;co(S2)),+if uL(I;co(S1)co(S2))(L(I;co(S1))L(I;co(S2))),

    with respect to the weak topology of L(I;R3), where fhom is defined in (3.17). Moreover if {un}nNL(I;co(S1)co(S2)) satisfies

    supnNEn(un)<+

    then, up to a subsequence, unu for some uL(I;co(S1))L(I;co(S2)).

    Proof. We first deal with case (ⅰ). We start by proving the compactness result. Let {un}nNL(I;co(S1)co(S2)) be such that

    supnNEn(un)<¯C, (3.18)

    for some ¯C>0. Thus we have that {un}nNPCλn. Let us consider the open partition Cn(un)={(Ij)n|j{1,,M(un)}} associated with un, where M(un)1=|DA(un)|(I)|v1v2|N. By formula (3.8) and by the definition of A, we compute

    En(un)=Hn(un)+Pn(un)λn(1+α28)#In(I)Pn(un)λn(1+α28)#In(I)=knλn|DA(un)|(I)λn(1+α28)#In(I)=knλn(M(un)1)|v1v2|λn(1+α28)#In(I)C(α)+knλn(M(un)1)|v1v2|, (3.19)

    for some constant C=C(α)>0, where the last inequality is obtained by observing that λn#In(I)=λn1λnλn1, as n+, and thus it is bounded. Therefore by formulae (3.18) and (3.19) we obtain that

    supnNM(un)<C(η,¯C,α,|v1v2|).

    Hence, the sequence {un}nN satisfies the hypotheses of Proposition 3.9 and so we deduce the existence of uD such that, up to a subsequence, unDu.

    Now we prove the liminf inequality. Let {un}nNL(I;co(S1)co(S2)) be such that unDuD. It is not restrictive to assume that {un}nNPCλn. By the liminf inequality of Theorem 3.10 we have

    lim infn+En(un)Ifhom(u(t))dt. (3.20)

    On the other hand, by the lower semicontinuity of the total variation respect the weak convergence in BV(I;{v1,v2}), we have

    lim infn+Pn(un)=lim infn+knλn|DA(un)|(I)η|DA(u)|(I). (3.21)

    Hence by formulae (3.20) and (3.21) we obtain

    lim infn+En(un)lim infn+En(un)+lim infn+Pn(un)Ifhom(t)dt+η|DA(u)|.

    We finally prove the limsup inequality. Let uL(I;co(S1)co(S2)). We may assume that uD. Since A(u)BV(I;co(S1)co(S2)), it is not restrictive to suppose that the number of jumps of u from one circle to the other is one, i.e., |DA(u)|(I)=|v1v2|. Furthermore, by the same density argument exploited in Proposition 3.2 and the locality of the construction, we may assume that

    u(t)={a1if t[0,12],a2if t(12,1],

    where a1co(S1) and a2co(S2). Let {vjn}nNL(I;Sj) be the recovery sequence for the constant function aj obtained by the Γ-convergence result in Remark 3.11 with 2λn as the spacing of the lattice, i.e., vjnaj and

    fhom(aj)=limn+En(vjn)=limn+2λn12λn2i=0[α(vjn)i(vjn)i+1+(vjn)i(vjn)i+2]. (3.22)

    We define

    un(t)={v1n(2t)if t[0,12],v2n(2t1)if t(12,1].

    Remarking that, for all nN,

    A(un)(t)=A(u)(t)={v1if t[0,12],v2if t(12,1],

    we deduce that unDu. We compute

    En(un)=1212λn2i=02λn[α(v1n)i(v1n)i+1+(v1n)i(v1n)i+2]+1212λn2i=02λn[α(v2n)i(v2n)i+1+(v2n)i(v2n)i+2]+12λni=12λn1λn[αuinui+1n+uinui+2n]. (3.23)

    We observe that

    |12λni=12λn1λn[αuinui+1n+uinui+2n]|C(α)λn0, (3.24)

    as n+. By formulae (3.22), (3.23), (3.24), we obtain that

    limn+En(un)=fhom(a1)+fhom(a2)2=Ifhom(u(t))dt. (3.25)

    Since |DA(un)|(I)=|DA(u)|(I)=|v1v2| we get

    limn+Pn(un)=limn+λnkn|v1v2|=η|v1v2|. (3.26)

    Combining (3.25) and (3.26), we deduce the limsup inequality.

    Now we deal with case (ⅱ). Firstly, we prove the compactness result. Let {un}nNL(I;co(S1)co(S2)) be such that

    supnNEn(un)<¯C,

    for some constant ¯C>0. Thus we have that {un}nNPCλn. With the same compactness argument used in the previous case, we deduce the existence of uD such that unDu. In particular unu. By the lower semicontinuity of the total variation respect the weak convergence in BV(I;{v1,v2}), remarking that EnC(α), for some positive constant C(α), we get

    0=lim infn+¯Cλnknlim infn+1λnkn[En(un)+λnkn|DA(un)|(I)]lim infn+(C(α)λnkn+|DA(un)|(I))|DA(u)|(I),

    hence uL(I;co(S1))L(I;co(S2)).

    Let us prove the liminf inequality. Let {un}nNL(I;co(S1)co(S2)) be such that unuL(I;co(S1)co(S2)) and suppose that

    lim infn+En(un)<+.

    Up to the extraction of a subsequence, we may assume that the previous lower limit is actually a limit. By compactness, we infer that unuL(I;co(S1))L(I;co(S2)). Hence, by Theorem 3.10, we obtain

    lim infn+En(un)lim infn+En(un)Ifhom(u(t))dt.

    We finally prove the limsup inequality. Let uL(I;co(S1)), the case uL(I;co(S2)) being fully analogous. The recovery sequence obtained from Remark 3.11, {un}nNL(I;S1), satisfies the limsup inequality.

    In this subsection and in the following one we study the system when it is close to the helimagnet/ferromagnet transition point as the number of particles diverges. In what follows we let α=αn and we assume that αn4, as n+, and that n is sufficiently large so that Proposition 3.5 holds true.

    Once again, with a slight abuse of notation, we extend the energies En, Pn and En to the space L(I;R3), setting their value as + in L(I;R3)PCλn. Similarly, we extend A from L(I;co(S1)co(S2)) to L(I;R3).

    The main result of this subsection, Theorem 3.16, concerns the phenomenon of magnetic anisotropy transitions. Having in mind Proposition 3.5 and (3.9), we define the functional

    Gn:=EnminwPCλnEn(w)=Enλn#In(I)[R2(1+α2n8)+(αn1)(1R2)].

    At this point we need to introduce modified spin fields in order to understand better the asymptotic behaviour of the energy Gn. Let uPCλn and let Cn(u)={Ij|j{1,,M(u)}} be the open partition associated with u, with Ij=(tj,tj+1), for j{1,,M(u)1}, and IM(u)=(tM(u),1). We set tM(u)+1:=λn1λn. Since u is piecewise constant on the edges of the lattice [0,1]λnZ, we have that t1=0 and t2,,tM(u)+1 are multiples of λn, so that tjλnN, for any j{2,,M(u)+1}.

    We define the auxiliary spin ˜uIj:¯IjS1S2 by

    ˜uIj(t)={u(t)if t[tj,tj+1),wjif t=tj+1,

    and we set ˜uIM(u)(t)=wM(u) for t(tM(u)+1,1], where wjS1S2 is a vector such that the following boundary condition is satisfied in ¯Ij:

    utj+1λn1wj=utjλnutjλn+1. (3.27)

    We prove the following decomposition lemma.

    Lemma 3.13 (Decomposition of Gn). Let uPCλn and let Cn(u)={Ij|j{1,,M(u)}} be the open partition associated with u. We have

    Gn(u)=M(u)j=1MMn(˜uIj)+M(u)1j=1(Rn)j(u)+(Rn)M(u)(u)+Rn(u), (3.28)

    where, for all j{1,,M(u)},

    MMn(˜uIj):=λniIn(Ij)(αn˜uiIj˜ui+1Ij+˜uiIj˜ui+2Ij)+λnR2(1+α2n8)#In(Ij)+λn(αn1)(1R2)(#In(I)M(u)+1)M(u),

    and, for all j{1,,M(u)1},

    (Rn)j(u):=λn(αnutj+1λn1utj+1λn+utj+1λn1utj+1λn+1+utj+1λn2utj+1λnutj+1λn2wj),
    (Rn)M(u)(u):=λn(utM(u)+1λn2utM(u)+1λnutM(u)+1λn2wM(u)),
    Rn(u):=λnR2(1+α2n8)(M(u)1)+λn(αn1)(1R2)(M(u)1).

    Proof. Remarking that

    In(Ij)={tjλn,tjλn+1,,tj+1λn2},j{1,,M(u)},

    we may write

    Gn(u)+minwPCλnEn(w)=λnM(u)1j=1iIn(Ij)(αnuiui+1+uiui+2)+M(u)1j=1(αnutj+1λn1utj+1λn+utj+1λn1utj+1λn+1)+iIn(IM(u))(αnuiui+1+uiui+2).

    After adding and subtracting the terms utj+1λn2wj, for any j{1,,M(u)}, we interchange utj+1λn2wj and utj+1λn2utj+1λn in the first and the third sums, for any j{1,,M(u)}, obtaining

    Gn(u)+minwPCλnEn(w)=λn[M(u)j=1iIn(Ij)(αn˜uiIj˜ui+1Ij+˜uiIj˜ui+2Ij)+M(u)1j=1(αnutj+1λn1utj+1λn+utj+1λn1utj+1λn+1+utj+1λn2utj+1λnutj+1λn2wj)+(utM(u)+1λn2utM(u)+1λnutM(u)+1λn2wM(u))]=M(u)j=1MMn(˜uIj)+M(u)1j=1(Rn)j(u)+(Rn)M(u)(u)λnR2(1+α2n8)M(u)j=1#In(Ij)λn(αn1)(1R2)(#In(I)M(u)+1).

    We conclude the proof by computing

    minwPCλnEn(w)λnR2(1+α2n8)M(u)j=1#In(Ij)λn(αn1)(1R2)(#In(I)M(u)+1)=λnR2(1+α2n8)[#In(I)M(u)j=1#In(Ij)]+λn(αn1)(1R2)(M(u)1)=λnR2(1+α2n8)(M(u)1)+λn(αn1)(1R2)(M(u)1)=Rn(u),

    where we used

    Mj=1#In(Ij)=#In(I)M(u)+1. (3.29)

    Remark 3.14. In the decomposition (3.28) of Gn(u) the functional MMn(˜uIj) represents the energy of the j-th modified spin field ˜uIj, which is localized in one circle. The remainders for such modifications, (Rn)j(u) and (Rn)M(u)(u), consist of the interactions between spins with values in two neighboring intervals, Ij and Ij+1. Furthermore, they contain an additional term linked to the boundary condition (3.27). The term Rn(u) contains a corrective addend.

    Remark 3.15. Following the same computations done in (3.7), we infer that MMn(˜uIj)0, for all j{1,,M(u)} and uPCλn.

    The next theorem shows that the correct scaling of the energy to capture spin fields' magnetic anisotropy transitions is λn. To this end, for MN, we set

    RM:=inf{lim infn+1λn[M1j=1(Rn)j(un)+(Rn)M(un)+Rn(un)]|{un}nNPCλn such that A(un)BVvBV(I;{v1,v2}), with M=|Dv|(I)|v1v2|+1N}.

    Theorem 3.16 (First order Γ-convergence of En). Assume that there exists limn+λnkn=:η(0,+). Then the following compactness and Γ-convergence results hold true:

    (i) (Compactness) If for {un}nNL(I;R3) there exists a constant C>0 independent of n such that

    supnNGn(un)λnCandsupnNPn(un)C, (3.30)

    then, up to subsequences, A(un)BVvBV(I;{v1,v2}).

    (ii) (liminf inequality) For all vBV(I;{v1,v2}) and {un}nNPCλn such that A(un)BVv and (3.30) holds for some constant C>0, then

    lim infn+Gn(un)λnRM,

    where M=|Dv|(I)|v1v2|+1N.

    (iii) (limsup inequality) For all vBV(I;{v1,v2}) there exists {un}nNPCλn such that A(un)BVv, (3.30) holds for some constant C>0 and

    limn+Gn(un)λn=RM,

    where M=|Dv|(I)|v1v2|+1N.

    Proof. We start by proving (ⅰ). Let {un}nNL(I;R3) be such that (3.30) holds true. It follows that {un}nNPCλn. Since η(0,+), by the second inequality of formula (3.30), we deduce that the sequence {|DA(un)|(I)}nN is bounded and so the sequence {A(un)}nN is bounded in the space BV(I;{v1,v2}). Thus, up to a subsequence, {A(un)}nN converges to a function vBV(I;{v1,v2}) weakly in BV(I;{v1,v2}).

    We prove (ⅱ). Let vBV(I;{v1,v2}) and {un}nNPCλn be such that A(un)BVv and (3.30) holds. By assumption, {|DA(un)|(I)}nN is bounded. Let Cn(un)={(Ij)n|j{1,,M(un)}} be the open partition associated with un. Up to subsequences, we may assume that M=M(un) is independent of n. By Lemma 3.13, Remark 3.15 and the definition of RM we have

    lim infn+Gn(un)λnlim infn+Mj=1MMn(˜un(Ij)n)λn+lim infn+[M1j=1(Rn)j(un)λn+(Rn)M(un)+Rn(un)λn]lim infn+[M1j=1(Rn)j(un)λn+(Rn)M(un)+Rn(un)λn]RM.

    We finally prove (ⅲ). Let vBV(I;{v1,v2}). It is not restrictive to assume that v=v1χ[0,12]+v2χ(12,1] and thus we can choose uPCλn such that A(u)=v. By the definition of RM and by [12,Theorem 4.2], we gain the existence of {un}nN such that A(un)BVA(u), unχ[0,12]S1,unχ(12,1]S2 and the following formulae are satisfied:

    limn+1λn[M1j=1(Rn)j(un)+(Rn)M(un)+Rn(un)]=RM,
    (1αn4)32MMn(unχ[0,12])λn<C,(1αn4)32MMn(unχ(12,1])λn<C.

    Therefore

    limn+Gn(un)λn=RM.

    We let α=αn:=4(1δn), where {δn}nN is a positive vanishing sequence.

    At the second order we split the global functional on the 2-dimensional sphere into finitely many functionals localized in circles, where we repeat the analysis lead in [12]. For each circle S we define a convenient order parameter.

    Let uPCλn. According to the notation introduced in Subsection 3.6, for j{1,,M(u)} and i{tjλn,tjλn+1,,tj+1λn1}, we consider the pair (˜uiIj,˜ui+1Ij) of vectors that take values in S, for some =j{1,2}. We associate each pair with the corresponding oriented angle θiIj[π,π) with vertex in the center of the circle S given by

    θiIj:=χ[πv(˜uiIj),πv(˜ui+1Ij)]arccos(πv(˜uiIj)πv(˜ui+1Ij)).

    We set

    wiIj:=84αnsinθiIj2=2δnsinθiIj2

    and

    w(t)=wiIj{for} tλn{i+[0,1)},i{tjλn,,tj+1λn1},j{1,,M(u)}.

    We extend w(t)=wtj+1λn1IM(u), for t[tM(u)+1,1], so that w is well-defined in the whole interval ¯I. Note that we can define a map Tn by setting

    Tn(u):=(w,A(u)),uPCλn,

    and we denote ~PCλn:=Tn(PCλn). We observe that if h=Tn(u)=Tn(v) then u(t) and v(t) belong to the same circle, for any t¯I, and u,v differ by a constant rotation. Furthermore, Gn(u)=Gn(v) and Pn(u)=Pn(v). The same identity holds for the functionals defined in Lemma 3.13. Therefore, with a slight abuse of notation, we now set

    Gn(h):={Gn(u)if h~PCλn,+otherwise,
    Pn(h):={Pn(u)if h~PCλn,+otherwise,
    MMn(h|Ij):={MMn(˜uIj)if h~PCλn,+otherwise,
    (Rn)j(h):={(Rn)j(u)if h~PCλn,+otherwise,
    (Rn)M(u)(h):={(Rn)M(u)(u)if h~PCλn,+otherwise,
    Rn(h):={Rn(u)if h~PCλn,+otherwise,

    for j{1,,M(h)}, where hL1(I;R×{v1,v2}), h=Tn(u) and M(h):=M(u).

    We want to study the convergence of the functional

    Gn(h)={Gn(h)M(h)1j=1(Rn)j(h)+(Rn)M(h)(h)+Rn(h)if h~PCλn,+otherwise,=M(h)j=1MMn(˜h|Ij)

    for hL1(I;R×{v1,v2}). In order to establish the related result, we need a notion of convergence.

    Definition 3.17. Let {hn}nN~PCλn and hL1(I;R×{v1,v2}). We say that hn θ-converges to h (we write hnθh) if and only if the following conditions are satisfied:

    ● there exist {un}nNPCλn and a positive constant C such that if Cn(un)={(Ij)n|j{1,,M(un)}} is the open partition associated with un, then

    hn=Tn(un) and Pn(hn)<C,

    M(un)MN as n+,

    (Ij)nIj in the Hausdorff sense, as n+, for any j{1,,M}.

    hnχ(Ij)nhχIj in L1(I;R×{v1,v2}), for all j{1,,M}.

    We point out that the intervals Ij of the previous definition may be also empty.

    The next theorem shows that the correct scaling of the energy to capture spin fields' chirality transitions is 2λnδ32n.

    Theorem 3.18 (Second order Γ-convergence of En). Assume that there exist limn+λnkn=:η(0,+) and l:=limn+λn(2δn)12[0,+].

    Then the following statements are true:

    (i) (Compactness) If for {hn}nNL1(I;R×{v1,v2}) there exists a constant C>0 such that

    supnNGn(hn)2λnδ32nCandsupnNPn(hn)C, (3.31)

    then, up to a subsequence, hnθh, where

    if l=0, hBV(I;{1,1}×{v1,v2});

    if l(0,+), h|IjH1|per|(Ij;R×{v1,v2}) for all j{1,,M(h)};

    if l=+, h is piecewise constant with values in R×{v1,v2}.

    The space H1|per|((a,b);R×{v1,v2}) is equal to

    {hH1((a,b);R×{v1,v2}):|w(a)|=|w(b)|whereh=(w,A(u))}.

    (ii) (liminf inequality)

    If l=0, for all h=(w,A(u))BV(I;{1,1}×{v1,v2}) and for all {hn}nN~PCλn such that hnθh and (3.31) holds true for some constant C>0, then

    lim infn+Gn(hn)2λnδ32n43R2M(h)j=1|Dw|(Ij).

    If l(0,+), for all h=(w,A(u))L1(I;R×{v1,v2}) such that h|IjH1|per|(Ij;R×{v1,v2}), for every j{1,,M(h)}, and for all {hn}nN~PCλn such that hnθh and (3.31) holds true for some constant C>0, then

    lim infn+Gn(hn)2λnδ32nM(h)j=1R2[1lIj(w2(x)1)2dx+lIj(w(x))2dx].

    If l=+, for all piecewise constant functions h:IR×{v1,v2} and for all {hn}nN~PCλn such that hnθh, and (3.31) holds true for some constant C>0, then

    lim infn+Gn(hn)2λnδ32n0.

    (iii) (limsup inequality)

    If l=0, for all h=(w,A(u))BV(I;{1,1}×{v1,v2}) there exists {hn}nN~PCλn such that hnθh, (3.31) holds true for some constant C>0 and

    limn+Gn(hn)2λnδ32n=43R2Mj=1|Dw|(Ij).

    If l(0,+), for all h=(w,A(u))L1(I;R×{v1,v2}) such that h|IjH1|per|(Ij;R×{v1,v2}) for all j{1,,M(h)}, there exists {hn}nN~PCλn such that hnθh, (3.31) holds true for some constant C>0 and

    limn+Gn(hn)2λnδ32n=Mj=1R2[1lIj(w2(x)1)2dx+lIj(w(x))2dx].

    If l=+, for all piecewise constant functions h:IR×{v1,v2} there exists {hn}nN~PCλn such that hnθh, (3.31) holds true for some constant C>0 and

    limn+Gn(hn)2λnδ32n=0.

    Proof. We prove the statement only in the case l=0, the other cases being fully analogous. We start by proving (ⅰ). Let {hn}nNL1(I;R×{v1,v2}) be such that (3.31) holds true for some constant C>0. By formula and Remark 3.15, we infer that

    MMn(hn|Inj)λnδ32nC,for all j{1,,M(hn)} and nN.

    It is easy to see that, up to subsequences, M=M(hn) is independent of nN and the interval (Ij)nIj=(tj1,tj), in the Hausdorff sense, for every j{1,,M(hn)} (it may happen that Ij=, for some j). In the following computations we drop for simplicity the dependence on n writing Ij in place of (Ij)n.

    Reasoning as in Proposition 3.5, thanks to (3.29), we compute

    Gn(hn)=Mj=1λniIn(Ij)(αnπ˜uinIjπ˜ui+1nIj+π˜uinIjπ˜ui+2nIj)+λnR2(1+α2n8)Mj=1#In(Ij)+λn(αn1)(1R2)(#In(I)M+1)λn(αn1)(1R2)Mj=1#In(Ij)=Mj=1λniIn(Ij)(αnπ˜uinIjπ˜ui+1nIj+π˜uinIjπ˜ui+2nIj)+λnR2(1+α2n8)(#In(I)M+1),

    where we set π˜uinIj:=πv˜uinIj, with =j{1,2} such that ˜uinIjS.

    By the definition of ˜uinIj and geometric and trigonometric identities, we observe that

    R2π˜uinIjπ˜ui+1nIj=2R2sin2(θiIj2),
    R2π˜uinIjπ˜ui+2nIj=R2[1cos(θiIj+θi+1Ij)],

    where, for simplicity of notation, we have dropped the dependence on n of the angles θiIj. Taking into account the previous formulae, we gain

    Gn(hn)=λnMj=1iIn(Ij){αn[R2π˜uinIjπ˜ui+1nIj][R2π˜uinIjπ˜ui+2nIj]}+λnR2(1+α2n8)(#In(I)M+1)+λnR2(1αn)Mj=1#In(Ij)=λnR2Mj=1iIn(Ij){2αnsin2(θiIj2)[1cos(θiIj+θi+1Ij)]}+λnR2(2αn+α2n8)(#In(I)M+1). (3.32)

    The proof can be carried out as in [12,Theorem 4.2]. For reader's convenience we give here its sketch. By trigonometric identities, it holds

    8sin2(θiIj2)2sin2(θiIj)=8sin4(θiIj2).

    Moreover, taking into account the boundary condition (3.27), we can find a vanishing sequence {γn}nNR such that

    iIn(Ij)[2sin2(θiIj)1+cos(θiIj+θi+1Ij)]2(1γn)iIn(Ij)(sin(θi+1Ij2)sin(θiIj2))2.

    We insert the previous two formulae in (3.32) and compute

    Gn(hn)=λnR2Mj=1iIn(Ij){8sin2(θiIj2)(82αn)sin2(θiIj2)+2(1αn4)2}λnR2Mj=1iIn(Ij)[1cos(θiIj+θi+1Ij)]2λnR2(1αn4)2(#In(I)M+1)+λnR2(2αn+α2n8)(#In(I)M+1)=λnR2Mj=1iIn(Ij){8sin2(θiIj2)2sin2(θiIj)(82αn)sin2(θiIj2)+2(1αn4)2}+λnR2Mj=1iIn(Ij)[2sin2(θiIj)1+cos(θiIj+θi+1Ij)]=8λnR2Mj=1iIn(Ij)[sin2(θiIj2)12(1αn4)]2+λnR2Mj=1iIn(Ij)[2sin2(θiIj)1+cos(θiIj+θi+1Ij)]λnR2Mj=1iIn(Ij){8[sin2(θiIj2)12(1αn4)]2+2(1γn)[sin(θi+1Ij2)sin(θiIj2)]2}.

    Dividing by 2λnδ32n and recalling that αn=4(1δn), we infer that

    Gn(hn)2λnδ32nR2{2δ12nλniIn(Ij)λn[(winIj)21]2+λn2δ12n(1γn)iIn(Ij)(wi+1nIjwinIjλn)2}. (3.33)

    If ε>0 is sufficiently small such that Iεj:=(tj+ε,tj+1ε)(Ij)n, for all nN, then

    MMn(wn|Iεj)λnδ32nC

    and (3.33) holds with Iεj in place of Ij, for any j{1,,M}. Therefore, applying [12,Theorem 2.2 and Remark 2.3] (see also [8]), {wnχIεj}nN converges, up to subsequences, to wBV(Ij) in L1. Thus we deduce the existence of hBV(I;{1,1}×{v1,v2}) such that hn:=(wn,A(un))θh.

    Now we prove (ⅱ). Let h=(w,A(u))BV(I;{1,1}×{v1,v2}) and {hn}nN~PCλn be such that hnθh and (3.31) holds true for some constant C>0. Up to a subsequence, M=M(hn) is independent of n. Moreover, denoting Ij=(tj,tj+1), for ε>0 sufficiently small, it holds that Iεj:=(tj+ε,tj+1ε)(Ij)n, for all j{1,,M(hn)} and nN. By the definition of Gn, we have

    lim infn+Gn(hn)2λnδ32n=lim infn+Mj=1MMn(hn|Ij)2λnδ32n43R2Mj=1|Dw|(Iεj),

    where in the last step we have used the liminf inequality of [12,Theorem 4.2]. Letting ε0, we obtain the liminf inequality.

    We finally prove (ⅲ). Let h=(w,A(u))BV(I;{1,1}×{v1,v2}). We can find M>0 and an open partition of I made by the intervals C={Ij}j{1,,M} such that h|Ij=(w|Ij,¯vj)BV(Ij;{1,1})×{v1,v2}). Thanks to the limsup inequality proved in [12,Theorem 4.2], for all j{1,,M} there exists a sequence {(zj)n}nNL1(Ij;R), such that (zj)nw|Ij in L1(Ij;R) and

    limn+MMn(hn|Ij)2λnδ32n=43R2|Dw|(Ij), (3.34)

    where hn|Ij:=((zj)n,¯vj). By the definition of Gn and (3.34) we gain

    limn+Gn(hn)2λnδ32n=limn+Mj=1MMn(hn|Ij)2λnδ32n=43R2Mj=1|Dw|(Ij),

    that is the thesis.

    In this section we analyze the problem in the two-dimensional case. Therefore we need to introduce proper notation and new definitions.

    Let {λn}nNR+ be a vanishing sequence of positive lattice spacings. Given i,jZ, we denote by Qλn(i,j):=(λni,λnj)+[0,λn)2 the half-open square with left-bottom corner in (λni,λnj). For a given set S, we introduce the class of spin fields with values in S which are piecewise constant on the squares of the lattice λnZ2:

    PCλn(R2;S):={u:R2S:u(x)=u(λni,λnj) for xQλn(i,j)}.

    We will identify a function uPCλn(R2;S) with the function defined on the points of the lattice λnZ2 given by (i,j)ui,j:=u(λni,λnj), for i,jZ. Conversely, given values ui,jS for i,jZ, we define uPCλn(R2;S) by setting u(x):=ui,j, for xQλn(i,j).

    Furthermore, we define the projection function A:PCλn(R2;S1S2)L(R2;{v1,v2}) by setting

    A(u)(x)={v1if u(x)S1,v2if u(x)S2,xR2.

    In this paper we will make use of the notion of BVG regularity. BVG domains and BVG functions have been introduced in [21] (see also [9,Section 3]).

    Definition 4.1. Let IR be an open set. We define the space of BVG functions by

    BVG(I):={ϕW1,(I):ϕBV(I)}.

    A bounded connected open set ΩR2 is called a BVG domain if Ω can be described locally at its boundary as the epigraph of a BVG function with respect to a suitable choice of the axes, i.e., if for every xΩ there exist a neighborhood UxR2, a function ψxBVG(R) and an isometry Rx:R2R2 satisfying

    Rx(ΩUx)={(y1,y2)R2:y1>ψx(y2)}Rx(Ux).

    We remark that smooth domains and polygons are BVG domains and BVG domains are Lipschitz domains.

    As in the one-dimensional case we observe that, if uPCλn(R2;S1S2), then a bounded connected open set ΩR2 can be uniquely partitioned in regions where the spin field u takes values only in one of the two circles. In other words, there exist M(u)N and a collection of connected open sets, {Cs}s{1,,M(u)}, such that

    {Cs}s{1,,M(u)} is an open partition of Ω, (4.1)
    either u(Cs)S1 or u(Cs)S2, for any s{1,,M(u)}, (4.2)
    if u(Cs1)×u(Cs2)S×S, for some s1,s2{1,,M(u)} and {1,2}, (4.3)
    then ¯Cs1¯Cs2 has at most a finite number of points.

    The last two properties imply that this partition is unique. We remark that the sets Cs are squares or union of squares. In particular, (4.3) ensures that u maps two confining sets of the open partition in different circles, if their intersection contain edges of squares.

    The following definition will be useful throughout the section.

    Definition 4.2. Let uPCλn(R2;S1S2) and ΩR2 be a bounded connected open set. We say that Cn(u)={Cs|s{1,,M(u)}} is the open partition of Ω associated with u if M(u)N and the collection {Cs}s{1,,M(u)} of open connected sets satisfies (4.1), (4.2) and (4.3). If Ω is a BVG domain, we call Cn(u) the open BVG partition of Ω associated with u if Cs is also a BVG domain, for all s{1,,M(u)}.

    Our model is an energy on discrete spin fields defined on square lattices inside a given domain ΩR2 belonging to the following class:

    A0:={ΩR2:Ω is a {simply connected BVG domain}}.

    To define the energies in our model, we introduce the set of indices

    In(Ω):={(i,j)Z2:¯Qλn(i,j),¯Qλn(i+1,j),¯Qλn(i,j+1)Ω},

    for ΩA0. Let αn:=4(1δn), where {δn}R+ is a vanishing sequence, and let {kn}nNR+ be a divergent sequence. In the following we shall assume that εn:=λnδn0 and λnknη(0,+), as n+.

    We consider the functionals Hn,Pn:L(R2;S1S2)×A0[0,+] defined by

    Hn(u;Ω):=12λnδ32n12λ2n(i,j)In(Ω)[|ui+2,jαn2ui+1,j+ui,j|2+|ui,j+2αn2ui,j+1+ui,j|2],
    Pn(u;Ω):=λnkn|DA(u)|(Ω),

    for uPCλn(R2;S1S2) and extended to + elsewhere.

    Similarly to the analysis at the first and second order in the one-dimensional case, we split the functional Hn as follows:

    Hn(u;Ω)=M(u)s=1[Hn(u;Cs)+(Rn)Cs(u)],

    where

    Hn(u;Cs):=Hn(u;Cs)+12λnδ32n2λ2n(αn1)(1R2)#In(Cs),
    (Rn)Cs(u):=12λnδ32n12λ2n(i,j)(CsIn(Ω))In(Cs)[|ui+2,jαn2ui+1,j+ui,j|2+|ui,j+2αn2ui,j+1+ui,j|2]12λnδ32n2λ2n(αn1)(1R2)#In(Cs),

    for any s{1,,M(u)}. The functionals (Rn)Cs collect the remainders associated with the decomposition of the energy in the open partition Cn(u)={Cs|s{1,,M(u)}}. They consist of the interactions between spin field's vectors located in different circles.

    In this subsection we introduce the chirality order parameter associated with a spin field. Let uPCλn(R2;S1S2) and let Cn(u)={Cs|s{1,,M(u)}} be the partition associated with u. For (i,j)In(Cs), we consider the pairs (ui,j,ui+1,j) and (ui,j,ui,j+1) of vectors that take values in S, for some =s{1,2}. We define the horizontal and vertical oriented angles between two adjacent spin vectors by

    ˜θi,jCs:=χ[πv(ui,j),πv(ui+1,j)]arccos(πv(ui,j)πv(ui+1,j))[π,π),
    ˇθi,jCs:=χ[πv(ui,j),πv(ui,j+1)]arccos(πv(ui,j)πv(ui,j+1))[π,π).

    We define the order parameter ((w,z),A(u))PCλn(R2;R2)×L(Ω;{v1,v2}) (we will write (w,z,A(u)) for simplicity) by setting

    wi,j:={2δnsin˜θi,jCs2if (i,j)In(Cs) for some s{1,,M(u)},0otherwise,
    zi,j:={2δnsinˇθi,jCs2if (i,j)In(Cs) for some s{1,,M(u)},0otherwise.

    It is convenient to introduce the transformation Tn:PCλn(R2;S1S2)PCλn(R2;R2)×L(Ω;{v1,v2}) given by

    Tn(u):=(w,z,A(u)).

    With a slight abuse of notation we define the functional Hn:L1loc(R2;R2×{v1,v2})×A0[0,+) by setting

    Hn(h;Ω)={Hn(u;Ω)if Tn(u)=h for some uPCλn(R2;S1S2),+otherwise. (4.4)

    Notice that Hn does not depend on the particular choice of u, since it is rotation-invariant. The same notation can be adopted for Pn, (Rn)Cs and Hn.

    We study the convergence of the functional

    Gn(h;Ω):={Hn(h,Ω)M(h)s=1(Rn)Cs(h)if Tn(u)=h for some uPCλn(R2;S1S2),+otherwise=M(h)s=1Hn(h;Cs).

    where M(h):=M(u). Hence, we introduce the functional G:L1loc(R2;R2×{v1,v2})×A0[0,+) by setting

    G(h;Ω):={43R2M(h)s=1(|D1w|(Cs)+|D2z|(Cs))if h=(w,z,α)Dom(G;Ω),+otherwise,

    where

    Dom(G;Ω):={(w,z,α)L1loc(R2;R2×{v1,v2}):{Cs}s{1,,M} open partition of Ω s.t. (w|Cs,z|Cs,α|Cs)BV(Cs;{1,1}2×{vs}),for some s{1,2},curl(w|Cs,z|Cs)=0 in D(Cs;R2)}.

    For hDom(G;Ω) we say that the collection {Cs}s{1,,M} existing in virtue of the definition of Dom(G;Ω) is the open partition associated with h.

    We have denoted by D(Cs;R2) the space of distributions and by curl the distribution curl defined by

    (curl(T))h,k,ξ:=Tk,hξ+Th,kξ,ξCc(Cs),TD(Cs;R2),

    for any h,k{1,2}.

    The following notion of convergence will be used.

    Definition 4.3. Let {hn}nNL1loc(R2;R2×{v1,v2}). We say that hn Θ-converges to hL1loc(R2;R2×{v1,v2}) (we write hnΘh) if the following conditions are satisfied:

    ● there exist {un}nNPCλn(R2;S1S2), a positive constant C such that

    hn=Tn(un) and Pn(un;Ω)<C,

    M(un)MN as n+,

    (Cs)nCs in the Hausdorff sense, as n+, for any s{1,,M},

    where Cn(un)={(Cs)n|s{1,,M(hn)}} is the open partition associated with un.

    hnχ(Cs)nhχCs in L1loc(R2;R2×{v1,v2}), for any s{1,,M}.

    As in formula (4.4) we define Pn(h;Ω):=Pn(u;Ω) for h=Tn(u) with uPCλn(R2;S1S2).

    We remark that in general it is not possible to prove a compactness result for a sequence {hn=Tn(un)}nNTn(PCλn(R2;S1S2)) satisfying only the following natural conditions:

    supnNGn(hn;Ω)<CandsupnNPn(hn;Ω)<C.

    Indeed, it could happen that the region {A(un)=v1} has an increasing number of holes vanishing in the limit so that {M(un)}nN is divergent. Neither the Hausdorff convergence of the sets of the open partition is ensured.

    In the following proposition we show that, if strong and technical conditions hold, then {hn}nN converges, up to subsequences, with respect to the Θ-convergence.

    Proposition 4.4. Let {hn=Tn(un)}nNTn(PCλn(R2;S1S2)) be a sequence such that

    supnNGn(hn;Ω)<CandsupnNPn(hn;Ω)<C, (4.5)

    for some constant C>0. Furthermore, we assume that the open partition associated with un, Cn(un)={(Cs)n|s{1,,M(un)}}, is such that

    M(un)MNasn+,
    (Cs)nCsintheHausdorffsense,asn+,s{1,,M}.

    Then there exists hDom(G;Ω) such that, up to a subsequence, hnΘh.

    Proof. Let {hn=(wn,zn,A(un))}nNTn(PCλn(R2;S1S2)) be a sequence satisfying (4.5). Since un|CsS, for some =s{1,2}, then, by geometric and trigonometric identities, we deduce that

    ui,jui+1,j=1R2+πui,jπui+1,j,
    ui,jui,j+1=1R2+πui,jπui,j+1,

    where πui,j:=πvui. Thus we may write

    Gn(hn;Ω)=Ms=1˜Hn(un;Cs),

    where

    ˜Hn(hn;Cs):=12λnδ32n12λ2n(i,j)In(Cs)[|πui+2,jnαn2πui+1,jn+πui,jn|2+|πui,j+2nαn2πui,j+1n+πui,jn|2].

    Fixing ε>0 sufficiently small, we have that for all nN, up to a subsequence, (Cs)ε:={xCs:dist(x,Cs)>ε}(Cs)n and un|(Cs)ε takes values only in one circle. We infer that

    Ms=1˜Hn(hn;(Cs)ε)Gn(hn;Ω)<C,

    which of course implies that ˜Hn(hn;(Cs)ε)<C, for all s{1,,M}. We are in position to apply [9,Theorem 2.1 ⅰ) and Remark 2.2] to deduce the existence of (w(Cs)ε,z(Cs)ε)BV((Cs)ε;{1,1}2) such that, up to subsequences, (wn,zn)(w(Cs)ε,z(Cs)ε) in L1loc((Cs)ε;R2) and curl(w(Cs)ε,z(Cs)ε)=0 in D((Cs)ε;R2). The couples (w(Cs)ε,z(Cs)ε) can be extended to 0 in Cs(Cs)ε. We preliminary observe that

    (w(Cs)ε2,z(Cs)ε2)=(w(Cs)ε1,z(Cs)ε1)a.e. on (Cs)ε2, (4.6)

    for any 0<ε1<ε2. Indeed, since (Cs)ε2(Cs)ε1, we have that

    (wn,zn)(w(Cs)ε1,z(Cs)ε1)inL1loc((Cs)ε2;R2).

    The uniqueness of the limit in the L1loc-topology implies (4.6). We now define the couples (wCs,zCs):CsR2 by

    (wCs,zCs):=limε0+(w(Cs)ε,z(Cs)ε).

    The definition is well-posed; indeed, since by (4.6),

    limε0+(w(Cs)ε,z(Cs)ε)=(w(Cs)1n,z(Cs)1n)a.e. in (Cs)1n,

    for all nN, then

    |{xCs:limε0+(w(Cs)ε(x),z(Cs)ε(x))}|=|+n=1{x(Cs)1n:limε0+(w(Cs)ε(x),z(Cs)ε(x))}|=0.

    Furthermore we define (w,z):ΩR2 by setting

    (w,z)(x)=(wCs,zCs)(x),

    for a.e. xΩ with xCs, for some s{1,,M}. Of course (w|Cs,z|Cs)=(wCs,zCs)BV(Cs;{1,1}2), as it is the limit of BV functions. In order to show the L1loc-convergence, we fix A⊂⊂Cs. Since dist(A,Cs)>0, there exists ε>0 such that A⊂⊂(Cs)ε. We obtain:

    (wn,zn)(wCs,zCs)L1(A;R2)=(wn,zn)(w(Cs)ε,z(Cs)ε)L1(A;R2),

    which vanishes as n+, up to subsequences. This leads to the convergence

    (wn,zn)(wCs,zCs)in L1loc(Cs;R2).

    Finally, we prove that curl(wCs,zCs)=0 in D(Cs;R2). If ξCc(Cs), then suppξ(Cs)ε for some ε>0 and so

    curl(wCs,zCs),ξ=(Cs)εw(Cs)ε2ξdx+(Cs)εz(Cs)ε1ξdx=curl(w(Cs)ε,z(Cs)ε),ξ=0.

    Now we state the main theorem of this section. The regularity assumption on Ω and on the open partition of h in the statement ⅱ) are required in order to apply [9,Theorem 2.1 ⅲ)] locally. As explained in [9] a simply connected BVG domain guaranties an extension property for BVG functions, which is needed to construct a recovery sequence for h. On the contrary, the proof of the liminf inequality ⅰ) actually works without assuming this kind of regularity (see [9,Remark 2.2]).

    Theorem 4.5. Let ΩA0. Then the following statements hold true:

    i) (liminf inequality) Let {hn}nNL1loc(R2;R2×{v1,v2}) and hL1loc(R2;R2×{v1,v2}). Assume that supnNPn(hn;Ω)<C for some constant C>0 and hnΘh. Then

    G(h;Ω)lim infn+Gn(hn;Ω).

    ii) (limsup inequality) Let hDom(G;Ω) be such that its open partition consists of BVG domains. Then there exists a sequence {hn}nNL1loc(R2;R2×{v1,v2}) such that hnΘh and

    lim supn+Gn(hn;Ω)G(h;Ω).

    Proof. We start by proving ⅰ). Let {hn}nNL1loc(R2;R2×{v1,v2}) and hL1loc(R2;R2×{v1,v2}) be such that supnNPn(hn;Ω)<C and hnΘh. Up to subsequences, we may assume that the lower limit in the right hand side of the liminf inequality is actually a limit. Furthermore we may assume that it is finite, the inequality being otherwise trivial. In particular, we have

    supnNGn(hn;Ω)<C,

    possibly with a larger C. By the definition of Θ-convergence, hn=(wn,zn,A(un))=Tn(un) for some unPCλn(R2;S1S2). Up to subsequences, M=M(hn) is independent of n and we may assume, for ε>0 sufficiently small, that (Cs)ε(Cs)n and un|(Cs)ε takes values only on one circle S, for all nN. Reasoning as in ⅰ), we infer

    Gn(hn;Ω)Ms=1˜Hn(hn;(Cs)ε),

    Since hnh in L1((Cs)ε;R2×{v}), as n+, we are in position to apply [9,Theorem 2.1 ⅱ) and Remark 2.2] so that, passing to the lower limit, we get

    lim infn+Gn(hn;Ω)Ms=1lim infn+˜Hn(hn;(Cs)ε)Ms=143R2[|D1w|((Cs)ε)+|D2z|((Cs)ε)],

    where h=(w,z,α). Letting ε0+ we get the thesis.

    Let us prove ⅱ). Let hDom(G;Ω). This implies that h=(w,z,α)L1loc(R2;R2×{v1,v2}) and the existence of an open partition of Ω, C={Cs|s{1,,M}} consisting of BVG domains such that, for some =s{1,2},

    (w|Cs,z|Cs,α|Cs)BV(Cs;{1,1}2×{v})andcurl(w|Cs,z|Cs)=0 in D(Cs;R2).

    Applying [9,Theorem 2.1 ⅲ)] to any (w|Cs,z|Cs), we get the existence of a sequence {((wn)Cs,(zn)Cs)}nNL1loc(R2;R2) such that ((wn)Cs,(zn)Cs)(w|Cs,z|Cs) in L1(Cs;R2) and

    lim supn+Hn((wn)Cs,(zn)Cs,v)43R2(|D1w|(Cs)+|D2z|(Cs))

    Defining (wn,zn,αn):R2R2×{v1,v2} by

    (wn,zn,αn)(x):=((wn)Cs(x),(zn)Cs(x),v),

    if xΩ such that xCs for some s{1,,M}, and arbitrarily extended outside Ω, and summing on s{1,,M} the previous inequality we obtain the thesis.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors wish to thank the reviewer for numerous suggestions that improved the paper. The authors warmly thank Prof. Marco Cicalese for the insightful discussions. L. Lamberti wishes to acknowledge the hospitality of the Faculty of Mathematics of the Technical University of Munich, where part of this research was carried out. The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). A. Kubin was supported by the DFG Collaborative Research Center TRR 109 "Discretization in Geometry and Dynamics". L. Lamberti was supported partially by the DFG Collaborative Research Center TRR 109 "Discretization in Geometry and Dynamics" and by COST Action 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology).

    The authors declare no conflict of interest.



    [1] R. Alicandro, M. Cicalese, Variational analysis of the asymptotics of the XY model, Arch. Rational Mech. Anal., 192 (2009), 501–536. https://doi.org/10.1007/s00205-008-0146-0 doi: 10.1007/s00205-008-0146-0
    [2] R. Alicandro, M. Cicalese, A. Gloria, Variational description of bulk energies for bounded and unbounded spin systems, Nonlinearity, 21 (2008), 1881–1910. http://doi.org/10.1088/0951-7715/21/8/008 doi: 10.1088/0951-7715/21/8/008
    [3] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, The Clarendon Press, 2000.
    [4] A. Bach, M. Cicalese, L. Kreutz, G. Orlando, The antiferromagnetic XY model on the triangular lattice: chirality transitions at the surface scaling, Calc. Var., 60 (2021), 149. https://doi.org/10.1007/s00526-021-02016-3 doi: 10.1007/s00526-021-02016-3
    [5] R. Badal, M. Cicalese, L. De Luca, M. Ponsiglione, Γ-convergence analysis of a generalized XY model: fractional vortices and string defects, Commun. Math. Phys., 358 (2018), 705–739. https://doi.org/10.1007/s00220-017-3026-3 doi: 10.1007/s00220-017-3026-3
    [6] A. Braides, Γ-convergence for beginners, Oxford: Oxford University Press, 2002. https://doi.org/10.1093/acprof: oso/9780198507840.001.0001
    [7] A. Braides, L. Truskinovsky, Asymptotic expansions by Γ-convergence, Continuum Mech. Thermodyn., 20 (2008), 21–62. https://doi.org/10.1007/s00161-008-0072-2 doi: 10.1007/s00161-008-0072-2
    [8] A. Braides, N. K. Yip, A quantitative description of mesh dependence for the discretization of singularly perturbed nonconvex problems, SIAM J. Numer. Anal., 50 (2012), 1883–1898. https://doi.org/10.1137/110822001 doi: 10.1137/110822001
    [9] M. Cicalese, M. Forster, G. Orlando, Variational analysis of a two-dimensional frustrated spin system: emergence and rigidity of chirality transitions, SIAM J. Math. Anal., 51 (2019), 4848–4893. https://doi.org/10.1137/19M1257305 doi: 10.1137/19M1257305
    [10] M. Cicalese, G. Orlando, M. Ruf, Emergence of concentration effects in the variational analysis of the N-clock model, Commun. Pure Appl. Anal., 75 (2019), 2279–2342. https://doi.org/10.1002/cpa.22033 doi: 10.1002/cpa.22033
    [11] M. Cicalese, M. Ruf, F. Solombrino, Chirality transitions in frustrated S2-valued spin systems, Math. Mod. Meth. Appl. Sci., 26 (2016), 1481–1529. https://doi.org/10.1142/S0218202516500366 doi: 10.1142/S0218202516500366
    [12] M. Cicalese, F. Solombrino, Frustrated ferromagnetic spin chains: a variational approach to chirality transitions, J. Nonlinear Sci., 25 (2015), 291–313. https://doi.org/10.1007/s00332-015-9230-4 doi: 10.1007/s00332-015-9230-4
    [13] H. T. Diep, Frustrated spin systems, World Scientific, 2005. https://doi.org/10.1142/5697
    [14] R. S. Dissanayaka Mudiyanselage, H. Wang, O. Vilella, M. Mourigal, G. Kotliar, W. Xie, LiYbSe2: Frustrated Magnetism in the Pyrochlore Lattice, J. Am. Chem. Soc., 144 (2022), 11933–11937. https://doi.org/10.1021/jacs.2c02839 doi: 10.1021/jacs.2c02839
    [15] D. V. Dmitriev, V. Ya Krivnov, Universal low-temperature properties of frustrated classical spin chain near the ferromagnet-helimagnet transition point, Eur. Phys. J. B, 82 (2011), 123–131. https://doi.org/10.1140/epjb/e2011-10664-6 doi: 10.1140/epjb/e2011-10664-6
    [16] S. L. Drechsler, O. Volkova, A. N. Vasiliev, N. Tristan, J. Richter, M. Schmitt, et al., Frustrated cuprate route from antiferromagnetic to ferromagnetic spin-12 Heisenberg chains: Li2ZrCuO4 as a missing link near the quantum critical point, Phys. Rev. Lett., 98 (2007), 077202. https://doi.org/10.1103/PhysRevLett.98.077202 doi: 10.1103/PhysRevLett.98.077202
    [17] M. J. P. Gingras, P. A. McClarty, Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets, Rep. Prog. Phys., 77 (2014), 056501. https://doi.org/10.1088/0034-4885/77/5/056501 doi: 10.1088/0034-4885/77/5/056501
    [18] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123–142. https://doi.org/10.1007/BF00251230 doi: 10.1007/BF00251230
    [19] L. Modica, S. Mortola, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici, (Italian), Boll. Un. Mat. Ital. A (5), 14 (1977), 526–529.
    [20] D. G. Nocera, B. M. Bartlett, D. Grohol, D. Papoutsakis, M. P. Shores, Spin frustration in 2D kagomé lattices: a problem for inorganic synthetic chemistry, Chem.-Eur. J., 10 (2004), 3850–3859. https://doi.org/10.1002/chem.200306074 doi: 10.1002/chem.200306074
    [21] A. Poliakovsky, Upper bounds for singular perturbation problems involving gradient fields, J. Eur. Math. Soc., 9 (2007), 1–43. http://doi.org/10.4171/JEMS/70 doi: 10.4171/JEMS/70
    [22] K. Yu. Povarov, L. Facheris, S. Velja, D. Blosser, Z. Yan, S. Gvasaliya, et al., Magnetization plateaux cascade in the frustrated quantum antiferromagnet Cs2CoBr4, Phys. Rev. Res., 2 (2020), 043384. https://doi.org/10.1103/PhysRevResearch.2.043384 doi: 10.1103/PhysRevResearch.2.043384
    [23] R. Skomski, Simple models of magnetism, Oxford University Press, 2008. https://doi.org/10.1093/acprof: oso/9780198570752.001.0001
    [24] R. Szymczak, P. Aleshkevych, C. P. Adams, S. N. Barilo, A. J. Berlinsky, J. P. Clancy, et al., Magnetic anisotropy in geometrically frustrated kagomé staircase lattices, J. Magn. Magn. Mater., 321 (2009), 793–795. https://doi.org/10.1016/j.jmmm.2008.11.076 doi: 10.1016/j.jmmm.2008.11.076
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