Periodic striped ground states in Ising models with competing interactions

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let $J$ be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent $p$ of the long range interaction is larger than $d+1$, with $d$ the space dimension, this happens for all values of $J$ smaller than a critical value $J_c(p)$, beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for $p>2d$ and $J$ in a left neighborhood of $J_c(p)$. In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes ($d=2$) or slabs ($d=3$), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.


Introduction and main results
The problem of proving the emergence of periodic patterns in systems with competing interactions is ubiquitous in several areas of physics, biology and material science [2], ranging from superconductor physics [12], micro-magnetism [10], polymer suspensions [23], martensitic phase transitions [26], quantum Hall systems [14], to metal-oxide-semiconductor field-effect transistors [37], nuclear matter [31], and many others [5,6]. In all these systems, experiments or simulations show evidence for the formation of remarkable patterns in suitable regions of the phase diagram. Examples are stripes, bubbles, zigzag patterns, and columnar phases. The fundamental understanding of these phenomena is still in a primitive stage, mostly based on variational computations and special assumptions on the structure of the low-energy states. There are just a few special cases where the periodicity of the ground state can be proved from first principles [13,24,25,39,40].
A particularly interesting and poorly understood phenomenon is that of periodic stripe formation [28,29,36,38]. In a series of papers, this phenomenon was studied in Ising and related models with short range attractive and long range repulsive interactions. The method of block reflection positivity led to rigorous proof the existence of periodic striped states both in one dimension [15,17,18] and in certain two-dimensional models, including a toy model model for martensitic phase transitions [21] and a model of in-plane spins with discrete orientations and dipolar interactions [16]. However, the physically interesting case of out-of-plane spins with dipolar, or dipolar-like, interactions, which is of great importance for the physics of thin magnetic films, eluded any rigorous treatment so far.
In a recent work [20], we succeeded in computing the specific ground state energy of such a system, with power-law decaying repulsive interactions and decay exponent p > 2d in d = 2, 3 space dimensions, asymptotically as the ferromagnetic transition line is approached. Our estimates allowed us to prove emergence of periodic stripe order in a suitable asymptotic sense, but they were not strong enough to fully control the ground state structure, or to prove breaking of rotational symmetry in the ground state. In this paper we extended the ideas of [19,20] and prove that periodic striped states of optimal width are exact infinite volume ground states. Moreover we give a characterization of infinite volume ground states that are invariant under translations by one (for d = 2) or two (for d = 3) independent fixed lattice vectors.
The setting is the following: consider Ising models defined by the formal Hamiltonian where σ ∈ {±1} Z d , d ≥ 2. The first sum in (1.1) ranges over nearest neighbor pairs in Z d , while the second over pairs of distinct sites in Z d . For different values of the exponent p, this model is used to describe the effects of frustration induced in magnetic films by the presence of dipolar inter-actions (p = 3) or in charged systems by the presence of an unscreened Coulomb interaction (p = 1), as well as many other frustrated systems [1,3,4,5,6,7,8,9,11,22,27,28,30,32,33,34,35,38,41].
In this paper, we choose the exponent p to satisfy the constraint p > 2d. As discussed in a previous work [20], if J > J c , with then there are exactly two ground states, σ x ≡ +1 ∀x ∈ Z d , and σ x ≡ −1 ∀x ∈ Z d . For J < J c , the ground state is not uniform, and for J close to J c it was conjectured to be a periodic striped configuration, i.e., a quasione-dimensional periodic configuration of the form (σ (h) ) x = sign(sin(π(x 1 + 1/2)/h)), or translations or rotations thereof, for a suitable stripe width h ∈ N. In this paper, we prove this conjecture, and characterize the set of infinite volume ground states invariant under translations generated by d − 1 lattice vectors. For simplicity, we restrict the discussion to d = 2 and p > 4 from now on. Similar considerations are valid in d = 3 and p > 6 (or, in fact, for any d ≥ 2 with p > 2d). In Appendix A we explain how to adapt the proof to dimension three and higher. Let e s (h) be the energy per site of σ (h) computed via (1.1). We let h * = argmin h∈N e s (h), which is uniquely defined for almost all 1 choices of J. We denote by σ * = σ (h * ) ∈ {±1} Z 2 , and we call it an optimal periodic striped configuration. Other 4h * − 1 optimal periodic striped configurations are obtained from σ * via translations and rotations.
In order to state our main result, we also need to introduce the following notions: the configuration s ∈ {±1} Z 2 is called an infinite volume ground state if it is energetically stable against compactly supported perturbations, that is, for any finite X ⊂ Z 2 , where s X is the restriction of s to X, and We shall say that two infinite volume ground states are equivalent, if they only differ on a finite set. The equivalence class of a given infinite volume ground state is called a sector. A sector is trivial, if it contains only one element. In terms of these notions, our main result can be summarized as follows.
Theorem 1. There exists ε > 0 such that, if J c − ε < J < J c , then the optimal periodic striped configurations are infinite volume ground states, and their sectors are trivial.
This result is a corollary of a quantitative lower bound on the energy of spin configurations, which will be formulated in Theorem 3 below, after having introduced a few more definitions. Our quantitative bounds also allow us to characterize the infinite volume ground states that are invariant under translation by a vector n ∈ Z 2 . Theorem 2. Under the same conditions as Theorem 1, any infinite volume ground state s that is invariant under translation by a vector n = (n 1 , n 2 ) ∈ Z 2 is characterized by the following property: there exists an "interface" of finite width, of the form I k 1 ,k 2 (n) = {x ∈ Z 2 : x · n ⊥ ∈ [k 1 , k 2 ]}, where n ⊥ = (−n 2 , n 1 ) and k 1 < k 2 are two integers, such that s coincides with two of the optimal striped configurations on the two infinite components of Let us now introduce a few more definitions, which are required for the formulation of our quantitative lower bound on the energy of a generic spin configuration.

Contours and corners
Given σ ∈ {±1} Z 2 , we let ∆ = {x ∈ Z 2 : (σ) x = −1}, and Γ(∆) be its boundary, i.e., the union of bonds of the dual lattice (Z 2 ) * separating a point x ∈ ∆ from a point y ∈ ∆ c . At every vertex of Γ(∆) ∩ (Z 2 ) * , there can be either 2 or 4 sides meeting. In the case of 4 sides, we deform the polygon slightly by "chopping off" the edge from the squares containing a − spin; see Fig. 1.
After the chopping, Γ(∆) splits into disconnected polygons Γ 1 , Γ 2 , . . . , which are called contours. The points where two orthogonal portions of a contour meet are called corners. The sites of the dual lattice where a non-trivial chopping operation took place correspond to two corners. We denote by N c (Γ i ) the number of corners of Γ i , and N c (∆) = i N c (Γ i ). Note Figure 1: In the case that 4 sides of the closed polygon Γ(∆) meet at a vertex v, we slightly deform Γ(∆) so that the two squares containing a − spin become disconnected from the vertex itself. Case (a) represents the situation where the minus spins are located at NE and SW of v, before and after the "chopping". Case (b) represents the situation where the minus spins are located at NW and SE of v, before and after the "chopping". that, if σ is a compactly supported perturbation of σ * , that is, if the set {x ∈ Z 2 : (σ) x = (σ * ) x } is finite, then N c (∆) < +∞.

Tiles
Given an integer , we pave Z 2 with tiles of side and, given a tile T , we denote by Γ T (∆) the union of the bonds in Γ(∆) that separate a point in ∆∩T from a point in ∆ c . Note that this convention assigns uniquely every bond in Γ(∆) to one of the tiles. The connected components of Γ T (∆) are contained in the contours Γ 1 , . . ., and are denoted by Γ T,i , i = 1, . . . , r T . Given a maximal straight portion of Γ T,i , we assign to each of its two endpoints a "number of corners", which can be either 1/2 or 0, depending on whether or not the given endpoint coincides with one of the corners in ∪ j≥1 Γ j . This assignment induces a notion of "number of corners in the tile T ", to be denoted by n c (T ), which is the sum of the number of corners of all the endpoints of the straight portions of Γ T,i , with i = 1, . . . , r T . Note that n c (T ) can be either integer or half-integer, and T n c (T ) = N c (∆).

Bad tiles and good regions
We now identify the tiles T i such that either n c (T i ) > 0, or they contain a square of side /5 completely contained in ∆ or in ∆ c , to be called hole. We call these tiles bad 2 , and we let N B be their number. For later convenience, we also let N hole B be the number of bad tiles containing a hole. The connected components of the complement of ∪ N B i=1 T i , are denoted by G i , i = 1, . . . , N G , and are called the good regions. By construction, any of these connected components contains portions of contours in Γ(∆) that are all straight with the same orientation, and have no corners. We denote by Γ G i the union of contours in Γ(∆) contained in G i . If the elements of Γ G i are all vertical, we will say that G i is vertically striped, and horizontally striped otherwise. Consider a good region G i that is vertically (resp. horizontally) striped. We say that R is a "rectangular portion of stripe" in G i , if R is a rectangle completely contained in ∆ or in ∆ c , with its two vertical (resp. horizontal) boundaries both belonging to Γ G i . We also define the distance between its two vertical (resp. horizontal) boundaries to be the width of R. Finally, we let A h (G i ) denote the area of the union of all the rectangular portions of stripes of width h in G i . Note that, if σ is a compactly supported perturbation of We are now in the position of stating our quantitative lower bound on the energy H X (s X |σ * ) of a spin configuration s with σ * boundary conditions. Theorem 3. There exist positive constants C 0 , C 1 , ε such that, if J c − ε < J < J c and C 0 h * ≤ ≤ (C 0 (J c − J)) −1 , then for every s ∈ {±1} Z 2 and every finite set X ⊂ Z 2 , 5) where N c , N hole B , and G i are, respectively, the number of corners, the number of bad tiles containing a hole, and the good regions, associated with the infinite spin configuration σ = (s X , σ * X c ) coinciding with s X on X and with σ * on X c , defined via tiling with squares of side length as described above.
(J c − J) −1 for small J c − J (compare with Eq. (2.30) below), the condition on can be fulfilled for J close to J c . Note also that e s (h * ) ∼ (J c − J) (p−2)/(p−3) for small J c − J, which agrees with the factor multiplying N hole B in the second term on the right side of (1.5).
Remark. The prefactor 1/2 in the second line of (1.5) can be replaced by any number less than 1, at the expense of modifying the constants C 0 and ε.
Theorem 3 implies, in particular, that σ * is an infinite volume ground state, and that every state σ that is a compactly supported perturbation of it, is not a ground state, simply because any such state necessarily has corners and, therefore, by (1.5), it has strictly larger energy than σ * . This immediately implies Theorem 1.
In order to see that also Theorem 2 is a consequence of Theorem 3, consider an infinite volume ground state s that is invariant under translations by an integer vector n. Let Λ ⊂ Z 2 be a square box of side L, and note that the energy price for changing the boundary conditions from s to σ * scales like the boundary, that is for any σ Λ . Using this inequality and the very definition of infinite volume ground state, we have

Now we apply Theorem 3, thus obtaining
and G i refer to the configuration (s Λ , σ * Λ c ). In particular, N c is bounded by (const.)L. Since every corner not at the boundary of Λ is repeated with period n, there can be at most a finite number of them (modulo translations by n) independently of L. This means that these corners are all contained in a finite strip I k 1 ,k 2 (n), as claimed in Theorem 2, with k 1 , k 2 independent of L. Similarly, we can argue that the holes and the stripes of width different from h * are all contained in a finite strip I k 1 ,k 2 (n). This concludes the proof of Theorem 2, in the case that h * is unique. As observed above, there are exceptional values of J for which e s (h) has two minimizers, h * and h * + 1. In these cases, the discussion above leaves open the possibility that on one of the connected components of I k 1 ,k 2 (n) c the stripes are not all of the same width. However, this cannot be the case: by proceeding as in [15, Section III.D], one can prove that each pair of neighboring stripes of widths h * , h * + 1 gives an extra positive contribution to the energy per unit stripe length. Therefore, pairs of stripes of different widths are all contained in a finite strip I k 1 ,k 2 (n), and Theorem 2 follows.
The rest of the paper is devoted to the proof of Theorem 3.

Proof of Theorem 3
The proof of Theorem 3 is divided into several steps, and uses many notations and ideas introduced in [20], which will be recalled here. As a preliminary step, we reduce to plus boundary conditions, which allows us to use the droplet formulation for the energy as in [20], see (2.3) below. Once the energy is expressed in terms of droplets, we can localize the energy in the bad tiles and good regions, by proceeding in a way analogous to [20]. The key technical novelty of this paper, as compared to [20], is an efficient way of estimating the energy in the good regions, which may have a complicated geometrical shape. The crucial estimate is summarized in Lemma 1 below, whose proof is given in Section 2.4. Before entering the proof, we recall some notation: we let τ = 2(J − J c ), which is assumed to be negative and small, in absolute value. We recall that h * is of the order |τ | −1/(p−3) and the specific energy of an optimal striped configuration, e s (h * ), is negative and of the order |τ | (p−2)/(p−3) (compare with (2.30) below).

Reduction to plus boundary conditions
The σ * boundary conditions, while natural in the perspective of proving uniqueness of the ground state, are not particularly convenient for using the droplet representation of [20]. However, a few simple algebraic manipulations allow us to reduce to the same boundary conditions of [20] (that is, plus boundary conditions) in a suitable enlarged box. To see this, rewrite H X (σ X |σ * ) as where Λ is a square box of side L, which we choose to be divisible by 2h * , (σ X , σ * Λ\X ) is the configuration on Λ whose restriction to X ⊂ Λ (resp. Λ \ X) coincides with σ X (resp. σ * Λ\X ), and H per can be further rewritten in terms of a Hamiltonian with plus boundary conditions: where Λ M is a square box of side LM , to be thought of as the union of M 2 copies of Λ, and (σ X , σ * Λ\X ) M 2 is a symbol for the configuration on Λ M obtained by juxtaposing M 2 copies of (σ X , σ * Λ\X ), one in each of the copies of Λ. Moreover, H + X (σ X ) = H X (σ X |σ + ) indicates the Hamiltonian with plus boundary conditions (here σ + is the uniform infinite spin configuration consisting of plus spins everywhere). Finally, for later reference, we introduce the shorthand u Λ M for the spin configuration (σ X , σ * Λ\X ) M 2 on Λ M , and u for the infinite one coinciding with u Λ M on Λ M and with σ + on the complement. From now on we shall consider u Λ M and u fixed once and for all.

Localization
We now re-express H + Λ M (u Λ M ) in terms of the droplets representation introduced in [20]. Using the notation introduced in Section 1.1.1, we let ∆ be the region of minus spins associated with u, Γ(∆) its boundary, and Γ 1 , . . . , Γ r the corresponding contours. We also denote by G(∆) the collection of contours, G(∆) = {Γ 1 , . . . , Γ r }. Note that ∆, Γ(∆), G(∆) and r are finite, because of the plus boundary conditions. As in [20], we denote by δ i the maximal connected components of ∆, and by D(∆) their collection. Given δ ∈ D(∆), we also let Γ(δ) be the boundary of δ, and N c (δ) its number of corners. In terms of these notations, we can re-express the energy of u Λ M as Let us consider the partition P of Λ M defined by the bad tiles and good regions of Λ M , in the sense of Section 1.1.3: We now localize the energy in the elements of P, by proceeding as in [20,Section 3]. More precisely, we derive a lower bound on the energy H + Λ M (u Λ M ) in the form of a sum of local energies E Q (B Q ), each depending only on the "bubble configuration" B Q within the region Q ∈ P. The notion of bubble configuration was introduced in [20, Section 3] and is recalled here: given Q ∈ P and δ ∈ D, we denote by Γ Q (δ) the portion of Γ(δ) belonging to belonging to the boundary ofδ , respectively. We shall refer to the pair (δ Q ) as to a bubble in Q originating from δ. We shall indicate Figure 2: On the left: a square region Q (in light grey, with dotted boundary) and the droplets having non-zero intersection with it. On the right: the corresponding bubble configuration B Q after localization in Q. Note that a single droplet can give rise after localization to more than one bubble: e.g., β 2 and β 7 originate from the same droplet, and similarly for β 3 and β 4 . Note also that the contour of a bubble does not necessarily coincide with the boundary of its droplet: in general, it is contained in it, and may even be disconnected (as in the case of β 5 , whose contour consists of two disconnected portions).
by B Q (δ) the set of bubbles in Q originating from δ, and by B Q = ∪ δ∈D B Q (δ) the total set of bubbles in Q. (See Fig. 2

.)
Given a bubble β = (δ β , Γ β ) ∈ B Q , we also define its localized self-energy as if there is such a bond, and is infinite, if b faces the boundary of Q. In terms of these notions, our localization bound takes the following form: 8) and the * on the last sum indicates the following constraint: if the bubbles in G i are all vertical (resp. horizontal) we only sum over pairs of bubbles that do not overlap after arbitrary translations in the vertical (resp. horizontal) direction.
Proof of (2.6). We start from (2.3). The goal is to bound it from below by a sum of terms, each of which is localized in an element Q of P. The first term on the right side of (2.3) is already local, i.e., it can be rewritten exactly as 2J Q∈P β∈B Q |Γ β |, which leads to the corresponding terms in (2.7) and (2.8). The interaction W (δ, δ ) on the right side of (2.3) can be rewritten as which is bounded from below by dropping the terms with Q = Q (recall that the interaction is positive), so that and the * * on the sum indicates the constraint that the two droplets δ, δ in D(∆), which δ β , δ β belong to, are different from each other, δ = δ . Regarding the second term on the right side of (2.3), we bound it from below by using [20, Eq.(2.9)], that is where d b (δ) is the distance between b and the bond b facing it in δ, and P(δ) is the set of unordered pairs of distinct sites in δ such that both C hv x→y and C vh x→y cross at least two bonds of Γ(δ). Here C hv x→y is the path on the lattice that goes from x to y consisting of two segments, the first horizontal and the second vertical. Similarly, C vh x→y is the path on the lattice that goes from x to y consisting of two segments, the first vertical and the second horizontal (note that the two paths can coincide, in the case that x i = y i for some i ∈ {1, 2}).
The first term on the right side of (2.10) can be bounded from below as which, after summation over δ, leads to the terms β∈B Q u Q (β) in (2.7) and (2.8). The second term on the right side of (2.10) is local and, after summation over δ, can be rewritten as 2 1− p 2 N B i=1 n c (T i ) (recall that the good regions have no corners), which leads to the last term on the right side of (2.7). Moreover, the sum over δ of the last term on the right side of (2.10) can be bounded from below by an expression similar to the right side of (2.9), namely where the † on the sum indicates the constraint that δ β , δ β belong to the same droplet δ ∈ D(∆), and all the pairs of points (x, y) in δ β × δ β are such that {x, y} ∈ P(δ). Combining the right sides of (2.9) and (2.12) we obtain where the † † on the sum indicates the constraint that: either δ β , δ β belong to different droplets in D(∆), or, if they belong to the same droplet in δ, they are such that all pairs of points (x, y) in δ β × δ β are in P(δ). Finally, note that: if Q is a bad tile (which is a convex region), then the constraint † † is automatically realized (i.e., it can be dropped), which leads to the second term on the right side of (2.7); if Q is a good region, then the constraint † † is easily seen to be weaker than the one indicated by * in (2.8), which leads to the last term on the right side of (2.8).

Lower bounds on the localized energies
In this subsection we state the key lower bounds on the localized energies in the bad and good regions, and prove that they imply Theorem 3. Recall the definitions of N B , N hole B and A h (G i ) given in Section 1.1.3. Recall also that is the side length of the tiles, which enters the definition of the partition P. Lemma 1. There exist positive constants c 0 , c 1 and ε such that, if −ε < τ < 0 and ≥ c 0 h * , then the energy E G of any good region G ∈ P satisfies There exist positive constants c 0 , c 2 and ε such that, if −ε < τ < 0 and c 0 h * ≤ ≤ (c 0 |τ |) −1 , then the energy E T of any bad tile T ∈ P satisfies

14)
where χ hole (T ) is equal to 1 if T contains a hole, and 0 otherwise.
Lemma 1 is one of the main technical novelties of this paper, and its proof is described in detail in the next section. The proof of Lemma 2 is simpler, it is an extension of the bounds worked out in [20], and its proof is postponed to Section 2.5.
Combining the two lemmas, we can easily derive Theorem 3. In fact, note that every portion of the boundary ∂G i of a good region G i ∈ P is adjacent to a bad tile, so that N G i=1 |∂G i | ≤ 4 N B . Using this observation, together with the fact that every bad tile either has a positive number of corners (at least 1/2) or a hole, and plugging (2.13) and (2.14) into (2.6), we obtain then the expression in parentheses that multiplies N B is positive, and we can drop it for the purpose of a lower bound. Via the use of (2.1)-(2.2), we thus arrive at Eq. (1.5).

Proof of Lemma 1 2.4.1 Deforming the good regions
We now discuss the proof of Lemma 1. Let us consider a good region G ∈ P, and let us assume without loss of generality that it only contains vertical stripes. We first want to slightly deform the domain G and correspondingly change the bubble configuration within, in order to make all the bubbles rectangular: here we call "rectangular" a bubble β = (δ β , Γ β ) such that δ β is a rectangle and Γ β is the union of its two vertical sides. Note that, in general, not all the bubbles in G are rectangular, due to boundary effects (in fact, the boundary can partially "cut" a portion of the rectangle, without disconnecting it, see e.g. the droplets δ 1 and δ 2 in Fig. 3).
In order to describe the deformation of G, think of the vertical boundary of G as a union of segments S i of length , induced by the tiling described in Sections 1.1.2 and 1.1.3. By construction, every boundary segment S i faces a portionS i of length of the boundary of a rectangular bubble in B G that is closer than 2 /5 to S i itself. We now deform the boundary ∂G continuously, Figure 4: The deformed good region corresponding to the good region of Fig. 3, after the moves of the vertical boundary segments indicated by the arrows. Note that after the deformation all the droplets are rectangular, and all the connected components of the complement of the droplets (the connected light grey regions) are rectangular as well.
by moving the segments S i towards the interior of G, in such a way that they coincide withS i ; see Fig. 4.
In this way we increase the boundary of G by at most |∂G| itself, and we increase the energy by at most |τ |·|∂G|, which is acceptable for our purposes. We shall denote by G the new region obtained from G by the deformations we just described, and the new bubble configuration B G . We have as just argued.

Slicing and bounding the energy of the good region
The deformed region G obtained in the previous step is a union of connected horizontal slices g j of height , with j = 1, . . . , N g , as shown in Fig. 5. To each slice we associate a sequence of integers (h 1 , w 1 , . . . , w n−1 , h n ), where n is the number of bubbles in B G intersecting the slice, h 1 , h 2 , ..., h n are their widths, ordered from left to right, and w 1 , w 2 , ... are the spacings between the first and second bubbles, second and third, etc. We also denote by s i , with i = 1, . . . , N s , the maximal connected segments in the intersection between the horizontal boundary of G and the boundaries of the droplets in B G (note that the boundary of a rectangular bubble does not coincide with its contour: rather, it consists of four segments, two horizontal and two vertical). Note that segments come in pairs: one can say that two segments form a pair if they belong to the boundary of the same droplet. Moreover, to each segment s j we associate its length h(s j ) and two spacings w 1 (s j ), w 2 (s j ), which are the horizontal distances from the next droplets ("next" by following the boundary of G ) to the left and to the right of s j . Note that if s j touches a corner of ∂G , say on its right side, then there may not be any droplet to its right: by following the boundary one may find that the next segment to the right could actually have the same horizontal coordinates, in which case we will assign the value +∞ to w 2 (s j ) (and similarly for w 1 (s j ) in the case of the next segment on the left); see Fig. 6.
The key ingredient for the proof of Lemma 1 is the following.
Lemma 3. Given G and B G as above, we have: n j is the sequence of widths and spacings associated to the slice g j , and the functions e ∞ and f are defined as follows:  Fig. 4 in the deformed good region G . The segments can be naturally grouped in pairs: two segments form a pair if they belong to the boundary of the same droplet. E.g., s 1 and s 17 are paired, s 11 and s 20 are paired, etc. Every segment s j comes with its length h(s j ), and with two spacings, the left spacing w 1 (s j ) and the right spacing w 2 (s j ), corresponding to the distances to the closest droplets to its left and to its right, by moving along the boundary. If one of the endpoints of s j is a corner of the boundary of G , then s j may not have any droplet to its left or right, in which case we let the corresponding spacing to be infinite. In the example in the figure, w 1 (s 1 ) = w 1 (s 17 ) = w 2 (s 8 ) = w 2 (s 9 ) = +∞, and all the other spacings are finite. Figure 7: A pictorial representation of the droplets involved in the definitions of f (w 1 , h, w 2 ), which represents the interaction energy of the infinite half strip of width h, with the two quarter spaces to its lower left and lower right. The spacings w 1 and w 2 represent the horizontal distances between the infinite half strip and the quarter space to its left and to its right, respectively.
Remark. e ∞ is the energy per unit vertical length of an infinite vertically striped configuration, and f (w 1 , h, w 2 ) is the interaction energy between the droplets in Fig. 7. n j ), i.e., to the one arising from the first two terms on the right side of (2.18). Therefore, all we have to prove is that the difference between the interaction terms in E G (B G ) and in Let us focus on a given slice g j and on the intersection of a given bubble β with this slice. The interaction energy of this portion of bubble with all the other bubbles in G , as it appears in E G (B G ), is where we recall that the * on the sum indicates the constraint that the bubbles β intersecting β after vertical translations should not be included in the sum. Eq. (2.20) can be bounded from below by summing only over the bubbles β having non zero intersection with g j : n j ) on the right side of (2.17) contains a term of that form, with the difference that δ β is replaced by the infinite vertical strip of the same width containing it. In fact, the interaction term in (2.18) satisfies i is the analogue of the set l i defined after (2.18), corresponding to the widths and spacings h n j associated to the slice g j . Therefore, what remains to be proved is that ,h(s j ),w 2 (s j ) ).
(2.22) Let us rewrite the left side of (2.22) as (2.23) Note that the horizontal boundary of L(δ β ) \ δ β consists of two segments s j , s k with h(s j ) = h(s k ) and s j above s k (these are the pairs of segments mentioned in Section 2.4.2; see Fig. 6), and there is a one-to-one correspondence between summing over β and summing over these pairs of segments s j , s k (which is of course the same as summing over all segments). Moreover, the set L(δ β ) \ δ β is equal to the union of two sets, each of which is a translation, and one also a reflection, of L + h(s j ) , that is for translations τ 1 β and τ 2 β , and r denoting reflection about the x 1 -axis.
To conclude, it is enough to note that i=1,...,Ng which implies the desired inequality (2.22), and thus completes the proof of Lemma 3.

Reflection positivity
We now show how to bound from below the right side of (2.17), and how to use the resulting estimate to conclude the proof of Lemma 1. The key step is to use reflection positivity to obtain a lower bound on the e ∞ term. This is an application of the block reflection positivity for one-dimensional spin systems worked out in [15]. The result is the following: To see this, note that e ∞ (h 1 , w 1 , . . . , h n ) = lim L→∞ H per Λ L (σ h 1 ,...,hn ), where: 1. given a spin configuration σ Λ L on Λ L = [1, L] ∩ Z, and σ L+1 ≡ σ 1 ; 2. the spins in the configuration σ h 1 ,...,hn are equal to −1 on the intervals ..,hn ) is a one-dimensional spin Hamiltonian with a reflection positive long-range interaction and periodic boundary conditions, of the class considered in [15,16]. Therefore, we can apply the chessboard estimate proved e.g. in the Appendix of [16]. As a result, using [16,Eqs. (A4)-(A5)] and recalling the fact that the spin configuration σ h 1 ,...,hn consists of blocks of alternating sign, of size h 1 , w 1 , . . . , h n , w n , with w n = w n (L) = L − (h 1 + w 1 + · · · + h n ), we get where e s (h) is the energy per site (as computed from H per Λ L , in the limit L → ∞) of the infinite periodic configuration consisting of blocks all of the same size h, and of alternating sign. Note that e s (h) is the same as the one defined in Section 1 for the two-dimensional model. Finally, to go from (2.29) to (2.26), observe that lim L→∞ w n (L)e s (w n (L)) = τ . This follows, e.g., from the explicit expression of e s (h), derived in [20, Appendix A]: Remark. From Eq. (2.30) it follows straight away that the optimal stripe width is h * = ((p − 2)A p |τ | −1 ) 1/(p−3) (1 + o(1)) as τ → 0, and also that e s (h * ) = p−3 p−2 τ h * (1 + o(1)).

Putting things together
Plugging (2.26) into (2.17) gives (2.31) Now, N g ≤ 1 2 |∂G | and it remains to show that the sum of the last two lines can be bounded from below by (const.)τ |∂G |+ 1 From the definition of f (w 1 , h, w 2 ) it easily follows that it can be bounded independently of h, as for a suitable constant C 2 . Moreover, from (2.30) it follows that for all w ≥ 1 and a suitable C 3 > 0. Note that the left side is non-negative, while the right side may be negative. A simple consequence of (2.33) is that Using this bound in (2.32) implies that the last line of (2.31) can be bounded as where the prime on the sum indicates the constraint that w i (s j ) < ∞. Now, every spacing w i (s j ) appears twice in the sum above (because every spacing is to the left or to the right of two different segments s j , s j ), hence To complete the proof of Lemma 1, note that |G | ≤ |G|, and |∂G | ≤ 2|∂G|, as already argued above.
Remark. The proof of (2.37) is valid for bubble configurations a bit more general than those considered here: in fact, we never used the fact that B G has no holes, in the sense explained in Section 1.1.3. The only property we really used is that the bubbles in B G are all rectangular with the same orientation.

Proof of Lemma 2
We proceed similarly to [20,Section 3]. The first step is to estimate the cost of erasing the bubbles with corners. Write n c (T ) = β∈B T ν c (β), with ν c (β) the number of corners associated with the bubble β, which may be an integer or a half-integer. Consider a bubble with ν c (β) > 0. Dropping the positive interaction of this bubble with the others, its contribution to the energy is bounded from below as Note that, in order for Γ β to be very long, the number of corners must be sufficiently large: in formulae (see [20,Eq.(3.10)] and following lines), If, as we are assuming, ν c (β) ≥ 1/2, then ν c (β) + 1 ≤ 3ν c (β), so that ν c (β) ≥ |Γ β |/(6 ). Inserting this back into (2.38) gives The first term on the right side is positive, and, therefore, can be dropped for a lower bound, if < (6 · 2 p/2 |τ |) −1 . Therefore, denoting by S T the subset of B T consisting of all the bubbles without corners, In order to estimate the energy of the corner-less configuration S T we proceed exactly as in the proof Lemma 1. Assume that the contours in S T are vertical. We deform the tile T by moving to the right the left vertical boundary of T , until it hits the left vertical contour of a bubble, and vice versa for the right vertical boundary. We call T and S T the new region and configuration obtained after the deformation. In passing from T, S T to T , S T we increase the energy by at most 2|τ | . Now we use the bound (2.37) which, as remarked after (2.37), is valid for all configurations consisting only of rectangular bubbles with the same orientation. The result is where h 1 , . . . , h n are the widths of the bubbles in S T , and w 1 , . . . , w n−1 their separations. It T contains a hole, then either one of the h i 's or w i 's is larger than /5, or, the width of T is smaller than 4 /5. In the first case, one of the terms h i e s (h i )−e s (h * ) or w i e s (w i )−e s (h * ) is larger than ( 2 /5)|e s (h * )|: to see this recall that ≥ c 0 h * , for a large enough constant c 0 , and use (2.30), which implies that e s (h) is positive for h ≥ /5 in this case. In the second case, the difference between e s (h * )|T | and e s (h * )|T | is larger than ( 2 /5)|e s (h * )|. In both cases, we get a gain at least ( 2 /5)|e s (h * )|, which is larger than (const.) 2 |τ | (p−2)/(p−3) . To conclude the proof, note that under the stated assumptions on (that is, c 0 h * ≤ ≤ (c 0 |τ |) −1 for a suitable constant c 0 ), the error term (2 + C)|τ | is smaller than c n c (T ) + |τ | (p−2)/(p−3) 2 χ hole (T ) , where c can be made as small as desired, by increasing c 0 (recall also that by definition of bad tile, either n c (T ) ≥ 1/2, or χ hole (T ) = 1). This concludes the proof.

A The higher-dimensional case
In this appendix we shall detail our main results in the case d ≥ 3, and explain the main differences in their proof as compared to the two-dimensional case. The starting point is a representation of the energy in terms of droplets as in (2.3), whose boundaries, separating plus spins from minus spins, consist now of d − 1 dimensional plaquettes. Tiles are now d-dimensional cubes of side length , and are used to divide space into good regions and bad cubes, with the good regions only containing "stripes" (i.e., quasi-one-dimensional regions of uniform spins, delimited by two flat parallel interfaces; they are slabs in d = 3), which can be oriented in d different directions.
Our first claim concerns the fact that the localization bound (2.6) still holds, with the obvious notion of "corner", namely d − 2 dimensional segments where two plaquettes with different orientation meet. The proof of the analogue of (2.6) in higher dimensions is essentially the same as in d = 2, and relies on the analogue of (2.10), whose proof is in [20,App. D].
The key bound in Lemma 1 for the good regions holds verbatim also for general d ≥ 2. After the modification from G to G , each stripe has a definite width h but will not be a cuboid, in general; it is bounded by a union of d − 1 dimensional cuboids s k with width h and all other dimensions equal to . As before, each s k comes with two numbers, w 1 (s k ) and w 2 (s k ), measuring the distance to the next slice in the direction perpendicular to the stripes. The analogues of the slices g j introduced in Sect. 2.4.2 are cylinders with base area d−1 and various heights, which are obtained by adding up the various stripe widths h i and their separation w i ; they are oriented perpendicular to the stripes. With these modified definitions, Eq. (2.17) still holds, with replaced by d−1 in front of the first term on the right side, and f (w 1 , h, w 2 ) now denoting the interaction energy as depicted in Fig. 7 where χ hole (T ) is equal to 1 if T contains a hole, and 0 otherwise.
Its proof is a rather straightforward adaptation of the one in d = 2, and we refer to [20,App. D], where the necessary changes were described in the case d = 3.
Since every portion of the boundary of a good region G i is adjacent to a bad tile, we have the bound N G i=1 |∂G i | ≤ 2d d−1 N B . In combination with the bounds above, this leads to the following generalization of Theorem 3. Theorem 3'. For given d ≥ 2, there exist positive constants C 0 , C 1 , ε such that, if J c − ε < J < J c and C 0 h * ≤ ≤ (C 0 (J c − J)) −1/(d−1) , then for every s ∈ {±1} Z d and every finite set X ⊂ Z d , (e s (h) − e s (h * ))A h (G i ), (A.5) where N c , N hole B , and G i are, respectively, the number of corners, the number of bad tiles containing a hole, and the good regions, associated with the infinite spin configuration σ = (s X , σ * X c ) coinciding with s X on X and with σ * on X c , defined via tiling with squares of side length as described above.
Theorem 3' implies the analogue of Theorem 1, i.e., the fact that striped configurations with stripe width h * are infinite volume ground states with trivial sectors for J close to J c , and also the analogue of Theorem 2, stating that all infinite volume ground states that are invariant under translations by d − 1 lattice vectors are characterized by the existence of an interface separating the cubic lattice Z d into two components, on each of which the configuration is perfectly striped.