The antiferromagnetic XY model on the triangular lattice: chirality transitions at the surface scaling

We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice. The system is fully frustrated and displays two families of ground states distinguished by the chirality of the spin field. We compute the {\Gamma}-limit of the energy in a regime which detects chirality transitions on one-dimensional interfaces between the two admissible chirality phases.


Introduction
Ordering problems in magnetism have been extensively studied by both the physics and the mathematics communities. Researchers have been attracted by the rich phase diagrams and critical behaviors of magnetic models which are often the result of difficult-to-detect optimization effects taking place at several energy and length scales. The reason for such a complex behavior can be traced back to the presence of many competing mechanisms which give rise to frustration. Frustration in the context of spin systems (here, as it is customary in the statistical mechanics literature, we will often refer to magnets as to spins) refers to the situation where spins cannot find an orientation that simultaneously minimizes all the pairwise exchange interactions. Such interactions are said to be ferromagnetic or antiferromagnetic if they favour alignment or antialignment, respectively. Often frustration occurs in those systems where spins are subject to conflicting short range ferromagnetic and long range antiferromagnetic interactions, as when modulated phases appear (see, e.g., the expository paper [38]). For antiferromagnetic lattice systems, that is systems of lattice spins subject to only antiferromagnetic interactions, frustration can also stem from the relative spatial arrangement of spins induced by the geometry of the lattice. In this case frustration is often referred to as geometric frustration. As a consequence of geometric frustration magnetic compounds show complex geometric patterns that induce often unexpected effects whose understanding is one of the primary subjects in statistical and condensed matter physics as it can help to better explain the nature of phase transitions in magnetic materials [28,33,34]. From a mathematical perspective, several interesting questions can be addressed. In this paper we are interested in the variational coarse graining of the system, in the line of what is by now addressed to as the "discrete-to-continuum variational analysis of discrete systems". Within this line of investigation the analysis of spin systems turns out to be a difficult nonlinear optimization problem requiring the combination of several methods ranging from simple discrete optimization procedures to sophisticated techniques in geometric measure theory and the calculus of variations. While models where frustration is induced by the competition of ferromagnetic/antiferromagnetic interactions have been already studied from a variational perspective (see, e.g., [1,30,31,32,22,10,37,18,26]), what we present here is the first discrete-to-continuum result for a geometrically frustrated system.
We carry out the discrete-to-continuum variational analysis (at zero temperature) of a geometrically frustrated spin model in a specific energetic regime and we characterize the effective behavior of its low-energy states, that is states that can deviate from the global minimizers (ground states) by a certain small amount of energy. More precisely we consider a 2-dimensional nearest-neighbors antiferromagnetic planar spin model on the triangular lattice, cf. [28,Chapter 1]. Despite being considered one of the most elementary geometrically frustrated spin models, its variational analysis turns out to be quite a delicate task. More in detail, we let ε > 0 be a small parameter and we consider the triangular lattice L ε with spacing ε (see Subsection 2.2 for the precise definition). To every spin field u : L ε → S 1 we associate the energy εσ,εσ ∈Lε |σ−σ |=1 u(εσ), u(εσ ) , (1.1) where ·, · denotes the scalar product. (Below, the energy will be restricted to bounded regions in the plane.) This model is antiferromagnetic since the interaction energy between two neighboring spins is minimized by two opposite vectors. Such an order in the magnetic alignment, also known as antiferromagnetic order, is frustrated by the geometry of the triangular lattice, which inhibits a configuration where each pair of neighboring spins are opposite or, equivalently, where each interaction is minimized. This suggests that the antiferromagnetic XY model depends substantially on the geometry of the lattice, which affects the structure of the ground states, the choice of the relevant variables and of the energy scalings. Notice, for example, that on a square lattice the system would not be frustrated, as opposite vectors distributed in a checkerboard structure minimize each interaction. In fact, on the square lattice a straightforward change of variable allows one to recast the antiferromagnetic XY model into the ferromagnetic XY model [2, Remark 3], which is driven by an energy with neighboring interactions − u(εσ), u(εσ ) . The latter model has been thoroughly investigated in the last decade both on the square lattice [2,3,5,19,20] and on the triangular lattice [16,27]. Independently of the geometry of the lattice, it has been proved that spin fields that deviate from the ground states by an amount of energy which diverges logarithmically as ε vanishes form of topological charges (vortex-like singularities of the spin field as those arising in the Ginzburg-Landau model [9,36]), when subject to boundary conditions or external magnetic fields. We now come back to our model (1.1). In order to identify the relevant variable of the system, we first need to characterize the ground states of the antiferromagnetic XY system in (1.1). To this end it is convenient to rearrange the indices of the sum in (1.1) and to recast the energy as a sum over all triangular plaquettes T with vertices εi, εj, εk ∈ L ε T u(εi), u(εj) + u(εj), u(εk) + u(εk), u(εi) = 1 2 T |u(εi) + u(εj) + u(εk)| 2 − 3 . (1. 2) In each triangle T the energy is minimized (and is equal to − 3 2 ) if and only if u(εi) + u(εj) + u(εk) = 0, namely, when the vectors of a triple (u(εi), u(εj), u(εk)) point at the vertices of an equilateral triangle. By the S 1 -symmetry, every rotation of a minimizing triple (u(εi), u(εj), u(εk)) is minimizing, too. The ground states in this model feature an additional symmetry, usually referred to as Z 2 -symmetry: triple obtained by from a minimizing triple via a permutation of negative sign as (u(εi), u(εk), u(εj)) is also minimizing. This determines two families of ground states, i.e., spin fields for which the energy is minimized in each plaquette, see Figure 1. These two families can be distinguished through the chirality, a scalar which quantifies the handedness of a certain spin structure. To define the chirality of a spin field u in a triangle T , we need a consistent ordering of its vertices εi, εj, εk. We assume that εi ∈ L 1 ε , εj ∈ L 2 ε , εk ∈ L 3 ε , where L 1 ε , L 2 ε , L 3 ε are the sublattices as in Figure 1, and we set (see (2.1) for the precise definition) where the symbol × stems for the cross product. We denote by χ(u) ∈ L ∞ (R 2 ) the function equal to χ(u, T ) on the interior of each plaquette T . The ground states are exactly those configurations u that satisfy either χ(u) ≡ 1 or Figure 1. A ground state u pos ε with positive chirality and a ground state u neg ε with negative chirality. Any other ground state of the system is obtained by composing one of these two configurations with a constant rotation. In the center: three points of the sublattices L 1 ε , L 2 ε , and L 3 ε in black, gray, and white, respectively.
In this paper we analyze the energy regime at which the two families of ground states coexist and at the same time the energy of the system concentrates at the interface between the two chiral phases {χ = 1} and {χ = −1}. We fix Ω ⊂ R 2 open, bounded, and with Lipschitz boundary and we consider the energy (1.2) restricted to Ω, i.e., computed only on plaquettes of L ε contained in Ω. We refer the energy to its minimum by removing the energy of the ground states (− 3 2 for each plaquette) and we divide it by the number of lattice points in Ω (of order 1/ε 2 ). We obtain (up to a multiplicative constant) the energy per particle given by We are interested to the asymptotic behavior of the energy above as ε → 0 on sequences of spin fields u ε : L ε → S 1 that can deviate from ground states yet satisfying a bound E ε (u ε ) ≤ Cε. To this end we define the energy F ε (u) := 1 ε E ε (u) and study sequences of spin fields with equibounded F ε energy. Due to the S 1 -symmetry, the energy at this regime cannot distinguish ground states with the same chirality, so that the relevant order parameter of the model is, in fact, not the spin field but its chirality: in Proposition 3.1 we prove that a sequence (u ε ) satisfying F ε (u ε ) ≤ C admits a subsequence (not relabeled) such that χ(u ε ) → χ strongly in L 1 (Ω) for some χ ∈ BV (Ω; {−1, 1}), i.e., the admissible chiralities in the continuum limit are −1 and 1 and the chirality phases {χ = −1} and {χ = 1} have finite perimeter in Ω. This suggests that the model shares similarities with systems having finitely many phases, such as Ising models [15,1,35] or Potts models [21]. However, a crucial difference consists in the fact that in our case the variable that shows a phase transition is not the spin variable itself, but the chirality, which depends on the spin field in a nonlinear way. This is a source of difficulties that will be explained below.
To describe the asymptotic behavior of the system it is convenient to introduce the functionals depending only on functions χ ∈ L 1 (Ω) defined (with a slight abuse of notation) by F ε (χ) := inf{F ε (u) : u : L ε → S 1 such that χ = χ(u, T ) on every T ⊂ Ω} (equal to +∞ if χ is not the chirality of a spin field). The main result in this paper is Theorem 2.5, where we prove that the Γ-limit of F ε with respect to the L 1 -convergence is an anisotropic surface energy given by extended to +∞ otherwise in L 1 (Ω), where J χ is the interface between {χ = −1} and {χ = 1} and ν χ is the normal to J χ . The density ϕ is given by the following asymptotic formula where Q ν is the square with one side orthogonal to ν, u pos ε and u neg ε are the ground states depicted in Figure 1, and ∂ ± ε Q ν are a discrete version of the top/bottom parts of ∂Q ν . Asymptotic formulas like (1.3) are common in discrete-to-continuum variational analyses and are often used to represent Γ-limits of discrete energies [6,10,13,12,8,29]. However, proving an asymptotic lower bound with the density (1.3) for this model requires additional care and is the technically most demanding contribution of this paper. We conclude this introduction by describing the main difficulties that arise in the proof.
Via a classical blow-up argument (see Proposition 4.1) we obtain an asymptotic lower bound with the surface density where χ ν is the pure-jump function which takes the values χ ν (x) = ±1 for ± x, ν > 0. Hence, the proof of the asymptotic lower bound boils down to the proof of the inequality ψ(ν) ≥ ϕ(ν).
To obtain the latter inequality, we need to modify sequences (u ε ) with χ(u ε ) → χ ν in L 1 (Q ν ) without increasing their energy in such a way that they attain the boundary conditions required in (1.3). A common approach to deal with this modification consists in selecting (via a well-known slicing/averaging argument due to De Giorgi) a low-energy frame contained in Q ν and close to ∂Q ν where the sequence can be modified using a cut-off function that interpolates to the boundary values. In our problem, instead, a cut-off modification of χ(u ε ) may generate a sequence of functions that are not chiralities of spin fields (and thus have infinite energy F ε ). Consequently, we have to operate directly on the sequence (u ε ), on whose convergence we have no information due to the invariance of the system under rotation of the spin field (the S 1 -symmetry discussed above). We turn however the S 1 -symmetry to our advantage to define the needed modification. Inside a onedimensional slice of L ε , a spin field close to a ground state in one triangle can be slowly rotated to reach any other ground state with the same chirality by paying an amount of energy proportional to the energy in the starting triangle (see Lemma 4.5). This one-dimensional construction can then be reproduced in the whole Q ν starting from triangles in a low-energy frame close to ∂Q ν in such a way that the modified spin field attains the fixed ground states u pos ε and u neg ε at the (discrete) boundary. However, for this procedure to be successful, the usual slicing/averaging method to find a low-energy frame close to ∂Q ν is not enough. We need to improve it and to find a frame with a better (smaller) energy bound. To this end, we proceed as follows. In Lemma 4.3 we show that ψ(ν) can be equivalently defined using in place of Q ν any rectangle coinciding with Q ν along the interface, but with arbitrarily small height (similar results appeared in different contexts, e.g., [14,15,23,17,24,25,29]). Hence the energy of any sequence (u ε ) admissible for (1.4) concentrates arbitrarily close to the jump set of χ ν , i.e., the interface { x, ν = 0}. With this result at hand, in Lemma 4.4 we can apply the averaging method with the advantage of knowing that in most of the space the total energy is going to vanish, thus finally deducing the existence of a frame close to ∂Q ν with the wished (small enough) energy bound. Even at this point, to reproduce the one-dimensional interpolation along this frame requires additional care. In fact, to conclude the argument one still needs to prove that the winding number of the spin field in the low-energy frame can be properly controlled (Step 3 of Proposition 4.2).

2.
Setting of the problem and statement of the main result 2.1. General notation. Throughout this paper Ω ⊂ R 2 is an open, bounded set with Lipschitz boundary. For every A ⊂ R 2 measurable we denote by |A| its 2-dimensional Lebesgue measure. With H 1 we indicate the 1-dimensional Hausdorff measure in R 2 . Given two points x, y ∈ R 2 we use the notation [x; y] := {λx + (1 − λ)y : λ ∈ [0, 1]} for the segment joining x and y. The set S 1 := {ν ∈ R 2 : |ν| = 1} is the set of all 2-dimensional unit vectors. For every such vector ν = (ν 1 , ν 2 ) ∈ S 1 we denote by ν ⊥ := (−ν 2 , ν 1 ) ∈ S 1 the unit vector orthogonal to ν obtained by rotating ν counterclockwise by π/2. Given v, w ∈ S 1 we denote by v, w their scalar product and by v×w = − v, w ⊥ their cross product. We denote by ι the imaginary unit in the complex plane. It will be often convenient to write vectors in S 1 as exp(ιθ), θ ∈ R. We denote by R ν ,h the rectangle of length > 0 and height h > 0 with two sides orthogonal to ν ∈ S 1 given by , extending the definition to the case = +∞ by setting R ν ∞,h := {x ∈ R 2 : | x, ν | < h/2}. Given ρ > 0 we define the cube centered at the origin with side length ρ and one face orthogonal to ν by Q ν ρ := R ν ρ,ρ . For ρ = 1 we simply write Q ν instead of Q ν 1 . By L ν := {x ∈ R 2 : x, ν = 0} we denote the line orthogonal to ν passing through the origin, while H ν + := {x ∈ R 2 : x, ν ≥ 0} and H ν − := R 2 \ H ν + stand for the two half spaces separated by L ν . Given 2.2. Triangular lattices and discrete energies. In this paragraph we define the discrete energy functionals we consider in this paper. To this end we first define the triangular lattice L. It is given by . For later use, we find it convenient here to introducê e 3 := 1 2 (−1, √ 3) as a further unit vector connecting points of L and to define three pairwise disjoint sublattices of L, denoted by L 1 , L 2 , and L 3 (see Figure 1), by L 1 := {z 1 (ê 1 +ê 2 ) + z 2 (ê 2 +ê 3 ) : z 1 , z 2 ∈ Z} , L 2 := L 1 +ê 1 , L 3 := L 1 +ê 2 .
Eventually, we define the family of triangles subordinated to the lattice L by setting T (R 2 ) := T = conv{i, j, k} : i, j, k ∈ L, |i − j| = |j − k| = |k − i| = 1 , where conv{i, j, k} denotes the closed convex hull of i, j, k. It is also convenient to introduce the families of upward/downward facing triangles For ε > 0, we consider rescaled versions of L and T (R 2 ) given by L ε := εL and T ε (R 2 ) := εT (R 2 ), T ± ε (R 2 ) := εT ± (R 2 ). With this notation every T ∈ T ε (R 2 ) has vertices εi, εj, εk ∈ L ε . The same notation applies to the sublattices, namely L α ε := εL α for α ∈ {1, 2, 3}. Given a Borel set A ⊂ R 2 we denote by T ε (A) := {T ∈ T ε (R 2 ) : T ⊂ A} the subfamily of triangles contained in A. Eventually, we introduce the set of admissible configurations as the set of all spin fields In the case ε = 1 we set SF := SF 1 . For u ∈ SF ε we now define the discrete energies F ε (u) as follows: for every T ∈ T ε (R 2 ) we set F ε (u, T ) := ε|u(εi) + u(εj) + u(εk)| 2 , and we extend the energy to any Borel set A ⊂ R 2 by setting If A = Ω we omit the dependence on the set and write F ε (u) := F ε (u, Ω).

Statement of the main result.
Notice that χ(u) ∈ L 1 (Ω). We then extend F ε to L 1 (Ω) by setting with the convention inf ∅ = +∞.
To state the main theorem we need to introduce two ground states, that we name u pos ε , u neg ε ∈ SF ε which have a uniform chirality equal to +1 and −1, respectively. They are given by ε , for every x ∈ L ε . We also set u pos := u pos 1 , u neg := u neg 1 . The ground states u pos and u neg will be used as boundary conditions on the discrete boundary of the square Q ν given by The proof of Theorem 2.5 will be carried out in Sections 4 and 5, where we prove separately the asymptotic lower bound (Proposition 4.1) and the asymptotic upper bound (Proposition 5.1), respectively.
Remark 2.6. By standard arguments in the analysis of asymptotic cell formulas (see e.g. [4,Proposition 4.6]) one can show that the limit in (2.11) actually exists, so that ϕ is well defined. Note that, by the symmetries of the interaction energies, there holds ϕ(−ν) = ϕ(ν). Moreover, one can show (cf. [4,Proposition 4.7]) that the one-homogeneous extension of ϕ is convex, hence continuous.
Remark 2.7. By a scaling argument we note that for all ρ > 0 there holds where ∂ ± ε Q ν ρ are defined according to (2.9) with Q ν ρ in place of Q ν .
Based on Lemma 3.2 we now prove Proposition 3.1.
Step 1. (Compactness of the auxiliary functions) Let ε > 0 and defineχ ε : Ω → {−1, 1} bŷ We claim that for every Ω ⊂⊂ Ω we have Then the uniform bound (3.1) together with [7, Theorem 3.39 and Remark 3.37] yields the existence of a function χ ∈ BV (Ω; {−1, 1}) and a subsequence (not relabelled) such thatχ ε → χ in L 1 (Ω). To prove (3.7) it is convenient to consider the class of triangles Let Ω ⊂⊂ Ω. By the very definition ofχ ε and of χ(u ε ) we have Estimating the H 1 -measure of the latter set in terms of the cardinality of T pos ε we thus infer (3.8) The last term in (3.8) can be bounded using Lemma 3.2. Indeed, from Lemma 3.2 we deduce that where the additional factor 3 comes from the fact that each triangle is counted 3 times. Thus, (3.7) follows from (3.8) and (3.9).

Lower Bound
In this section we start proving the main result of our paper, namely Theorem 2.5 by presenting the optimal lower bound estimate on the energy F ε , the technically most demanding part of our contribution. We begin with a blow-up argument that gives us a first asymptotic lower bound.
Proposition 4.1. Let F ε be as in (2.8). Then for every χ ∈ L 1 (Ω) we have where F is given by (2.10) and the Γ-lim inf is with respect to the strong topology in L 1 (Ω).
Proof. Let χ ε → χ in L 1 (Ω). We assume that lim inf ε F ε (χ ε ) < +∞, otherwise we have nothing to prove. Moreover, upon extracting a (not relabeled) subsequence we can assume the liminf to be a limit and hence sup ε F ε (χ ε ) < +∞. In view of Remark 2.4 we can find a sequence of spin fields u ε ∈ SF ε with χ(u ε ) = χ ε and F ε (χ ε ) = F ε (u ε ). In particular, sup ε F ε (u ε ) < +∞. Thus, from Proposition 3.1 we deduce that χ ∈ BV (Ω; {−1, 1}). As a consequence, to prove the statement of the proposition it suffices to show that where ϕ is as in (2.11). To prove (4.1) we consider the sequence of non-negative finite Radon measures µ ε given by where δ εi denotes the Dirac delta in εi. From the condition sup ε F ε (u ε ) < +∞ it follows that sup ε µ ε (Ω) < +∞, hence there exists a non-negative finite Radon measure µ such that up to subsequences (not relabeled) µ ε * µ. By the Radon-Nikodým Theorem the measure µ can be decomposed in the sum of two mutually singular non-negative measures as Then, to establish (4.1) it is sufficient to show that where ν χ (x 0 ) denotes the measure theoretic normal to J χ at x 0 . To verify (4.2) we choose x 0 ∈ J χ satisfying and we notice that (i) and (ii) are satisfied for H 1 -a.e. x 0 ∈ J χ thanks to the Besicovitch derivation Theorem and the definition of approximate jump point, respectively. Moreover, since µ is a finite Radon measure, we can choose a sequence ρ n → 0 along which µ(∂Q ν ρn (x 0 )) = 0. Thanks to [7, Proposition 1.62 (a)], the convergence µ ε * µ together with (i) implies that where the last inequality follows from the positivity of the energy. Notice that for every n ∈ N there exist sequences (ρ ε n ) and (x ε 0 ) with lim ε ρ ε n = ρ n , lim ε x ε 0 = x 0 , x ε 0 ∈ L ε , and ) . In fact, if we write x 0 in terms of the basisê 1 ,ê 2 as x 0 = a 1ê1 + a 2ê2 for some a 1 , a 2 ∈ R, we obtain the required sequence (x ε 0 ) by setting Then, upon noticing that |x ε 0 −x 0 | ≤ 2ε, it suffices to set ρ ε n : so that for any x ∈ T there also holds and similarly | x − x 0 , ν ⊥ | < ρ n /2, hence T ∈ T ε (Q ν ρn (x 0 )). As a consequence, we obtain the following estimate 1 where we have set σ ε n := ε/ρ ε n and v ε,n (z) := u ε (x ε 0 +ρ ε n z) for every z ∈ L σ ε n . Let χ ν : R 2 → {−1, 1} be given by Then (ii) ensures that χ(v ε,n ) → χ ν in L 1 (Q ν ) as first ε → 0 and then n → +∞. Thus, gathering (4.3)-(4.4) and applying a diagonal argument we find a sequence σ m := ε m /ρ nm converging to 0 as m → +∞ such that for v m := v εm,nm there holds χ(v m ) → χ ν in L 1 (Q ν ) and For , h > 0 let us finally introduce the minimization problem  To prove Proposition 4.2 it is necessary to modify admissible sequences for the infimum problem defining ψ(1, 1, ν) in such a way that they satisfy the boundary conditions required in the minimum problem defining ϕ(ν), without essentially increasing the energy. This will be done by a careful interpolation procedure based on several auxiliary results and estimates that we prefer to state in separate lemmas below. As a first step towards the proof of Proposition 4.2 we show that ψ( , h, ν) is independent of and h, which in turn will allow us to conclude that the energy of admissible functions for ψ(1, 1, ν) concentrates close to the line segment L ν (see Lemma 4.4 below). Lemma 4.3. Let ψ : (0, +∞)×(0, +∞)×S 1 → [0, +∞] be given by (4.5); then ψ(·, ·, ν) is independent of , h for every ν ∈ S 1 .
On account of Lemma 4.3 we show that for a sequence (u ε ) realizing the infimum in the definition of ψ(1, 1, ν) the energy concentrates close to the line L ν . As a consequence, we obtain that outside a small neighborhood of L ν there exists a suitable strip on which the energy is of order o(ε). To be more precise, for fixed ν ∈ S 1 , δ > 0, and every ε > 0 we introduce the class S ν ε,δ of strips (4.14) We denote the elements of S ν ε,δ by S ε,r . Then the following result holds true.
We are now in a position to start with the interpolation procedure mentioned before. The final interpolation procedure will be based on a one-dimensional construction that we introduce below.
One-dimensional interpolation. To define the one-dimensional interpolation we consider slices in the triangular lattice. To this end, letê 1 ,ê 2 , andê 3 be as in Section 2.2. Given α ∈ {1, 2, 3} we consider the orthogonal vectorê ⊥ α toê α and we define the slice in the directionê α by Σ α := sê α + tê ⊥ α : s ∈ R , t ∈ [0, √ 3 Given z ∈ Z, we define Finally, for every ε we set Σ α,z ε := εΣ α,z . (4.17) We shall define the one-dimensional interpolation in a slice Σ α starting from a triangle T 0 ∈ T (R 2 ) such that T 0 ⊂ Σ α . Let us denote by i 0 ∈ L 1 , j 0 ∈ L 2 , k 0 ∈ L 3 the vertices of T 0 . Note We define the lattice points i h ∈ L 1 , j h ∈ L 2 , k h ∈ L 3 and the triangle T h with the following recursive formula: we set τ (0) := 1, τ ( , the analogous equality being true also for j h and k h . Moreover, T 2h = T 0 + 3hê α . We define the half-slice Σ α (T 0 ) of the lattice L starting from T 0 by Σ α (T 0 ) := conv{T h : h ∈ N} . (4.19) Given u : L → S 1 and N, m ∈ N, we now define in the half-slice Σ α (T 0 ) a one-parameter family (parametrized by m) of spin fields which coincides with u on T 0 and with the fixed ground state u pos on T h for h ≥ N . We construct the interpolation in such a way that the configuration of spins rotates a fixed amount of times by 2π. To make the construction precise, we first say that the three angles θ(i 0 ) ∈ R (not necessarily in [0, 2π)), θ(j 0 ) ∈ [θ(i 0 ) − π, θ(i 0 ) + π) and θ(k 0 ) ∈ [θ(j 0 ) − π, θ(j 0 ) + π) represent a lifting of u in T 0 if u(i 0 ) = exp(ιθ(i 0 )), u(j 0 ) = exp(ιθ(j 0 )) and u(k 0 ) = exp(ιθ(k 0 )). We then define the interpolated angles θ(i h ), θ(j h ), θ(k h ) for h = 0, . . . , N by  Note that u N,m = u pos on T h for h ≥ N . Figure 3. Example of interpolation from u to u pos in the slice Σ α (T 0 ) starting from the triangle T 0 (in grey).
In the next lemma we estimate the energy of the interpolation on Σ α (T 0 ) in terms of the energy on the initial triangle T 0 plus an error depending on the number of steps N and on m. We assume that the configuration of spins in the initial triangle is sufficiently close to a ground state with chirality 1 (not necessarily u pos ).
Lemma 4.5. Let T 0 ∈ T (R 2 ) be a triangle of vertices i 0 ∈ L 1 , j 0 ∈ L 2 , and k 0 ∈ L 3 . Let u : L → S 1 and let θ(i 0 ) ∈ R, θ(j 0 ) ∈ [θ(i 0 ) − π, θ(i 0 ) + π) and θ(k 0 ) ∈ [θ(j 0 ) − π, θ(j 0 ) + π) be three angles representing a lifting of u in T 0 satisfying Let N, m ∈ N and assume that 2πm ≥ |θ(i 0 )| + 2π . Let u N,m be the interpolation on Σ α (T 0 ) defined according to (4.21). Then there exists a constant C > 0 independent of N and m such that Proof. It is not restrictive to assume that j 0 − i 0 =ê α as in Figure 3. We shall estimate each of the terms in the sum where we used that for h ≥ N we have that being u pos a ground state. Adopting the notation for the angles used in the construction in (4.21), we recast the energy in the first term of the sum as Note that, by (4.20) and (4.22), (4.26) By Taylor's formula, there exists ζ ∈ [φ, 2π/3] such that 1 As a result we obtain the estimates Analogously, Then by (4.25), (4.26), and the two previous estimates we infer that
We are now in a position to prove Proposition 4.2 and thus conclude the proof of the lower bound in Proposition 4.1.
The construction of the modified sequence ( u ε ) is divided in several steps.
Step 1. (Choosing a strip with low energy). We begin the construction by exploiting the property that the energy of (u ε ) concentrates close to the interface Q ν ∩ L ν in order to choose a strip with low energy. Given δ ∈ (0, 1 3 ), we consider the family of strips S ν ε,δ defined in (4.14) and we apply Lemma 4.4 to deduce the existence of a strip S ε = S ε, where σ ε → 0. The modification u ε of u ε will coincide with u pos ε and u neg ε in Q ν \ (Q ν 1−δ ∪ R ν 1,δ ) (notice that the square Q ν 1−δ contains the closure of S ε , cf. (4.14)). In the triangles contained in S ε the energy is low and thus u ε is close to ground states, yet not necessarily u pos ε or u neg ε . There u ε will start to interpolate from the configuration u ε until it reaches the fixed ground state u pos ε or u neg ε close to the boundary.
We shall describe in detail how to define u ε in the top part of the cube given by Q ν + = Q ν ∩ {x : x, ν > 0}, where the chirality of u ε converges to 1. The construction in Q ν − ∩ {x : x, ν < 0} is completely analogous.
Step 2. (Choosing triangles with low energy). We show here how to choose the triangles with low energy where to start the modification of u ε . Let us consider the line L ε := {x ∈ R 2 : x, ν = rε 2 + 3ε} , which cuts in two the top part of the strip given by the rectangle We describe now how to start the modification in S top ε . The modification in the other parts cf. Figure 4, will be only sketched since it is completely analogous. We consider now the slices (Σ α,z ε ) z∈Z of the ε-triangular lattice defined in (4.17). We choose α ∈ {1, 2, 3} such that | ê α , ν | ≥ 2 .) We can find a chain of closed triangles which intersect L ε such that each slice in the directionê α contains only one triangle of the chain. Specifically, there exist (T z ) z∈Z , satisfying for every z ∈ Z, cf. Figure 5. We prove this statement in Lemma 4.6 below, since the geometric argument is irrelevant for the present discussion. cf. (4.30), x ∈ (L ε + B 3ε (0)) ∩ R ν rε+6ε,1 ⊂ S ε . Let us show that , then we let T be the third triangle in conv{T z , T z+1 }. The triangle T is either conv{εi z , εj z , εk z+1 } or conv{εi z+1 , εj z , εk z } and is always contained in H z , see Figure 6. Letting εk be its vertex in L 3 ε (either εk z or εk z+1 ) we have that Then we estimate the last sum in (4.34) using (4.35) by for some positive constant C. In conclusion, by (4.34) we have that we define the lattice points i h which contradicts σ ε → 0. In conclusion, χ(u ε ) > 0 in S top ε . Let now z ∈ Z top ε . We have Since χ(u ε ) > 0 in T z , for ε small enough u ε is close to a ground state with chirality 1 and therefore, using (2.3) and Lemma 2.1 (see also (2.5)), Definition of the interpolation. We are in a position to define the interpolation. We reproduce the one-dimensional construction of Lemma 4.5 by suitably translating and scaling it, providing the precise notation as it will be useful for later estimates. We shall define the interpolation only on slices starting from every other triangle T z , for the constructions on two slices Σ α,z ε and Σ α,z+2 ε completely determine the values of the modified spin configuration in Σ α,z+1 ε . For this reason, let z ∈ Z top ε be such that z ≡ z 0 mod 2. We then define the interpolated angles θ(εi h z ), θ(εj h z ), θ(εk h z ) for h = 0, . . . , N ε as in (4.20) by (recall that i 0 Final estimate on top part. By (4.44), (4.43), summing over z and by (4.29), (4.32), (4.36), and (4.39) we conclude that 1 (4.45) Step 5. (Definition of modification in remaining parts of the square). The modification starting from S left ε and S right ε is completely analogous. We recall that β ∈ {1, 2, 3} is such that | ê β , ν ⊥ | ≥  We are finally in a position to define u ε in Q ν + . We fix δ ∈ (0, 1 8 ) and we consider the two-barred cross-shaped set (the white region in Figure 7) 1 In this estimate it becomes evident that it was crucial to prove that the energy concentrates close to the interface.
A classical averaging/slicing argument would only provide a bound on the strip Sε of the type Fε(uε, Sε) ≤ Cε. This would not suffice to conclude that the modified sequence does not increase the energy, as the right-hand side in this estimate would end up to be a constant.
Given T ∈ T ε (R 2 ) such that T ⊂ Q ν , we distinguish some cases.
Case T ⊂ R ν 1−5δ,1 \ Q ν rε (part of the cross-shaped set P δ aligned with ν): We give the definition in the case T ⊂ Q ν + (the case T ⊂ Q ν − being analogous). Let y 0 ∈ T . Let us consider the slice Σ α,z ε such that T ⊂ Σ α,z ε and let us show that z ∈ Z top ε . Let x ∈ T z and first of all note that x ∈ L ε + B 3ε (0). Since T and T z are contained in the same slice, by definition of Σ α,z ε we can find s ∈ R such that x 0 := y 0 + sê α ∈ T z . Since The definition is consistent with the previous case: if T ⊂ Q ν rε+6ε \ Q ν rε , then T is not contained in any half-slice Σ α,z ε (T z ) (because T z ∩∂Q ν rε+6ε = ∅) and thus u ε | T = u ε | T , in accordance with (4.46). If T ⊂ R ν 1−5δ,1 \ Q ν rε but T is not contained in any half-slice Σ α,z ε (T z ), then T ⊂ S ε . In particular, by (4.45) and (4.29) we infer that ν rε (part of the cross-shaped set P δ aligned with ν ⊥ ): As in the previous case, assuming T ⊂ Q ν + , we define u ε | T := u left ε | T if T is contained in a half-slice starting from a triangle in S left ε , u ε | T := u right ε | T if T is contained in a half-slice starting from a triangle in S right ε , and u ε | T := u ε | T otherwise. As before, the definition is compatible with (4.46). Similarly to (4.48) we obtain that Case T ∩ (R 2 \ P δ ) = ∅: let x be a vertex of T and assume that x is not the vertex of a triangle T covered by the previous cases. Then we set u ε (x) := u pos (4.50) We remark that and #{T ⊂ Q ν \ P δ : T ∩ L ν = ∅} ≤ C δ ε . (4.52) Let us check that u ε attains the desired boundary conditions (4.27). 3δ )), then we are in the case covered by (4.47). By (4.42) we have u ε | T = u top ε | T = u pos ε | T . Otherwise, if T ∩ (R 2 \ P δ ) = ∅, let x be a vertex of T and assume that x is not the vertex of a triangle T covered by the previous cases. Then, by definition, u ε (x) := u pos ε (x). We argue analogously if T ⊂ Q ν − \ Q ν 1−δ . Finally, if T ∩ L ν = ∅, then T ⊂ L ν + B 2ε (0) and thus it is not relevant for the boundary conditions by the definition of discrete boundary ∂ ± ε Q ν .
In the proof of Proposition 4.2 we applied the following lemma.
Lemma 4.6. Let Σ α,z ε be the slices of the triangular lattice defined in (4.17). Let L be a line in R 2 orthogonal to ν and assume that | ê α , ν ⊥ | ≤ 1 2 . Then there exists a chain of triangles (T z ) z∈Z satisfying for every z ∈ Z Proof. It is enough to prove the following: With the proven claim at hand it is immediate to define a chain of triangles (T z ) z∈Z which satisfies the properties in (4.53) by initializing the construction from a triangle T z0 ∈ T + ε (R 2 ) which satisfies T z0 ∩ L = ∅ and T z0 ⊂ Σ α,z0 ε . Such a triangle always exists since the set R 2 \ T ∈T + ε (R 2 ) T is the union of disjoint open triangles, thus cannot contain L.

Upper Bound
It remains to prove the Γ-limsup inequality to complete the proof of Theorem 2.5.

1)
where F is given by (2.10) and the Γ-lim sup is with respect to the strong topology in L 1 (Ω).