Clock Model to the Xy Model: Emergence of Concentration Effects in the Variational Analysis

FROM THE N -CLOCK MODEL TO THE XY MODEL: EMERGENCE OF CONCENTRATION EFFECTS IN THE VARIATIONAL ANALYSIS

MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

Abstract. We investigate the relationship between the N -clock model (also known as planar Potts model or ZN -model) and the XY model (at zero temperature) through a Γ-convergence analysis as both the number of particles and N diverge. By suitably rescaling the energy of the N -clock model, we illustrate how its thermodynamic limit strongly depends on the rate of divergence of N with respect to the number of particles. The N -clock model turns out to be a good approximation of the XY model only for N suﬃciently large; in other regimes of N , we show with the aid of cartesian currents that its asymptotic behavior can be described by an energy which may concentrate on geometric objects of various dimensions.

Keywords: Γ-convergence, XY model, N -clock model, cartesian currents, topological singularities.

MSC 2010: 49J45, 49Q15, 26B30, 82B20.

1. Introduction Classical ferromagnetic spin systems on lattices represent fundamental models to understand phase transition phenomena. On the one hand, the study of their properties has motivated the introduction of new mathematical tools which have provided useful insights for a number of problems arising in different fields. On the other hand, many techniques borrowed from probability theory, mathematical analysis, topology, and geometry have con- tributed to a better understanding of the properties of these systems. In this paper we make use of fine concepts in geometric measure theory and in the theory of cartesian currents to understand the relationship between the XY -model and the N -clock model (also known as planar Potts model or ZN -model) within a variational framework. The N -clock model is a two-dimensional nearest neighbors ferromagnetic spin model on the square lattice in which the spin field is constrained to take values in a set of N equi-spaced points of S1 . For N large enough, it is usually considered as an approximation of the XY (planar rotator) model, for which the spin field is allowed to attain all the values of S1 . The asymptotic behavior of the N -clock model for large N has been considered by Fröhlich and Spencer in the seminal paper [30]. There the authors have proved that both the N -clock model (for N large enough) and the XY model present Berezinskii-Kosterlitz-Thouless transitions, i.e., phase transitions mediated by the formation and interaction of topological 1 2 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF singularities. The microscopic picture leading to the emergence of such topological phase transitions (first introduced in [14, 41, 42]) is a result of a nontrivial interplay between entropic and energetic effects that takes place at different length scales. This paper contributes to precisely relating the N -clock model and the XY model at zero temperature. Specifically, we show that the enhancement of symmetry, from the discrete one of the N -clock model to the continuous one of the XY model, comes along with concentration of energy on geometric objects of various dimension. This is achieved by studying several rescaled versions of the energy of the N -clock model as N diverges, through a coarse graining procedure which is made rigorous by Γ-convergence. A crucial step of this analysis is the choice of the topologies which best identify the relevant variables of the coarse grained model and lead to the effective description of the microscopic/mesoscopic geometry of the spin field. In contrast to the XY model, the sole study of the distributional Jacobian of the spin field turns out to provide not enough information on the concentration effects of the energy; we shall see how these effects can be detected by cartesian currents, for the first time introduced in the context of lattice spin models. In what follows we present the model and our main results. We consider a bounded, open set with Lipschitz boundary Ω ⊂ R2 . Given a small parameter ε > 0, we consider the 2 2 square lattice εZ and we define Ωε := Ω ∩ εZ . The classical XY model is defined on spin 1 fields u:Ωε → S by X − ε2u(εi) · u(εj) , (1.1) hi,ji where a · b denotes the scalar product between a, b ∈ R2 and the sum is taken over ordered 2 2 pairs of nearest neighbor hi, ji, i.e., (i, j) ∈ Z ×Z such that |i − j| = 1 and εi, εj ∈ Ωε . The variational analysis of the XY model is part of a larger program devoted to the study of systems of spins with continuous symmetry [4, 7, 8, 13, 23, 26, 27, 49, 25]. 2π Here we consider an additional parameter Nε ∈ or, equivalently, θε := . The N Nε 1 integer Nε represents the number of points in the discretization Sε of the codomain S , while θε is the smallest non-zero angle between two neighboring spin values. More precisely, we set

Sε := {exp(ιkθε): k = 0,...,Nε − 1} , where ι is the imaginary unit. The admissible spin ﬁelds we consider here are only those 1 taking values in the discrete set Sε , i.e., we consider the energy deﬁned for every u:Ωε → S by  X 2 − ε u(εi) · u(εj) if u:Ωε → Sε , Fε(u) := hi,ji +∞ otherwise.

For Nε = N ∈ N, with N independent of ε, the spin system described by the energy Fε is usually referred to as N -clock model, cf. [30]. The particular case where N = 2 is the so-called Ising system. The analysis of Ising-type systems with short-range interaction has been the object of many recent papers in analysis and statistical mechanics [38, 17, 24, 22, 3]. See also [2, 5, 9, 19] for the long-range case. The minimum of the functional deﬁned in (1.1) is achieved on constant spin ﬁelds. For this reason it is customary in this setting to refer the energy to its minimum (cf. [4, Theorem 2]) and to introduce the functional X 1 X XY (u) = − ε2 u(εi) · u(εj) − 1 = ε2|u(εi) − u(εj)|2. ε 2 hi,ji hi,ji

This suggests to study the excess energy Eε(u) := Fε(u) − min Fε , under those scalings 1 κε → 0 for which Eε has a nontrivial variational limit in the sense of Γ-convergence, κε cf. [28, 18]. In order to understand such scalings, it is convenient to recast the energy as ( XYε(u) if u:Ωε → Sε , Eε(u) := (1.2) +∞ otherwise. FROM THE N -CLOCK MODEL TO THE XY MODEL 3

One can expect that the relevant scaling κε is affected by Nε , as it emerges in the 1 two limiting scenarios Nε = 2 and Sε = S (formally corresponding to Nε = +∞). In fact, for the Ising system, i.e., Nε = 2, it has been shown in [3] that, choosing κε = ε, a 1 configuration uε with equibounded energies ε Eε(uε) can be identified, as ε → 0, with a partition of Ω in two regions, in each of which the spin is constantly equal to one of the 1 two values (1, 0),(−1, 0). Moreover, the energy ε Eε(uε) approximates an interfacial-type energy, whose anisotropy reflects the symmetries of the underlying lattice. In contrast, for 1 the XY system, i.e., Sε = S , it has been shown in [4, Example 1] that no interfacial-type 2 energy emerges at any scaling κε ε . To show this, the authors provide an example that 1 we recall here. In this example, let Ω = B1/2(0) be the ball of radius 2 centered at 0, let 1 v1 = exp(ιϕ1), v2 = exp(ιϕ2) ∈ S , and let us define for x = (x1, x2) ( v if x ≤ 0 , u(x) := 1 1 (1.3) v2 if x1 > 0 . 1 1 1 2 Then we can construct uε → u in L (Ω; ) such that XYε(uε) → 0 for all κε ε in S κε the following way: for εi = (εi1, εi2) ∈ Ωε we define  v1 if εi1 ≤ 0 ,  εi1 uε(εi) := exp ι (ϕ1 − ϕ2) 1 − + ϕ2 if 0 < εi1 ≤ ηε , (1.4) ηε  v2 if εi1 > ηε , where ηε → 0 is chosen below. Then 2 1 ε ηε ε XYε(uε) ∼ 1 − cos (ϕ1 − ϕ2) ∼ . (1.5) κε ηε κε ηεκε 2 ε2 Thanks to the assumption κε ε , it is always possible to find ηε → 0 such that ηε . κε As a consequence, the angle between two neighboring spins uε(ε(i + e1)) and uε(εi) has ε κε 1 modulus |ϕ1 − ϕ2| . This construction, always possible in the case Sε = , may ηε ε S 1 κε not be feasible when Sε 6= S if the minimal angle θε satisfies θε & ε . Hence one cannot exclude the possibility of finding a nontrivial scaling κε such that interfacial-type energies 1 arise in the limit of Eε(uε). κε To better understand the behavior of Eε in the constrained setting uε :Ωε → Sε , we ε modify the construction in (1.4) by choosing ηε so that |ϕ1 − ϕ2| = θε , i.e., ηε = ηε ε |ϕ1 − ϕ2| , see Figure 1. As a result θε 1 ε εθε Eε(uε) ∼ 1 − cos(θε) |ϕ1 − ϕ2| ∼ |ϕ1 − ϕ2| . κε θεκε κε

This suggests that the nontrivial scaling κε = εθε leads to a ﬁnite energy proportional to |ϕ1 − ϕ2|. The construction can be optimized by choosing the angles ϕ1 and ϕ2 in 1 such a way that |ϕ1 − ϕ2| equals the geodesic distance on S between v1 and v2 , namely 1 dS (v1, v2).

v1 = exp(ιϕ1) θε v2 = exp(ιϕ2)

ε ηε = |ϕ2 − ϕ1| θε

Figure 1. Construction which shows that 1 E approximates the geodesic distance between εθε ε the two values v1 and v2 of a pure-jump function. During the transition between v1 and v2 in the strip of size η = |ϕ − ϕ | ε the minimal angle between two adjacent vectors is θ . ε 1 2 θε ε 4 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

1 The fact that dS (v1, v2) is the total variation (in the sense of [11, Formula (2.11)]) of the 1 S -valued pure-jump function u defined in (1.3) suggests that at the scaling κε = εθε the 1 1 1 Γ-limit of Eε might be finite on the class BV (Ω; ) of -valued functions of bounded εθε S S variation. This intuition is confirmed in Proposition 4.1, where we prove that a sequence uε :Ωε → Sε (here and in what follows we identify it with its piecewise constant extension to Ω) satisfying the bound 1 Eε(uε) ≤ C εθε 1 1 is compact in BVloc(Ω; S ). More precisely, there exists a u ∈ BV (Ω; S ) such that, up 1 1 to a subsequence, uε → u strongly in L (Ω; S ). This compactness goes along with the 1 first result of the paper, Theorem 1.1 below, which states that Eε(uε) Γ-converges to a εθε functional that can be expressed on BV (Ω; S1) as an anisotropic total variation reflecting the symmetries of the underlying lattice. As it will be explained below, this result is true only in the regime ε| log ε| θε 1. To state the theorem, we need to introduce some notation, cf. [12]. Given a function u ∈ BV (Ω; S1), its distributional derivative Du can be decomposed as Du = ∇uL2 + (c) + − 1 2 D u + (u − u ) ⊗ νuH Ju , where ∇u denotes the approximate gradient, L is the Lebesgue measure in R2 ,D(c)u is the Cantor part of Du, H1 is the 1-dimensional Hausdorff 1 measure, Ju is the H -countably rectifiable jump set of u oriented by the normal νu , and + − u and u are the traces of u on Ju . By | · |1 we denote the 1-norm on vectors and by | · |2,1 the anisotropic norm on matrices given by the sum of the Euclidean norms of the columns.

Theorem 1.1 (Regime ε| log ε| θε 1). Assume that ε| log ε| θε 1. Then the following results hold:

i) (Compactness) Let uε :Ωε → Sε be such that 1 Eε(uε) ≤ C. εθε Then there exists a subsequence (not relabeled) and a function u ∈ BV (Ω; S1) such 1 2 that uε → u in L (Ω; R ). 1 ii)( Γ-liminf inequality) Assume that uε :Ωε → Sε and u ∈ BV (Ω; S ) satisfy uε → u in L1(Ω; R2). Then Z Z (c) − + 1 1 1 |∇u|2,1 dx + |D u|2,1(Ω) + dS (u , u )|νu|1 dH ≤ lim inf Eε(uε) . ε→0 εθε Ω Ju 1 iii)( Γ-limsup inequality) Let u ∈ BV (Ω; S ). Then there exists a sequence uε :Ωε → Sε 1 2 such that uε → u in L (Ω; R ) and Z Z 1 (c) − + 1 1 lim sup Eε(uε) ≤ |∇u|2,1 dx + |D u|2,1(Ω) + dS (u , u )|νu|1 dH . ε→0 εθε Ω Ju

The optimality of the regime ε| log ε| θε can be explained as follows. If θε ε| log ε| and uε :Ωε → Sε is a sequence with equibounded energies 1 Eε(uε) ≤ C, εθε then, by (1.2), we have

1 εθε 1 θε 2 XYε(uε) = 2 Eε(uε) ∼ ∼ 0 . (1.6) ε | log ε| ε | log ε| εθε ε| log ε| This entails that the formation of vortices is not allowed in this regime. Indeed, it has been 1 proven in [4] that the Γ-limit of the functionals ε2| log ε| XYε agrees with the total variation of a measure µ supported on ﬁnitely many points, usually referred to as vortices. (Such result drew inspiration from the literature on the Ginzburg-Landau functional, cf. [16, 46, 39, 40, 47, 1, 48]; a variational equivalence between the two models has been proven in [7].) FROM THE N -CLOCK MODEL TO THE XY MODEL 5

εi

µuε = δεi

2 Figure 2. Example of discrete vorticity measure equal to a Dirac delta on the point εi ∈ εZ . By following a closed path on the square of the lattice with the top-right corner in εi , the spin 1 ﬁeld covers the whole S . The discrete vorticity measure can only have weights in {−1, 0, 1} .

1 A bound ε2| log ε| XYε(uε) ≤ C yields compactness for the discrete vorticity measure µuε associated to the spin field uε , where µuε counts the winding number of uε at each point of 2 f the lattice εZ (see Figure 2 and cf. (2.7) for the precise definition). More precisely, µuε * µ up to a subsequence in the flat convergence (i.e., in duality with Lipschitz functions with PN compact support, see (2.8)), where µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z. The measure µ represents the vortex-like singularities of the spin field uε as ε goes to zero. Moreover, the Γ-liminf inequality in [4, Theorem 1] yields 1 2π|µ|(Ω) ≤ lim inf XYε(uε) . (1.7) ε→0 ε2| log ε| The inequality above is optimal. This can be proven using the fact that the energy of 1 x−x0 the discretization uε :Ωε → of the prototypical vortex-like function , x0 ∈ Ω, S |x−x0| behaves as 1 XY (u ) ∼ 2π . (1.8) ε2| log ε| ε ε Combining inequality (1.7) with (1.6) we conclude that µ is the zero measure, i.e., f µuε * 0 . (1.9)

x−x0 This agrees with the fact that the the discretization vε :Ωε → Sε of carries an energy |x−x0| 1 E (v ) ∼ 2π , (1.10) ε2| log ε| ε ε cf. (4.78) for the rigorous bound. Notice that, in this case, the discretization vε is meant both in the domain and in the codomain. The emergence of the additional constraint on the absence of vortices in the limit suggests the possibility that the lower bound in Theorem 1.1- (ii) might be improved. This phenomenon does not occur in the regime ε| log ε| θε , where one has no information on the flat limit of µuε . Since smooth maps u:Ω → S1 provide the simplest example of S1 -valued maps with no 1 vortices, the problem of finding a better lower bound for Eε in the regime θε ε| log ε| εθε reminds that of finding the L1 -lower semicontinuous envelope of the functional with linear growth Z 1 1 1,1 1 |∇u| dx , u ∈ C (Ω; S ) ∩ W (Ω; S ) . (1.11) Ω Even though in this paper we are interested in anisotropic energies as the one obtained in Theorem 1.1, we notice here that they share the same features as (1.11). For this reason, not to overburden the reader with additional notation, in what follows we recall the analysis done for (1.11) in [33] (see also [34] for a generalization to manifold-valued maps). There the authors show that the functional Z Z (c) − + 1 1 1 |∇u| dx + |D u|(Ω) + dS (u , u ) dH , u ∈ BV (Ω; S ) ,

Ω Ju 6 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF is, in general, strictly smaller than the L1 -lower semicontinuous envelope of (1.11). For x example, let Ω = B be the ball of radius 1 centered at 0 and u(x) = |x| ; then it is x 1 1 not possible to find a sequence of smooth maps uk such that uk → |x| in L (Ω; S ) and R R x 1 Ω |∇uk| → Ω ∇ |x| . Nonetheless, it is possible to find a sequence of smooth functions x uk → |x| such that Z Z x |∇uk| dx → ∇ |x| dx + 2π , (1.12) Ω Ω where 2π is reminiscent of the energy due to the necessary concentration of ∇uk on a radius of Ω = B . The correct framework to carry out this analysis in its full generality is that of cartesian currents cart(Ω×S1) (cf. [31, 32]), which can be characterized as those currents obtained as weak limits of graphs of smooth maps from Ω to S1 (see Theorem 3.3). Given a sequence 1 1 1,1 1 1 ∗ 2 uk ∈ C (Ω; S ) ∩ W (Ω; S ) and u ∈ BV (Ω; S ) with uk * u in BV (Ω; R ) there exists a 1 T ∈ cart(Ω×S ) such that the graphs Guk of uk converge weakly (in the sense of currents) to T . The currents associated to the graphs of smooth maps are boundaryless. Such a property is inherited by the limit object T , i.e., ∂T = 0. For this reason, in general, the current T is different from the graph Gu of the limit map u, which may have a nontrivial boundary. Nevertheless, T can be represented as

1 T = Gu + L× S , (1.13) J K where S1 is the current defined by integration over S1 and L is an integer multiplicity 1- rectifiableJ K current, which keeps track of the possible concentration of ∇uk on 1-dimensional sets (see also Figure 3). The L1 -lower semicontinuous envelope of (1.11) is then (see, e.g., [34, Theorem 7.4]) Z (c) 1 |∇u| dx + |D u|(Ω) + I(u; Ω) , u ∈ BV (Ω; S ) , (1.14) Ω where I(u; Ω) is a surface energy depending on the jump part of Du and obtained by optimizing in the 1-current L. The latter is reminiscent of the energy due to concentration effects of ∇uk and it satisfies Z − + 1 1 dS (u , u ) dH ≤ I(u; Ω) .

(See [37] to interpret the energy (1.14) in terms of the optimal lifting problem in BV .) x Note that in the example (1.12) I(u; Ω) 6= 0, even if |x| has no jumps, and, more precisely, I(u; Ω) = 2π , obtained with L supported on a radius of Ω = B . If L = 0, then the inequality above is actually an equality.

1 It can be defined, e.g., by uk(x) = exp(ιϕk(x)), where ϕk(x) is the smooth function defined as follows. Define the function ψek in polar coordinates (ρ, ϑ) by ψek(ρ, ϑ) = ϑ, if ϑ ∈ [0, 2π − 1/k], and ψek(ρ, ϑ) = ∞ (2πk − 1)(2π − ϑ), if ϑ ∈ [2π − 1/k, 2π]. Let ϕek be a suitable regularization of ψek . Let ζk ∈ Cc (B2/k) be a cut-off function such that 0 ≤ ζk ≤ 1, ζk ≡ 1 on B1/k , |∇ζk| ≤ 2k and define ϕk = ϕek(1 − ζk). Then

Z Z Z 1 Z 2π−1/k Z 1 Z 2π |∂ϑψek| |∇uk| dx ∼ dx = dϑ dρ + (2πk − 1) dϑ dρ → 2π + 2π , B\B2/k B\B2/k ρ 2/k 0 2/k 2π−1/k Z Z |∇uk| dx ≤ |∇ϕek|(1 + ζk) + |ϕek||∇ζk| dx → 0 . B2/k\B1/k B2/k\B1/k FROM THE N -CLOCK MODEL TO THE XY MODEL 7

− S1 J K

u− L

u 0

Ω

1 Figure 3. Depiction of a cartesian current of the form T = Gu − L× S . In the picture, Ω is the unit disc centered at the origin. The function u has no jumpsJ andK presents a vortex- like singularity, turning once counterclockwise around the origin. In particular, the graph 1 Gu has a hole, namely, ∂Gu = −δ0× S . The current T features a concentration part 1 J K −L× S . It is supported on a radius of the ball and is characterized by a vertical part (in J K 1 − + gray) that connects clockwise in S the (equal) traces u and u of u on the two sides of the radius. Note that the vertical part is not given by the geodesic connecting u− and u+ . The concentration part is needed to compensate the boundary of the graph Gu , so that ∂T = 0 . 1 1 Indeed, −∂L× S = δ0× S = −∂Gu inside Ω . In conclusion, the current T does not turn J K J K 1 around the origin. In this ﬁgure, for H -a.e. x in the support of L , the length `T (x) is 2π .

1 We now discuss the asymptotic behavior of Eε when ε θε ε| log ε|. (The εθε importance of the lower bound ε θε will be highlighted after the statement of Theorem 1.3 1 below.) In this regime, the limit of Eε shares strong similarities with (1.14). To a spin εθε 1 field uε :Ωε → Sε with equibounded energy Eε(uε) ≤ C we associate a current Gu , εθε ε 1 whose main feature is its relation with the discrete vorticity measure ∂Guε = −µuε × S , cf. J K Proposition 3.11. Thanks to the latter property and (1.9), the limit T of the currents Guε satisfies ∂T = 0 and, more precisely, T ∈ cart(Ω×S1). 1 To define the energy obtained as Γ-limit of Eε in this regime, we need to introduce εθε further quantities. It will feature an anisotropic surface term J (u; Ω), playing the same role of I(u; Ω) in (1.14). More precisely, for u ∈ BV (Ω; S1) we consider (see (4.15) for the precise definition) Z 1 1 J (u; Ω) = inf `T (x)|νT (x)|1 dH (x): T ∈ cart(Ω×S ) ,T as in (1.13) for some L .

JT (1.15) Here JT is the 1-dimensional jump-concentration set of T oriented by the normal νT , accounting for both the jump set of u and the support of the concentration part L in the decomposition (1.13). At each point x ∈ JT , the current T has a vertical part, given by a 1 curve in S which connects the traces of u on the two sides of JT ; `T (x) is its length. We are now in a position to state the theorem.

Theorem 1.2 (Regime ε θε ε| log ε| – no vortices). Assume that ε θε ε| log ε|. Then the following results hold:

i) (Compactness) Let uε :Ωε → Sε be such that 1 Eε(uε) ≤ C. εθε f Then µuε * 0. In addition, there exists a subsequence (not relabeled) and a function 1 1 2 u ∈ BV (Ω; S ) such that uε → u in L (Ω; R ). 8 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

1 ii)( Γ-liminf inequality) Assume that uε :Ωε → Sε and u ∈ BV (Ω; S ) are such that 1 2 uε → u in L (Ω; R ). Then Z (c) 1 |∇u|2,1 dx + |D u|2,1(Ω) + J (u; Ω) ≤ lim inf Eε(uε) . ε→0 εθε Ω 1 iii)( Γ-limsup inequality) Let u ∈ BV (Ω; S ). Then there exists a sequence uε :Ωε → Sε 1 2 such that uε → u in L (Ω; R ) and Z 1 (c) lim sup Eε(uε) ≤ |∇u|2,1 dx + |D u|2,1(Ω) + J (u; Ω) . ε→0 εθε Ω 1 In Theorem 1.2 it is proven that the bound Eε(uε) ≤ C forces the absence of vortices εθε in the limit allowing, instead, for BV -type concentration effects. This raises the natural 1 question of finding a bound on Eε(uε) which allows for configurations that also concen- εθε trate energy on finitely many vortices. The energy needed for the formation of a single vortex of type x−x0 can be deduced |x−x0| from (1.10) (cf. (4.78) for the rigorous bound). Indeed, for the discretization vε :Ωε → Sε of x−x0 we have that |x−x0| 1 ε Eε(vε) ∼ 2π| log ε| . (1.16) εθε θε Notice that, since θε ε| log ε|, the right-hand side diverges as ε → 0. Estimate (1.16) suggests that a bound of the type 1 ε Eε(uε) ≤ C| log ε| (1.17) εθε θε is compatible with the formation of finitely many vortices. Indeed, assuming (1.17), we f PN shall prove in Proposition 4.10 that µuε * µ (up to a subsequence), with µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z, and 2π|µ|(Ω) ≤ C . A finer analysis can be done by assuming the more precise bound 1 ε Eε(uε) − 2πM| log ε| ≤ C, (1.18) εθε θε 1 namely, assuming finiteness of the excess energy obtained by removing from Eε(uε) the εθε leading order term corresponding to a vortex configuration µ with |µ|(Ω) = M . Condition (1.18) suggests that a sequence uε may display in the limit simultaneously vortex-type and BV -type concentration effects. More precisely, (1.18) implies an energy f PN bounded as in (1.17) and, in particular, that µuε * µ, with µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z, and |µ|(Ω) ≤ M . In fact, the interesting situation to study is |µ|(Ω) = M . In that case, once the diverging energy 2πM| log ε| ε has been saturated through the formation θε 1 of the vortices µ, a finite energy Eε is still accessible to the system and it might lead εθε to BV -type concentration effects as in Theorem 1.2. As already discussed, to a spin field 1 uε :Ωε → Sε we associate a current Guε with ∂Guε = −µuε × S . This induces a nontrivial J K 1 constraint on the current T given by the limit of the Guε , namely ∂T = −µ× S . This condition couples the vortex-type and BV -type concentration effects displayed byJ theK spin field. 1 The next theorem shows that the limit of Eε features a coupling term J (µ, u; Ω) εθε generalizing J (u; Ω) in (1.15) given by Z 1 J (µ, u; Ω) := inf `T (x)|νT (x)|1 dH (x): T ∈ Adm(µ, u; Ω) ,

JT where Adm(µ, u; Ω), deﬁned in (4.22), is a suitable class of currents T satisfying, in particular, the constraint ∂T = −µ× S1 . Observe that J (µ, u; Ω) reduces to J (u; Ω) for µ = 0. J K

Theorem 1.3 (Regime ε θε ε| log ε| – M vortices). Assume that ε θε ε| log ε|. Then the following results hold: FROM THE N -CLOCK MODEL TO THE XY MODEL 9

i) (Compactness) Let M ∈ N and let uε :Ωε → Sε be such that 1 ε Eε(uε) − 2πM| log ε| ≤ C. εθε θε PN Then there exists a measure µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z, such that (up f to a subsequence) µuε * µ and |µ|(Ω) ≤ M . If, in addition, |µ|(Ω) = M , then 1 there exists a function u ∈ BV (Ω; S ) such that (up to a subsequence) uε → u in L1(Ω; R2). PN ii)( Γ-liminf inequality) Let uε :Ωε → Sε and let µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z f 1 with |µ|(Ω) = M . Assume that µuε * µ. Let u ∈ BV (Ω; S ) be such that uε → u in L1(Ω; R2). Then Z (c) 1 ε |∇u|2,1 dx + |D u|2,1(Ω) + J (µ, u; Ω) ≤ lim inf Eε(uε) − 2πM| log ε| . ε→0 εθε θε Ω PN iii)( Γ-limsup inequality) Let µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z with |µ|(Ω) = M and 1 f let u ∈ BV (Ω; S ). Then there exists a sequence uε :Ωε → Sε such that µuε * µ, 1 2 uε → u in L (Ω; R ), and Z 1 ε (c) lim sup Eε(uε) − 2πM| log ε| ≤ |∇u|2,1 dx + |D u|2,1(Ω) + J (µ, u; Ω) . ε→0 εθε θε Ω

To motivate the condition ε θε , it is convenient to recall the first order analysis of the XY model developed in [8, Section 4] (see also [16, 48, 10] for the Ginzburg-Landau model). f PM Let us suppose that µuε * µ, where µ = h=1 dhδxh , with dh = ±1. By (1.7)–(1.8), the least energy needed to form M vortices is 2πM| log ε| and additional information on the physical system are obtained by carrying out a finer analysis under the bound on the excess energy 1 XY (u ) − 2πM| log ε| ≤ C. ε2 ε ε Specifically, in [8, Theorem 4.2] the following asymptotic expansion of the energy 1 XY (u ) ∼ 2πM| log ε| + (µ) + Mγ (1.19) ε2 ε ε W is rigorously proven in a variational sense. In the formula, W is a Coulomb-type interaction potential referred to as renormalized energy, while γ is the core energy carried by each vortex, cf. (7.2) and (7.4) for the precise definitions. Analogously, the result stated in Theorem 1.3 leads to the formal expansion 1 θ Z E (u ) ∼ 2πM| log ε| + ε |∇u| dx + |D(c)u| (Ω) + J (µ, u; Ω) . (1.20) ε2 ε ε ε 2,1 2,1 Ω

If ε θε , the BV -type energy in (1.20) has a divergent pre-factor, in contrast to the ﬁnite term W(µ) + Mγ in (1.19). If, instead, θε ε, we expect the following formal asymptotic behavior to hold true: 1 E (u ) ∼ 2πM| log ε| + (µ) + Mγ . ε2 ε ε W This intuition is conﬁrmed in the next theorem.

Theorem 1.4 (Regime θε ε). Assume that θε ε. Then the following results hold: i) (Compactness) Let M ∈ N and let uε :Ωε → Sε be such that 1 E (u ) − 2πM| log ε| ≤ C. ε2 ε ε PN Then there exists a measure µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z, with |µ|(Ω) ≤ M f such that (up to a subsequence) µuε * µ. Moreover, if |µ|(Ω) = M , then |dh| = 1. 10 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

PM ii)( Γ-liminf inequality) Let µ = h=1 dhδxh , xh ∈ Ω, |dh| = 1, and let uε :Ωε → Sε f be such that µuε * µ. Then 1 W(µ) + Mγ ≤ lim inf Eε(uε) − 2πM| log ε| . ε→0 ε2 PM iii)( Γ-limsup inequality) Let µ = h=1 dhδxh , xh ∈ Ω, |dh| = 1. Then there exists f uε :Ωε → Sε such that µuε * µ and 1 lim sup 2 Eε(uε) − 2πM| log ε| ≤ W(µ) + Mγ . ε→0 ε We remark that, in contrast to Theorem 1.3, only vortices with multiplicity ±1 are allowed in Theorem 1.4. The proof of the Γ-limsup inequality in Theorem 1.4 can be obtained by a careful adap- tation of some of the techniques adopted in the literature about the first order analysis of the Ginzburg-Landau [16, 48, 10] and of the classical XY model [8, 23]. In contrast, the construction of the recovery sequence in the proof of Theorems 1.1, 1.2, 1.3 turns out to be a significantly more delicate task. We highlight here some of its main technical difficulties, referring to Section 4 for a detailed explanation. Given u ∈ BV (Ω; S1), we define its recovery sequence following a gradual approximation procedure, which involves a series of steps of increasing complexity. At each of these steps, the map u is modified without essentially changing the energy. The first main issue is to regularize the map u. It is known that, in general, a map belonging to BV (Ω; S1) cannot be approximated in energy by means of S1 -valued smooth functions. Nonetheless, the result in [15] (see also [6]) guarantees the density of S1 -valued maps that are smooth outside finitely many point-singularities (whose number and position are however not provided). Such a regularization suffices to proceed with the construction in Theorem 1.1; in contrast, a more precise approximation is necessary for the purposes of Theorems 1.2, 1.3. Indeed, in that case the limit energy depends on the vorticity measure µ, which needs to be precisely related to singularities emerging in the regularization of u. This is achieved using the Approximation Theorem for cartesian currents, cf. Lemma 4.17. The next main issue is to construct a recovery sequence uε for such a regularization of u. Close to each singularity, uε is defined by discretizing on the ε-lattice a proper translation x of |x| and then by projecting its values on Sε . The energy carried by this discrete spin field close to a singularity is sensitive to the regime of θε : if ε| log ε| θε 1 as in Theorem 1.1, its energy is negligible; if, instead, ε θε ε| log ε| as in Theorems 1.2, 1.3, then the energy diverges according to (1.16). Far from the singularities, the problem reduces to the construction of a recovery sequence for a smooth S1 -valued map. This can be further simplified to the case of a piecewise constant S1 -valued map by introducing a mesoscopic scale into the problem, see Lemma 4.13. For such maps, the construction is a refinement of that in the example presented above and depicted in Figure 1. The most delicate step is to merge the different parts of the recovery sequence close to and far from the singularities. This is achieved in the proof of Proposition 4.22 (Step 2) by a careful interpolation on dyadic layers of mesoscopic squares, whose size is chosen to x be smaller for layers closer to the singularity. At each layer generation, |x| is sampled at a different mesoscopic length-scale. The latter is optimized in order to provide the correct control on the energy in progressing from each layer to the next one. We collect here the results proved in this paper. Theorems 1.1, 1.2, 1.3, and 1.4 cover the regimes of θε from θε finite to θε ε. For each regime we have scaled the Nε -clock energy Eε to highlight concentration effects of the spin field. Accordingly, we have shown that the energy may concentrate on sets of various dimensions. On the one hand, if θε is finite, the physical system is an Ising-type system with finitely many phases, for which it is well known that the energy concentrates on 1-dimensional domain walls separating phases of constant spin, cf. [19, 45]. On the other hand, we prove in Theorem 1.4 that, for θε ε, the Nε -clock model approximates the XY model (in FROM THE N -CLOCK MODEL TO THE XY MODEL 11 the sense of the asymptotic expansion (1.19)). In particular, the spin field develops 0- dimensional singularities, the vortices. For ε θε 1, we observed two intermediate 1 phenomena. When θε 1, every value of S is admissible for the limit spin field, and the limit energy detects both the formation of 1-dimensional domain walls and diffuse transitions. As shown in Theorem 1.1, this completes the description of the limit energy in the case ε| log ε| θε 1. If, instead, ε θε ε| log ε|, the previous phenomenon is coupled in a nontrivial way with the possible formation of vortices. The precise behavior of the energy is described in Theorems 1.2–1.3. All our results are summarized in Table 1. We remark that the coexistence of singularities of two different dimensions has been already included in the framework of XY -type models in [13] and of Ginzburg-Landau-type energies in [35]. While in the aforementioned papers the presence of singularities of various dimensions is enforced by suitably modifying the multi-well potential or by adding a surface term to the usual energy, here they appear as a result of the dependence on ε of the codomain.

Regime Energy bound Limit of µuε Energy behavior

1 θε ﬁnite ε Eε ≤ C not relevant interfaces

1 ε| log ε| θε Eε ≤ C not relevant BV εθε

1 f ε θε ε| log ε| Eε ≤ C µu * 0 BV + concentration εθε ε

vortices 1 ε f ε θε ε| log ε| Eε − 2πM| log ε| ≤ C µu * µ + εθε θε ε BV + concentration

1 f θε ε ε2 Eε − 2πM| log ε| ≤ C µuε * µ XY

Table 1. In this table we summarize our results. By “Ising” we mean that the energy concentrates on 1 -dimensional domain walls that separate the different phases, while “ BV ” denotes a BV -type total variation. The expression “ BV +concentration” indicates the presence in a BV -type energy of a surface term of the form J (u; Ω) or J (µ, u; Ω) which accounts for concentration effects on 1-dimensional surfaces. By “vortices” we mean that a diverging energy is carried by the system for the creation of vortex-like singularities in the limit. Finally, “ XY ” expresses the fact that the energy is a good approximation (at first order) of the classical XY model.

In this paper we also study the critical regime θε ∼ ε| log ε| in Section 5. In this case, 1 ε 1 0 a bound Eε(uε) − 2M| log ε| ≤ C is equivalent to both 2 Eε(uε) ≤ C and εθε θε ε | log ε| 1 0 0 Eε(uε) ≤ C for some other constant C . The former bound is compatible with the εθε f formation of vortices and, in fact, µuε * µ up to a subsequence; the latter bound implies compactness for uε in BV and for the currents Guε . In this setting, the excess energy 1 ε Eε(uε) − 2M| log ε| cannot diverge to −∞, thus we do not need to require additional εθε θε assumptions on the mass of µ. Indeed, we prove the following result.

Theorem 1.5 (Regime θε ∼ ε| log ε|). Assume that θε = ε| log ε|. Then the following results hold:

i) (Compactness) Let uε :Ωε → Sε be such that 1 E (u ) ≤ C. ε2| log ε| ε ε

PN Then there exists a measure µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z, such that (up to a f 1 subsequence) µuε * µ and there exists a function u ∈ BV (Ω; S ) such that (up to 1 2 a subsequence) uε → u in L (Ω; R ). 12 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

PN ii)( Γ-liminf inequality) Let uε :Ωε → Sε , let µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z, and 1 f 1 2 let u ∈ BV (Ω; S ). Assume that µuε * µ and uε → u in L (Ω; R ). Then Z (c) 1 |∇u|2,1 dx + |D u|2,1(Ω) + J (µ, u; Ω) + 2π|µ|(Ω) ≤ lim inf Eε(uε) . ε→0 ε2| log ε| Ω PN 1 iii)( Γ-limsup inequality) Let µ = h=1 dhδxh , xh ∈ Ω, dh ∈ Z and let u ∈ BV (Ω; S ). f 1 2 Then there exists a sequence uε :Ωε → Sε such that µuε * µ, uε → u in L (Ω; R ), and Z 1 (c) lim sup 2 Eε(uε) ≤ |∇u|2,1 dx + |D u|2,1(Ω) + J (µ, u; Ω) + 2π|µ|(Ω) . ε→0 ε | log ε| Ω

In the critical regime θε ∼ ε one can adapt Step 1 in the proof of Proposition 4.22 to 1 show that the variational limit of ε2| log ε| Eε is given by 2π|µ|(Ω), where µ is the vorticity 1 measure. When the energy is scaled as ε2 Eε one may expect from (1.19)–(1.20) a nontrivial interplay between the BV -type energy and the renormalized and core energies. This appears to deserve a separate analysis and is not treated in this paper.

2. Notation and preliminaries We denote the imaginary unit by ι. We shall often identify R2 with the complex plane C. 2 Given a vector a = (a1, a2) ∈ R , its 1-norm is |a|1 = |a1| + |a2|. We define the (2, 1)-norm 2×2 of a matrix A = (aij) ∈ R as the sum of the Euclidean norms of its columns, i.e., 2 2 1/2 2 2 1/2 |A|2,1 := a11 + a21 + a12 + a22 . 1 1 1 If u, v ∈ S , their geodesic distance on S is denoted by dS (u, v). It is given by the angle 1 in [0, π] between the vectors u and v , i.e., dS (u, v) = arccos(u · v). Observe that 1 1 1 2 |u − v| = sin 2 dS (u, v) , (2.1) and, in particular, we have the equivalence of distances in the sense that π |u − v| ≤ d 1 (u, v) ≤ |u − v| . (2.2) S 2 αε Given two sequences αε and βε , we write αε βε if limε→0 = 0. We will use the βε notation deg(u)(x0) to denote the topological degree of a continuous map u ∈ C(Bρ(x0) \ 1 {x0}; S ), i.e., the topological degree of its restriction u|∂Br (x0) , independent of r < ρ. We denote by Iλ(x) the half-open squares given by 2 Iλ(x) = x + [0, λ) . (2.3) 2.1. BV-functions. In this section we recall basic facts about functions of bounded variation. For more details we refer to the monograph [12]. Let O ⊂ Rd be an open set. A function u ∈ L1(O; Rn) is a function of bounded variation if its distributional derivative Du is given by a finite matrix-valued Radon measure on O. In that case, we write u ∈ BV (O; Rn). n n The space BVloc(O; R ) is defined as usual. The space BV (O; R ) becomes a Banach space when endowed with the norm kukBV (O) = kukL1(O) +|Du|(O), where |Du| denotes the total variation measure of Du. The total variation with respect to the anisotropic norm | · |2,1 is n denoted by |Du|2,1 . When O is a bounded Lipschitz domain, then BV (O; R ) is compactly 1 n ∗ n embedded in L (O; R ). We say that a sequence un converges weakly in BV (O; R ) to 1 n ∗ u if un → u in L (O; R ) and Dun * Du in the sense of measures. We state some fine properties of BV -functions. To this end, we need some definitions. A function u ∈ L1(O; Rn) is said to have an approximate limit at x ∈ O whenever there exists z ∈ Rn such that 1 Z lim |u(y) − z| dy = 0 . ρ→0 d ρ Bρ(x) FROM THE N -CLOCK MODEL TO THE XY MODEL 13

Next we introduce so-called approximate jump points. Given x ∈ O and ν ∈ Sd−1 we set ± Bρ (x, ν) = {y ∈ Bρ(x): ±(y − x) · ν > 0} .

We say that x ∈ O is an approximate jump point of u if there exist a 6= b ∈ Rn and ν ∈ Sd−1 such that 1 Z 1 Z lim |u(y) − a| dy = lim |u(y) − b| dy = 0 . ρ→0 d + ρ→0 d − ρ Bρ (x,ν) ρ Bρ (x,ν) The triplet (a, b, ν) is determined uniquely up to the change to (b, a, −ν). We denote it by + − (u (x), u (x), νu(x)) and let Ju be the set of approximate jump points of u. The triplet + − (u , u , νu) can be chosen as a Borel function on the Borel set Ju . Denoting by ∇u the approximate gradient of u, we can decompose the measure Du as Z Z + − d−1 (c) Du(B) = ∇u dx + (u (x) − u (x)) ⊗ νu(x) dH + D u(B) , B Ju∩B

(c) (j) + − d−1 where D u is the so-called Cantor part and D u = (u − u ) ⊗ νuH Ju is the so-called jump part. We will need the slicing properties of BV -functions. Let d ≥ 2. Given a unit vector ξ ∈ Sd−1 , we denote by Πξ the hyperplane orthogonal to ξ . For every set E ⊂ Rd and z ∈ Πξ , ξ the section of E corresponding to z is the set Ez := {t ∈ R : z + tξ ∈ E}. Accordingly, n ξ ξ n ξ for any function u: E → R , the function uz : Ez → R is deﬁned by uz(t) := u(z + tξ). We recall a characterization of BV functions by slicing [12, Remark 3.104]. Let us ﬁx an open set O ⊂ Rd and u ∈ L1(O; Rn). Then u ∈ BV (O; Rn) if and only if for every d−1 ξ ξ n d−1 ξ ξ ∈ S we have uz ∈ BV (Oz; R ) for H -a.e. z ∈ Π and Z ξ ξ d−1 |Duz|(Oz) dH (z) < ∞ . Πξ Moreover it is possible to reconstruct the distributional gradient Du from the gradients of ξ d−1 ξ ξ the slices Duz through the formula Du ξ = H Π ⊗ Duz , i.e., Z ξ ξ d−1 Du ξ(B) = Duz(Bz ) dH (z) , Πξ for every Borel set B ⊂ Rd . More precisely, the same decomposition holds true for each part of the decomposition of Du, namely Z Z ξ ξ d−1 ∇u ξ dx = ∇uz(Bz ) dH (z) , B Πξ Z (c) (c) ξ ξ d−1 D u ξ(B) = D uz(Bz ) dH (z) , Πξ Z (j) (j) ξ ξ d−1 D u ξ(B) = D uz(Bz ) dH (z) , Πξ

d ξ d−1 ξ ξ ± for every Borel set B ⊂ . Moreover, J ξ = (Ju) for H -a.e. z ∈ Π and (u ) (t) = R uz z z ± ξ ∓ ξ ξ (u )z(t) (= (u )z(t), respectively) for every t ∈ (Ju)z if ξ·νu(z+tξ) > 0 (if ξ·νu(z+tξ) < 0, respectively). If O is a bounded open set, we define the space BV (O; S1) as the space of those functions u ∈ BV (O; R2) such that u(x) ∈ S1 almost everywhere. We remark that a slightly different approach to define the set BV (O; S1) is taken in [11] using S1 as a metric space. The results of this paper (see, for instance, the limit functionals in Theorem 1.1) fit more in that abstract framework, but we decided to take the above definition for simplicity. 14 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

2.2. Results for the classical XY model. We recall here some results about the classi- 1 cal XY model, namely when the spin field uε :Ωε → S is not constrained to take values in a discrete set. Following [7], in order to define the discrete vorticity of the spin variable, it is convenient to introduce the projection Q: R → 2πZ defined by Q(t) := argmin{|t − s| : s ∈ 2πZ} , (2.4) with the convention that, if the argmin is not unique, then we choose the one with minimal modulus. Then for every t ∈ R we define (see Figure 4) Ψ(t) := t − Q(t) ∈ [−π, π] . (2.5)

−2π −π 0 π 2π

−π

Figure 4. Graph of the function Ψ for t ∈ (−2π, 2π) . Observe that Ψ is an odd function.

2 1 Let u: εZ → S and let ϕ:Ωε → [0, 2π) be the phase of u deﬁned by the relation u = exp(ιϕ). The discrete vorticity of u is deﬁned for every εi ∈ εZ2 by 1 h du(εi) := Ψ ϕ(εi + εe1) − ϕ(εi) + Ψ ϕ(εi + εe1 + εe2) − ϕ(εi + εe1) 2π (2.6) i + Ψ ϕ(εi + εe2) − ϕ(εi + εe1 + εe2) + Ψ ϕ(εi) − ϕ(εi + εe2) .

As already noted in [7], the discrete vorticity du only takes values in {−1, 0, 1}, i.e., only singular vortices can be present in the discrete setting. We introduce the discrete measure representing all vortices of the discrete spin ﬁeld deﬁned as X µu := du(εi)δεi+(ε,ε) . (2.7) 2 εi∈εZ

Remark 2.1. In [7, 8] the vorticity measure ˚µu is supported in the centers of the squares completely contained in Ω, i.e., X ˚µu = du(εi)δεi+1/2(ε,ε) . 2 εi∈εZ εi+[0,ε]2⊂Ω In this paper we prefer definition (2.7) since it fits well with our definition of discrete currents in Section 3.5 on the whole set Ω. However, as we will borrow some results from [7, 8], we have to ensure that these definitions are asymptotically equivalent with respect to the flat convergence defined below. Definition 2.2 (Flat convergence). Let O ⊂ R2 be an open set. A sequence of measures µj ∈ Mb(O) converges flat to µ ∈ Mb(O) if Z Z 0,1 ψ dµj → ψ dµ for every ψ ∈ Cc (O) . (2.8) O O f In that case, we denote the convergence by µj * µ. Observe that the flat convergence is weaker than the weak* convergence. The two notions are equivalent when the measures µj have equibounded total variations. The two vorticity measures µu and ˚µu are then close in the following sense. 2 1 1 Lemma 2.3. Assume that uε : εZ → S is a sequence such that ε2 Eε(uε) ≤ C| log ε|. f Then µuε Ω − ˚µuε * 0. FROM THE N -CLOCK MODEL TO THE XY MODEL 15

0,1 Proof. Note that, for any ψ ∈ Cc (Ω) with kψkW 1,∞ ≤ 1, for ε > 0 small enough (depending on the support of ψ ) we have X ε ε 1 |hµ Ω − ˚µ , ψi| ≤ |d (εi)|√ ≤ √ |˚µ |(Ω) ≤ Cε E (u) ≤ Cε| log ε| , uε uε u u ε2 ε 2 2 2 εi∈εZ εi+[0,ε]2⊂Ω where in the last but one inequality we used [7, Remark 3.4]. This proves the claim. We recall the following compactness and lower bound for the XY model.

1 Proposition 2.4. Let uε :Ωε → S and assume that there is a constant C > 0 such that 1 PN ε2| log ε| XYε(uε) ≤ C . Then there exists a measure µ ∈ Mb(Ω) of the form µ = h=1 dhδxh f with dh ∈ Z and xh ∈ Ω, and a subsequence (not relabeled) such that µuε Ω * µ. Moreover 1 2π|µ|(Ω) ≤ lim inf XYε(uε) . ε→0 ε2| log ε| Proof. In [8, Theorem 3.1-(i)] it is proven that (up to a subsequence) the discrete vorticity measures ˚µuε converge ﬂat to a measure of the claimed form satisfying also the lower bound. The claim follows by applying Lemma 2.3.

Remark 2.5. Observe that in the regime θε ε| log ε| the bound 1 Eε(uε) ≤ C εθε f and Proposition 2.4 imply that µuε Ω * 0.

3. Currents For the theory of currents and cartesian currents we refer to the books [31, 32]. We recall here some notation, deﬁnitions, and basic facts about currents. We additionally prove some technical lemmata.

3.1. Definitions and basic facts. Given an open set O ⊂ Rd , we denote by Dk(O) the space of k -forms ω : O 7→ ΛkRd that are C∞ with compact support in O.A k -current k T ∈ Dk(O) is an element of the dual of D (O). The duality between a k -current and a k -form ω will be denoted by T (ω). The boundary of a k -current T is the (k−1)-current k−1 ∂T ∈ Dk−1(O) defined by ∂T (ω) := T (dω) for every ω ∈ D (O) (or ∂T := 0 if k = 0). As for distributions, the support of a current T is the smallest relatively closed set K in O 0 such that T (ω) = 0 if ω is supported outside K . Given a smooth map f : O → O0 ⊂ RN such that f is proper2, f #ω ∈ Dk(O) denotes the pull-back of a k -form ω ∈ Dk(O0) through 0 f . The push-forward of a k -current T ∈ Dk(O) is the k -current f#T ∈ Dk(O ) defined by # k f#T (ω) := T (f ω). Given a k -form ω ∈ D (O), we can write it via its components

X α ∞ ω = ωα dx , ωα ∈ Cc (O) , |α|=k where the expression |α| = k denotes all multi-indices α = (α1, . . . , αk) with 1 ≤ αi ≤ d, and dxα = dxα1∧...∧dxαk . The norm of ω(x) is denoted by |ω(x)| and it is the Euclidean norm of the vector with components (ωα(x))|α|=k . The total variation of a k -current T ∈ Dk(O) is deﬁned by |T |(O) := sup{T (ω): ω ∈ Dk(O), |ω(x)| ≤ 1} .

If T ∈ Dk(O) with |T |(Ω) < ∞, then we can deﬁne the measure |T | ∈ Mb(O) |T |(ψ) := sup{T (ω): ω ∈ Dk(O), |ω(x)| ≤ ψ(x)}

2that means, f −1(K) is compact in O for all compact sets K ⊂ O0 . 16 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

for every ψ ∈ C0(O), ψ ≥ 0. As a consequence of Riesz’s Representation Theorem (see [31, ~ d ~ 2.2.3, Theorem 1]) there exists a |T |-measurable function T : O 7→ ΛkR with |T (x)| = 1 for |T |-a.e. x ∈ O such that Z T (ω) = hω(x), T~(x)i d|T |(x) (3.1)

O for every ω ∈ Dk(O). We note that if T has finite total variation, then it can be extended to a linear functional acting on all forms with bounded, Borel-measurable coefficients via the dominated convergence theorem. In particular, in this case the push-forward f#T can be defined also for f ∈ C1(O,O0) with bounded derivatives, cf. the discussion in [31, p. 132]). A set M ⊂ O is a countably Hk -rectifiable set if it can be covered, up to an Hk -negligible subset, by countably many k -manifolds of class C1 . As such, it admits at Hk -a.e. x ∈ M a tangent space Tan(M, x) in a measure theoretic sense. A current T ∈ Dk(O) is an integer multiplicity (i.m.) rectifiable current if it is representable as Z T (ω) = hω(x), ξ(x)iθ(x) dHk(x) , for ω ∈ Dk(O) , (3.2)

M where M ⊂ O is a Hk -measurable and countably Hk -rectifiable set, θ : M → Z is locally k d k H M-summable, and ξ : M → ΛkR is a H -measurable map such that ξ(x) spans Tan(M, x) and |ξ(x)| = 1 for Hk -a.e. x ∈ M. We use the short-hand notation T = τ(M, θ, ξ). One can always replace M by the set M ∩ θ−1({0}), so that we may always assume that θ 6= 0. Then the triple (M, θ, ξ) is uniquely determined up to Hk -negligible modifications. Moreover, one can show, according to the Riesz’s representation in (3.1), that T~ = ξ and the total variation3 is given by |T | = |θ|Hk M. If Tj are i.m. rectifiable currents and Tj *T in Dk(O) with supj(|Tj|(V ) + |∂Tj|(V )) < ∞ for every V ⊂⊂ O, then by the Closure Theorem [31, 2.2.4, Theorem 1] T is an i.m. rectifiable current, too. By M we denote the current defined by integration over M. J K 3.2. Currents in product spaces. Let us introduce some notation for currents defined on the product space Rd1 ×Rd2 . We will denote by (x, y) the points in this space. The standard d1 basis of the first space R is {e1, . . . , ed1 }, while {e¯1,..., e¯d2 } is the standard basis of the d2 d1 d2 second space R . Given O1 ⊂ R ,O2 ⊂ R open sets, T1 ∈ Dk1 (O1), T2 ∈ Dk2 (O2) and k1+k2 a (k1 + k2)-form ω ∈ D (O1×O2) of the type X α β ω(x, y) = ωαβ(x, y) dx ∧dy ,

|α|=k1 |β|=k2 the product current T1 × T2 ∈ Dk1+k2 (O1×O2) is deﬁned by X X β α T1×T2(ω) := T1 T2 ωαβ(x, y) dy dx ,

|α|=k1 |β|=k2 α β while T1×T2(φ dx ∧dy ) = 0 if |α| + |β| = k1 + k2 but |α|= 6 k1 , |β|= 6 k2 .

3.3. Graphs. Let O ⊂ Rd be an open set and u:Ω → R2 a Lipschitz map. Then we can consider the d-current associated to the graph of u given by Gu := (id×u)# O ∈ 2 2 J K D2(O×R ), where id×u: O → O×R is the map (id×u)(x) = (x, u(x)). Note that by deﬁnition we have Z Gu(ω) = hω(x, u(x)),M(∇u(x))i dx O for all ω ∈ Dd(O × R2), with the d-vector 1 2 1 2 M(∇u) = (e1 + ∂x1 u e¯1 + ∂x1 u e¯2) ∧ ... ∧ (ed + ∂xd u e¯1 + ∂xd u e¯2) . (3.3)

3For i.m. rectiﬁable currents, the total variation coincides with the so-called mass. Hence, we will not distinguish between these two concepts. FROM THE N -CLOCK MODEL TO THE XY MODEL 17

The above formula can be extended to the class A1(O; R2) deﬁned by 1 2 1 2 1 A (O; R ) := {u ∈ L (O; R ): u approx. diﬀ. a.e. and all minors of ∇u are in L (O)} . 1,2 2 1 2 2 Remark 3.1. We recall that ∂Gu|Ω×R = 0 when u ∈ W (O; R ) ⊂ A (O; R ), see [31, 3.2.1, Proposition 3]. This property however fails for general functions u ∈ A1(O; R2). In Lemma 3.4 we need to interpret the graphs of W 1,1(O; S1) as currents. This can be done because of the following observation.

Lemma 3.2. Let O ⊂ Rd be an open, bounded set. Then W 1,1(O; S1) ⊂ A1(O; R2). Proof. It is well-known that Sobolev functions are approximately diﬀerentiable a.e. More- over, all 1-minors of ∇u are in L1(O). We argue that all 2-minors vanish at a.e. point. To this end, denote by P : R2 \{0} → R2 the smooth mapping P (x) = x/|x|. Since for u ∈ W 1,1(O; S1) we have u = P ◦ u almost everywhere, for a.e. x ∈ O the chain rule for approximate diﬀerentials yields ∇u(x) = ∇P (u(x))∇u(x). Since ∇P (u(x)) has at most rank 1, also ∇u(x) has at most rank 1 and therefore all 2-minors have to vanish as claimed. Later on we use the orientation of the graph of a smooth function u: O ⊂ R2 → S1 (cf. 2 [31, 2.2.4]). For such maps we have |Gu| = H M, where M = (id×u)(Ω), and p 2 1 + |∇u(x)| G~ u(x, y) = e1 ∧ e2 1 2 + ∂x2 u (x)e1 ∧ e¯1 + ∂x2 u (x)e1 ∧ e¯2 (3.4) 1 2 − ∂x1 u (x)e2 ∧ e¯1 − ∂x1 u (x)e2 ∧ e¯2 , for every (x, y) ∈ M.

3.4. Cartesian currents. Let O ⊂ Rd be a bounded, open set. We recall that the class of cartesian currents in O×R2 is defined by 2 2 2 cart(O×R ) := {T ∈ Dd(O×R ): T is i.m. rectifiable, ∂T |O×R = 0, O π#T = O ,T | dx ≥ 0 , |T | < ∞ , kT k1 < ∞} , J K O 2 where π : O×R → O denotes the projection on the first component, T | dx ≥ 0 means ∞ 2 that T (φ(x, y) dx) ≥ 0 for every φ ∈ Cc (O×R ) with φ ≥ 0, and ∞ 2 kT k1 = sup{T (φ(x, y)|y| dx): φ ∈ Cc (O×R ) , |φ| ≤ 1} . Note that, if for some function u Z Z T (φ(x, y) dx) = φ(x, u(x)) dx then kT k1 = |u| dx . (3.5) O O The class of cartesian currents in O×S1 is 1 2 1 cart(O×S ) := {T ∈ cart(O×R ) : supp(T ) ⊂ O×S } , (cf. [32, 6.2.2] for this definition). We recall the following approximation theorem which explains that cartesian currents in O×S1 are precisely those currents that arise as limits of graphs of S1 -valued smooth maps. The proof, based on a regularization argument on the lifting of T , can be found in [33, Theorem 7].4

Theorem 3.3 (Approximation Theorem). Let T ∈ cart(O×S1). Then there exists a se- ∞ 1 quence of smooth maps uh ∈ C (O; S ) such that 2 Guh *T in Dd(O×R ) and 2 2 |Guh |(O×R ) → |T |(O×R ) .

4Notice that some results in [33] require O to have smooth boundary. This is not the case for this theorem, which is based on a local construction. 18 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

Using the above approximation result, we now prove an extension result for cartesian currents, which we could not ﬁnd in the literature.

Lemma 3.4 (Extension of cartesian currents). Let O ⊂ Rd be a bounded, open set with Lipschitz boundary and let T ∈ cart(O×S1). Then there exist an open set Oe ⊃⊃ O and a 1 2 2 current T ∈ cart(Oe×S ) such that Te|O×R = T and |Te|(∂O×R ) = 0. ∞ 1 Proof. Applying Theorem 3.3 we ﬁnd a sequence uk ∈ C (O; S ) such that Guk *T in 2 2 2 2 Dd(O×R ) and |Guk |(O×R ) → |T |(O×R ). In particular, the sequence |Guk |(O×R ) is bounded, which implies that Z sup |∇uk|dx < C . (3.6) k O

Next we extend the functions uk . To this end, note that there exists t > 0 and a bi- Lipschitz map Γ: (∂O×(−t, t)) → Γ(∂O×(−t, t)) such that Γ(x, 0) = x for all x ∈ ∂O, Γ(∂O×(−t, t)) is an open neighborhood of ∂O and

2 Γ(∂O×(−t, 0)) ⊂ O, Γ(∂O × (0, t)) ⊂ R \ O. (3.7) This result is a consequence of [44, Theorem 7.4 & Corollary 7.5]; details can be found for instance in [43, Theorem 2.3]. The extension of uk is then achieved via reflection. More 0 0 0 precisely, for a sufficiently small t > 0 we define it on O with O = O + Bt0 (0) by

( −1 uk(Γ(P (Γ (x)))) if x∈ / O, uek(x) = (3.8) uk(x) otherwise,

1,1 0 1 where P (x, τ) = (x, −τ). Since Γ is bi-Lipschitz, we have that uek ∈ W (O ; S ) and by a change of variables we can bound the L1 -norm of its gradient via Z Z Z Z −1 |∇uek| dx ≤ |∇uk| dx + CΓ |(∇uk) ◦ Γ ◦ P ◦ Γ | dx ≤ CΓ |∇uk| dx , (3.9) O0 O O0\O O where the constant CΓ depends only on the bi-Lipschitz properties of Γ and the dimension. Due to Lemma 3.2 we obtain that u ∈ A1(O0; 2). In particular, the current G ∈ ek R uek 0 2 Dd(O × R ) is well-defined in the sense of Z G (ω) = hω(x, u (x)),M(∇u (x))i dx , uek ek ek O0 with M(∇u ) given by (3.3). We next prove that G ∈ cart(O0 × 1). First note that ek uek S whenever ω ∈ Dd(O0 ×R2) is a form with supp(ω) ⊂⊂ O0 ×R2 \(O0 ×S1), then the definition yields G (ω) = 0. By standard arguments (apply for instance [31, 3.2.1 Proposition 5] to uek a constant sequence) it then suffices to prove that ∂G | 0 2 = 0. We will argue locally. uek O ×R 0 For each x ∈ O we choose a rotation Qx , radii rx > 0, and heights hx > 0 such that the d−1 cylinders Cx := x + Qx (−rx, rx) × (−hx, hx) satisfy

(i) Cx ⊂⊂ O if x ∈ O; 0 0 (ii) Cx ⊂⊂ O \ O if x ∈ O \ O; 0 (iii) Cx ⊂⊂ O if x ∈ ∂O and 0 d 0 d−1 0 Cx ∩ O = Cx ∩ x + Qx{(x , xd) ∈ R : x ∈ (−rx, rx) : −hx < xd < ψ(x )} for some ψ ∈ Lip(Rd−1). ∞ 1 For x ∈ O we have ∂G | 2 = ∂G | 2 = 0 since u ∈ C (C ; ). Next consider uek Cx×R uk Cx×R k x S 0 0 the second case, namely x ∈ O \ O. Since Cx ⊂⊂ O \ O, the properties in (3.7) imply −1 that Γ ◦ P ◦ Γ (Cx) ⊂⊂ O. In particular, by the smoothness of uk on O we have that 1,∞ u ∈ W (C ), so that by Remark 3.1 again ∂G | 2 = 0. Finally, we consider ek x uek Cx×R x ∈ ∂O. Since Cx ∩ Ω is (up to a rigid motion) the subgraph of a Lipschitz function, it is in particular simply connected. By classical lifting theory, we ﬁnd a sequence of scalar functions FROM THE N -CLOCK MODEL TO THE XY MODEL 19

∞ ϕk ∈ C (Cx ∩ O) such that uk(x) = exp(ιϕk(x)). In particular, using the chain rule we 1,1 see that ϕk ∈ W (Cx ∩ O). Now fix 0 < δ < rx small enough such that Bδx (x) ⊂ Cx and −1 0 (Γ ◦ P ◦ Γ )(Bδx (x) ∩ (O \ O)) ⊂ Cx ∩ O, which can be realized due to (3.7). We then extend the lifting ϕk to a function ϕek ∈ 1,1 W (Bδx (x)) via the same reflection construction as in (3.8), which is well-defined due to the above inclusions. Observe that this definition guarantees that uek(y) = exp(ιϕek(y)) for almost every y ∈ Bδx (x). Expressed in terms of currents this means that

G | 2 = χ G | 2 , uek Bδx (x)×R # ϕek Bδx (x)×R where G ∈ D (B (x)× ) is the current associated to the graph of ϕ and χ: d× → ϕek d δx R ek R R Rd×S1 is the covering map deﬁned by χ(x, ϑ) := (x, cos(ϑ), sin(ϑ)). In particular, by [33, 1 Theorem 2, p. 97 & Proposition 1 (i), p. 100] we have G | 2 ∈ cart(B (x) × ), uek Bδx (x)×R δx S so that by the deﬁnition of cartesian currents we have ∂G | 2 = 0. uek Bδx (x)×R 0 0 Thus we have shown that for every x ∈ O there exists a ball Bδx (x) ⊂ O such that ∂G | 2 = 0. Using a partition of unity to localize the support of any form ω ∈ uek Bδx (x)×R d−1 0 2 D (O × ) with respect to the x-variable, we conclude that ∂G | 0 2 = 0 and R uek O ×R therefore G ∈ cart(O0 × 1). As seen in the proof of Lemma 3.2, all 2-minors of Du uek S vanish, so that the bounds (3.6) and (3.9) yield

Z Z 1 |G |(O0× 2) = |M(∇u )| dx ≤ 1 + |∇u |2 2 dx ≤ C. uek R ek ek O0 O0 Hence, up to a subsequence, we can assume that G * T in D (O0× 2), see [31, 2.2.4 uek e d R 0 2 Theorem 2]. From [31, 4.2.2. Theorem 1] it follows that Te ∈ cart(O ×R ). Since uek = uk 2 2 on O, we ﬁnd that Te|O×R = T . It remains to show that |Te|(∂O × R ) = 0. To this end, note that for 0 < η < η0 < 1, by the bi-Lipschitz continuity of Γ and (3.7) we have that −1 out in (Γ ◦ P ◦ Γ )(Oη ) ⊂ Oη0 , out in where the sets Oη and Oη0 are deﬁned as out 0 in 0 Oη := {x ∈ O \ O : dist(x, ∂O) < η},Oη0 = {x ∈ O : dist(x, ∂O) < η } . Hence, similar to (3.9) we obtain that

Z 1 2 2 2 in 2 |G |((∂O + B (0))× ) ≤ C 1 + |∇u | dx = C |G |(O 0 × ) . (3.10) uek η R Γ k Γ uk η R Oin η0 2 2 0 Since |Guk |(O×R ) → |T |(O×R ) and |T | is a finite measure, there exists a sequence η → 0 in 2 in 2 such that |Guk |(Oη0 ×R ) → |T |(Oη0 ×R ). Applying the lower semicontinuity of the mass with respect to weak convergence of currents in (3.10), we infer that 2 in 2 |Te|((∂O + Bη(0))×R ) ≤ CΓ|T |(Oη0 ×R ) . 0 2 Sending first η → 0 and then η → 0 we conclude that |Te|(∂O×R ) = 0 as claimed. We will also use the structure theorem for cartesian currents in O×S1 that has been proven in [33, Section 3, Theorems 1, 5, 6].5 However, to simplify notation, from now on we focus on dimension two. Recall that Ω ⊂ R2 is a bounded, open set with Lipschitz boundary. To state the theorem, we recall the following decomposition for a current T ∈ cart(Ω×S1). Letting M the countably H2 -rectifiable set where T is concentrated, we denote by M(a) the set of points (x, y) ∈ M at which the tangent plane Tan(M, (x, y)) does not contain vertical vectors (namely, the Jacobian of the projection πΩ restricted to Tan(M, (x, y)) has Ω (jc) (a) 1 dπ#|T | maximal rank), by M := (M\M )∩(JT ×S ), where JT := {x ∈ Ω: dH1 (x) > 0}, and by M(c) := M\ (M(a) ∪ M(jc)). Then we can split the current via T = T (a) + T (c) + T (jc) ,

5As for the Approximation Theorem, no boundary regularity is required for this result. 20 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF where T (a) := T M(a) , T (c) := T M(c) , T (jc) := T M(jc) are mutually singular measures, and we denote by the restriction of the Radon measure T . Hereafter we use 1 2 2 1 the notation xb = x and xb = dx . Theorem 3.5 (Structure Theorem for cart(Ω×S1)). Let T ∈ cart(Ω×S1). Then there 1 exists a unique map uT ∈ BV (Ω; S ) and an (not unique) i.m. rectifiable 1-current LT = ~ (jc) (j) 1 τ(L, k, LT ) ∈ D1(Ω) such that T = T + LT × S and J Z K (a) T (φ(x, y) dx) = T (φ(x, y) dx) = φ(x, uT (x)) dx , (3.11) Ω Z T (a)(φ(x, y) dxl∧dym) = (−1)2−l φ(x, u (x))∂(a)um(x) dx , (3.12) b T xl T Ω Z T (c)(φ(x, y) dxl∧dym) = (−1)2−l φ(x, u (x)) d∂(c)um(x) , (3.13) b eT xl T Ω Z Z T (j)(φ(x, y) dxl∧dym) = (−1)2−l φ(x, y) dym νl (x) dH1(x) (3.14) b uT γ JuT x ∞ 2 1 − for every φ ∈ Cc (Ω×R ), γx being the (oriented) geodesic arc in S that connects uT (x) + to uT (x) and ueT being the precise representative of uT . Remark 3.6. In [33, Theorem 6] the structure of T (j) is formulated in a slightly different way, − − + + using the counter-clockwise arc γϕ−,ϕ+ between (cos(ϕ ), sin(ϕ )) and (cos(ϕ ), sin(ϕ )) and replacing JuT by Jϕ , where ϕ ∈ BV (Ω) is a local lifting of T . More precisely, the 2 2 1 notion of lifting is understood in the sense that T = χ#Gϕ , where χ: R ×R → R ×S is the covering map (x, ϑ) 7→ (x, cos(ϑ), sin(ϑ)) and Gϕ ∈ cart(Ω×R) is the cartesian current given by the boundary of the subgraph of ϕ (hence the push-forward via χ is well-defined as Gϕ has finite mass, see Section 3.1). To explain how to deduce (3.14), we recall the local + 0 construction in [33]: for every x ∈ Jϕ one chooses p (x) ≥ 0 and k (x) ∈ N ∪ {0} such that ϕ+(x) = p+(x) + 2πk0(x), 0 ≤ p+(x) − ϕ−(x) < 2π , where we recall that in the scalar case the traces (and the normal to the jump set) are − + 0 arranged to satisfy ϕ < ϕ on Jϕ . Then, locally, the 1-current LT in [33, Theorem 6] 0 0 0 ~ 0 0 0 is given by LT = τ(L , k (x), LT ), where L ⊂ Jϕ denotes the set of points with k (x) ≥ 1 ~ 0 0 ~ 0 2 1 and LT is the orientation of L defined via LT = νϕe1 −νϕe2 . To obtain the representation via geodesics, we let ( (p+(x), k0(x)) if p+(x) − ϕ−(x) < π , (q+(x), k(x)) = (p+(x) − 2π, k0(x) + 1) if p+(x) − ϕ−(x) > π , The case p+(x) − ϕ−(x) = π , i.e, antipodal points, needs special care. In this case we + ± ± define q (x) and k(x) according to the following rule: let ϕe (x) := ϕ (x) mod 2π ∈ [0, 2π). Then

( + 0 + − + (p (x), k (x)) if Ψ(ϕe (x) − ϕe (x)) = π , (q (x), k(x)) = + 0 + − (p (x) − 2π, k (x) + 1) if Ψ(ϕe (x) − ϕe (x)) = −π , with the function Ψ deﬁned in (2.5). Replacing (p+(x), k0(x)) by (q+(x), k(x)), the modiﬁed structure of T (j) can be proven following exactly the lines of [33, p.107-108], noting that by + − the chain rule in BV [12, Theorem 3.96] we have Jϕ = JuT ∪ {x ∈ Jϕ : q (x) = ϕ (x)}. In particular, ~ ~ 2 1 LT = τ(L, k, LT ) , L = {x ∈ Jϕ : k(x) ≥ 1} , LT = νϕe1 − νϕe2 (3.15) still depend on the local lifting ϕ, but in (3.14) the curves γϕ−,ϕ+ are replaced by the − − − + more intrinsic geodesic arcs γx connecting uT (x) = (cos(ϕ (x)), sin(ϕ (x))) to uT (x) = + + (cos(ϕ (x)), sin(ϕ (x))) (these formulas are consistent with the choice νuT (x) = νϕ(x)). FROM THE N -CLOCK MODEL TO THE XY MODEL 21

− + In particular, exchanging uT (x) and uT (x) will change the orientation of the arc (also in 6 (j) the case of antipodal points) and of the normal νuT (x), so that the formula for T is invariant, hence well-deﬁned without the use of local liftings. The structure theorem highlights a peculiar property of cartesian currents in Ω×S1 : the current T cannot be written only in terms of its associated BV function uT because of 1 the presence of the concentration term LT × S . This feature can be seen in the typical 1 J K x example of a current T ∈ cart(Ω×S ) whose associated BV function is uT (x) = |x| a.e. 1 in B1(0). Such a current must be of the form T = G x + LT × S , where LT is an i.m. |x| J K rectiﬁable 1-current with ∂LT |B1(0) = δ0 . E.g., LT can be concentrated on any curve L which connects the point 0 to the boundary. The current LT is therefore necessary to compensate the possible presence of holes in the graph associated to uT . It is convenient to recast the jump-concentration part of T ∈ cart(Ω×S1) in the following 1 way. Let LT = τ(L, k, L~ T ) as in Theorem 3.5. We introduce for H -a.e. x ∈ JT the normal

νT (x) to the 1-rectiﬁable set JT = JuT ∪ L as ( ν (x) if x ∈ J , ν (x) = uT uT (3.16) T ~ 2 ~ 1 (−LT (x), LT (x)) if x ∈ L \ JuT , ~ 2 ~ 1 1 where we choose νuT (x) = (−LT (x), LT (x)) if x ∈ L∩JuT . For H -a.e. x ∈ JT we consider T − + the curve γx given by: the (oriented) geodesic arc γx which connects uT (x) to uT (x) if 1 x ∈ JuT \L (in the sense of Remark 3.6 in case of antipodal points); the whole S turning k(x) times if x ∈ L \ JuT , k(x) being the integer multiplicity of LT ; the sum (in the sense 7 1 of currents) of the oriented geodesic arc γx and of S with multiplicity k(x) if x ∈ JuT ∩L. Then Z Z (jc) l m 2−l n mo l 1 T (φ(x, y) dxb∧dy ) = (−1) φ(x, y) dy νT (x) dH (x) . (3.17) T JT γx T m The integration over γx with respect to the form dy in the formula above is intended T 1 with the correct multiplicity of the curve γx deﬁned for H -a.e. x ∈ JT by the integer number  ±1 , if x ∈ Ju \L , y ∈ supp(γx) ,  T k(x) , if x ∈ L \ J , y ∈ 1, m(x, y) := uT S (3.18) k(x) ± 1 , if x ∈ L ∩ J , y ∈ supp(γ ) ,  uT x  T k(x) , if x ∈ L ∩ JuT , y ∈ supp(γx ) \ supp(γx) , where ± = +/− if the geodesic arc γx is oriented counterclockwise/clockwise, respectively. More precisely, Z Z m m m 1 φ(y) dy = (−1) φ(y)yb m(x, y) dH (y) . (3.19) T T γx supp(γx )

Remark 3.7. Note that we constructed m(x, y) based on the orientation (3.16) of νT . As discussed in Remark 3.6, changing the orientation of νuT changes the orientation of the geodesic γx , while a change of the orientation of L~ T switches the sign of k(x). Hence changing the orientation of νT (x) changes m(x, y) into −m(x, y). If we choose locally νT = νϕ as in Remark 3.6, our construction above yields m(x, y) ≥ 0. In the proposition below, we derive an explicit formula for the vector T~ of a cartesian current. It seems that this result is well-known to experts, but since we could not ﬁnd a precise reference, we include a proof for the reader’s convenience.

6 1 More precisely, assume that u1, u2, ν ∈ S and assume that the geodesic arc from u1 to u2 is coun- + − terclockwise. If (uT (x), uT (x), νuT (x)) = (u2, u1, ν), then γx is oriented counterclockwise. If, instead, + − (uT (x), uT (x), νuT (x)) = (u1, u2, −ν) (equivalent to the first choice, according to the definition of jump point), then γx is oriented clockwise. 7 T 1 In this case, a more elementary way of defining γx is the following: let γx : [0, 1] → S be the geodesic arc, and let ϕx : [0, 1] → R be a continuous function (unique up to translations of an integer multiple of 2π ) T such that γx(t) = exp(ιϕx(t)). Then γx (t) = exp ι(1 − t)ϕx(0) + ιt(ϕx(1) + 2πk(x)) . 22 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

1 Proposition 3.8. Let T ∈ cart(Ω×S ), let uT be the BV function associated to T . Then |T (a)| = H2 M(a) , |T (c)| = H2 M(c) , |T (jc)| = |m|H2 M(jc) , and

p 2 1 + |∇uT (x)| T~(x, y) = e1 ∧ e2 (a) 1 (a) 2 + ∂x2 uT (x)e1 ∧ e¯1 + ∂x2 uT (x)e1 ∧ e¯2 (3.20) (a) 1 (a) 2 − ∂x1 uT (x)e2 ∧ e¯1 − ∂x1 uT (x)e2 ∧ e¯2 , for H2 -a.e. (x, y) ∈ M(a) , d∂(c)u1 d∂(c)u2 T~(x, y) = x2 T (x)e ∧ e¯ + x2 T (x)e ∧ e¯ d|D(c)u | 1 1 d|D(c)u | 1 2 T T (3.21) d∂(c)u1 d∂(c)u2 − x1 T (x)e ∧ e¯ − x1 T (x)e ∧ e¯ , (c) 2 1 (c) 2 2 d|D uT | d|D uT | for H2 -a.e. (x, y) ∈ M(c) , and 2 2 2 1 sign(m(x, y))T~(x, y) = − ν (x)y e1 ∧ e¯1 + ν (x)y e1 ∧ e¯2 T T (3.22) 1 2 1 1 + νT (x)y e2 ∧ e¯1 − νT (x)y e2 ∧ e¯2 , for H2 -a.e. (x, y) ∈ M(jc) , where m(x, y) is the integer defined in (3.18). Proof. Assume Ω simply connected (if not, the following arguments can be repeated locally). Let us consider the covering map χ:Ω×R → Ω×S1 defined by χ(x, ϑ) := (x, cos(ϑ), sin(ϑ)). By [33, Corollary 1, p. 105] there exists a lifting of T , i.e., there is a function ϕ ∈ BV (Ω; R) such that T = χ#Gϕ , where Gϕ ∈ cart(Ω×R) is the cartesian current given by the boundary of the subgraph of ϕ. The fine structure of such currents is well known, compare [29, Theorem 4.5.9], [31, 4.1.5 & 4.2.4]. We recall here that, if we consider the subgraph SGϕ := {(x, y) ∈ Ω×R : y < ϕ(x)}, then SGϕ is a set of finite perimeter; Gϕ is the current Gϕ = ∂ SGϕ . The interior normal to SGϕ is given by J K d(Dϕ, −L2) n(x, ϕ(x)) = (x) , for x ∈ Ω \ Jϕ , d|(Dϕ, −L2)| (3.23) − + n(x, ϑ) = (νϕ(x), 0) , for x ∈ Jϕ , ϑ ∈ [ϕ (x), ϕ (x)] , where νϕ is the normal to the jump set Jϕ . Moreover, the current Gϕ can be represented − as Gϕ = G~ ϕ|Gϕ| where |Gϕ| is concentrated on the reduced boundary ∂ SGu , |Gϕ| = 2 − ~ 3 ~ H ∂ SGu , and Gϕ is the 2-vector in R such that −Gϕ(x, ϑ) ∧ n(x, ϑ) = e1 ∧ e2 ∧ e3 , i.e., 3 2 1 G~ ϕ = −n e1 ∧ e2 + n e1 ∧ e3 − n e2 ∧ e3 . Finally, letting (a) 3 Σ := {(x, ϕe(x)) : x ∈ Ω \ Jϕ, n (x, u(x)) 6= 0} , (c) 3 Σ := {(x, ϕe(x)) : x ∈ Ω \ Jϕ, n (x, u(x)) = 0} , (j) − + 3 Σ := {(x, ϑ): x ∈ Jϕ, ϑ ∈ [ϕ (x), ϕ (x)], n (x, ϑ) = 0} ,

− (a) (c) (j) (a) (a) (c) (c) we have ∂ SGϕ = Σ ∪ Σ ∪ Σ and, denoting Gϕ = Gϕ Σ , Gϕ = Gϕ Σ , (j) (j) Gϕ = Gϕ Σ , by [33, formulas (2) and (16)] we have on the one hand that uT = (cos(ϕ), sin(ϕ)) a.e. and (a) (a) (c) (c) (j) (jc) χ#Gϕ = T , χ#Gϕ = T , χ#Gϕ = T . (3.24) − 4 On the other hand, observe that the Jacobian of dχ: Tan(∂ SGϕ, x) 7→ R equals 1 (indeed dχ maps any pair of orthonormal vectors of R3 to a pair of orthonormal vectors in R4 ). Hence by the area formula, for σ ∈ {a, c, j} we obtain Z Z (σ) # ~ 2 X ~ 2 χ#Gϕ (ω) = hχ ω, Gϕi dH (x, ϑ) = hω(x, y), dχ(x, ϑ)Gϕ(x, ϑ)i dH (x, y) . (x,ϑ)∈χ−1(x,y) Σ(σ) χ(Σ(σ)) (3.25) FROM THE N -CLOCK MODEL TO THE XY MODEL 23

Next, note that for σ ∈ {a, c} the map χ:Σ(σ) → χ(Σ(σ)) is one-to-one and for any (a) (c) (x, ϕe(x)) ∈ Σ ∪ Σ we have ~ 3 dχ(x, ϕe(x))Gϕ(x, ϕe(x)) = − n (x, ϕe(x))e1 ∧ e2 2 2 − n (x, ϕe(x)) sin(ϕe(x))e1 ∧ e¯1 + n (x, ϕe(x)) cos(ϕe(x))e1 ∧ e¯2 1 1 + n (x, ϕe(x)) sin(ϕe(x))e2 ∧ e¯1 − n (x, ϕe(x)) cos(ϕe(x)e2 ∧ e¯2 . (3.26)

~ 2 Since |n| = 1 we see that | dχ(x, ϕe(x))Gϕ(x, ϕe(x))| = 1, too. Moreover, for H -a.e. (x, y) ∈ (σ) ~ χ(Σ ) the vector dχ(x, ϕe(x))Gϕ(x, ϕe(x)) orients the tangent space at (x, y). Hence (3.24) and the uniqueness of the representation of i.m. rectiﬁable currents (cf. Section 3.1) implies χ(Σ(σ)) = M(σ) up to a H2 -negligible set, |T (σ)| = H2 M(σ) , and ~ ~ T (χ(x, ϕe(x))) = dχ(x, ue(x))Gϕ(x, ϕe(x)) 2 (σ) for H -almost every (x, y) = χ(x, ϕe(x)) ∈ Σ . By the chain rule in BV [12, Theorem 3.96] we deduce that 2 2 −uT (c) −ueT (c) ∇uT = 1 ⊗ ∇ϕ , D uT = 1 ⊗ D ϕ . uT ueT Combined with the formula for n given by (3.23), the formulas (3.20) and (3.21) then follow from (3.26) by a straightforward calculation. In order to treat the case σ = j , note that due to (3.23) we have for any (x, y) = χ(x, ϑ) ∈ χ(Σ(j))

~ 2 2 2 1 dχ(x, ϑ)Gϕ(x, ϑ) = − νϕ(x)y e1 ∧ e¯1 + νϕ(x)y e1 ∧ e¯2 1 2 1 1 + νϕ(x)y e2 ∧ e¯1 − νϕ(x)y e2 ∧ e¯2 =: ξ(x, y) .

Again |ξ(x, y)| = 1 and ξ(x, y) orients the tangent space at H2 -a.e. (x, y) ∈ χ(Σ(j)). Thus (3.25) and the uniqueness of the representation of i.m. rectiﬁable currents imply (up to H2 -negligible sets) that M(jc) = χ(Σ(j)), T~ = ξ on M(jc) , and |T (jc)| = N(x, y)H2 M(jc) , with N(x, y) = #{ϑ ∈ [ϕ−(x), ϕ+(x)] : (cos(ϑ), sin(ϑ)) = y} .

To conclude, we have to relate m(x, y) to N(x, y) and νT (x) to νϕ(x). First note that the proof of the structure theorem (sketched in Remark 3.6) yields JuT ∪L = Jϕ and, combined T − + T with the deﬁnition of the curves γx (cf. (3.17)), implies that χ[ϕ (x), ϕ (x)] = supp(γx ) for x ∈ Jϕ . Hence

(jc) (j) 2 T M = χ(Σ ) = {(x, y) ∈ Ω×R : x ∈ JuT ∪ L , y ∈ supp(γx )} . (3.27)

Moreover, provided we orient JuT the same way as Jϕ and L according to (3.15), equation (3.16) also yields νT = νϕ and m(x, y) = N(x, y) (a detailed proof of the latter requires to distinguish diﬀerent cases, which we omit here).8 Inserting this equality in (3.27) concludes the proof of (3.22).

Finally, we recall the following result, proven in [33, Section 4].

1 1 Proposition 3.9. If u ∈ BV (Ω; S ), then there exists a T ∈ cart(Ω×S ) such that uT = u a.e. in Ω.

8 As noted in Remark 3.7, the choice νT (x) = νϕ(x) always yields m(x, y) ≥ 0. The factor sign(m(x, y)) in (3.22) makes the formula invariant under the change of νT (x). 24 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

3.5. Currents associated to discrete spin fields. We introduce the piecewise constant interpolations of spin fields. For every set S , we put 2 2 2 PCε(S) := {u: R → S : u(x) = u(εi) if x ∈ εi + [0, ε) for some i ∈ εZ } 1 Given u:Ωε → S , we can always identify it with its piecewise constant interpolation belong- 1 2 ing to PCε(S ), arbitrarily extended to R . Note that the piecewise constant interpolation of u coincides with u on the bottom-left corners of the squares of the lattice εZ2 . 1 2 Given u ∈ PCε(S ), we will associate to u the current Gu ∈ D2(Ω×R ) defined by Z 1 2 Gu(φ(x, y) dx ∧dx ) := φ(x, u(x)) dx , (3.28) Ω Z Z l m 2−l m l 1 Gu(φ(x, y) dxb∧dy ) := (−1) φ(x, y) dy νu(x) dH (x) , (3.29)

Ju γx 1 2 Gu(φ(x, y) dy ∧dy ) := 0 , (3.30)

∞ 2 for every φ ∈ Cc (Ω×R ), where Ju is the jump set of u, νu(x) is the normal to Ju at x, 1 − + and γx ⊂ S is the (oriented) geodesic arc which connects the two traces u (x) and u (x). + − 1 If u (x) and u (x) are opposite vectors, the choice of the geodesic arc γx ⊂ S is done consistently with the choice made in (2.4) for the values Ψ(π) and Ψ(−π) as follows: let ± ± + − ϕ (x) ∈ [0, 2π) be the phase of u (x); if Ψ(ϕ (x) − ϕ (x)) = π , then γx is the arc that − + + − connects u (x) to u (x) counterclockwise; if Ψ(ϕ (x) − ϕ (x)) = −π , then γx is the arc − + that connects u (x) to u (x) clockwise. Note that the choice of the arc γx is independent of the orientation of the normal νu(x). 1 We deﬁne for H -a.e. x ∈ Ju the integer number m(x) = ±1, where ± = +/− if the geodesic arc γx is oriented counterclockwise/clockwise, respectively. Then Z Z m m m 1 φ(y) dy = (−1) m(x) φ(y)yb dH (y) . (3.31)

γx supp(γx)

u+ εi u− εi + εe1

Figure 5. The current Gu has vertical parts concentrated on the jump set Ju , where a transition from u− to u+ occurs.

1 2 Proposition 3.10. Let u ∈ PCε(S ) and let Gu ∈ D2(Ω×R ) be the current deﬁned in (3.28)–(3.30). Then Gu is an i.m. rectiﬁable current and, according to the representation 2 formula (3.1), Gu = G~ u|Gu|, where |Gu| = H M, (a) (j) M = M ∪ M = {(x, u(x)) : x ∈ Ω \ Ju} ∪ {(x, y): x ∈ Ju, y ∈ γx} , and

G~ u(x, y) = e1 ∧ e2 (3.32) for H2 -a.e. (x, y) ∈ M(a) and

2 2 2 1 G~ u(x, y) = sign(m(x)) − ν (x)y e1 ∧ e¯1 + ν (x)y e1 ∧ e¯2 u u (3.33) 1 2 1 1 + νu(x)y e2 ∧ e¯1 − νu(x)y e2 ∧ e¯2 for H2 -a.e. (x, y) ∈ M(j) . FROM THE N -CLOCK MODEL TO THE XY MODEL 25

Proof. First note that the set M is countably H2 -rectiﬁable. Since u is piecewise constant, for horizontal forms we have Z Z 2 (a) Gu(φ(x, y) dx) = φ(x, u(x)) dx = φ(x, y) dH M (x, y) .

2 Ω Ω×R By (3.31) we deduce that for l, m = 1, 2 Z Z l m 2−l m l 1 Gu(φ(x, y) dxb∧dy ) = (−1) φ(x, y) dy νu(x) dH (x)

Ju γx Z Z 2−l+m m 1 l 1 = (−1) φ(x, y)yb dH (y) νu(x)m(x) dH (x)

Ju supp(γx) Z 2−l+m m l 2 (j) = (−1) φ(x, y)yb νu(x)m(x) dH M (x, y) . 2 Ω×R 2 Then for every ω ∈ D2(Ω×R ) we have Z 2 Gu(ω) = hω, G~ ui dH M

2 Ω×R for G~ u deﬁned as in (3.32)–(3.33) and moreover G~ u(x, y) is associated to the tangent space ~ 2 at (x, y) ∈ M. Since also |Gu(x, y)| = 1 for |Gu|-a.e. (x, y) ∈ Ω×R , we conclude the proof.

1 2 Proposition 3.11. Let u ∈ PCε(S ) and let Gu ∈ D2(Ω×R ) be the current defined in (3.28)–(3.30). Then 1 ∂Gu|Ω× 2 = −µu× S , R J K 2 1 2 where µu is the discrete vorticity measure defined in (2.7) for u|εZ : εZ → S . 1 2 Proof. Let us fix 0 < ρ < min{ε/4, dist(Ωε, ∂Ω)} and η ∈ D (Ω×R ). With a partition of unity we can split η into the sum of 1-forms depending on their supports. We discuss here all the possibilities for the supports. Case 1: supp(η) ⊂ (εi + (0, ε)2)×R2 for some i ∈ Z2 . Since u is constant in (εi + 2 2 (0, ε) )×R , we get automatically ∂Gu(η) = 0 by Remark 3.1. Case 2: Let H be the side of the square εi + [0, ε]2 connecting two vertices p, q ∈ εZ2 2 and let U be the ρ/2-neighborhood of H \ Bρ(p)∪Bρ(q) . Assume that supp(η) ⊂ U×R . We claim that

∂Gu(η) = 0 . (3.34) To prove this, we approximate the pure-jump function u by means of a sequence of Lipschitz ± functions uj . Let u be the traces of u on the two sides of H and let νH be the normal ± ± ± ± to H oriented as νu . We let ϕb ∈ [0, 2π) be the phases of u deﬁned by u = exp(ιϕb ). − − + − + − We set ϕ := ϕb and ϕ := ϕb + Ψ(ϕb − ϕb ) ∈ (−π, 3π), where Ψ is the function given by (2.5). We then deﬁne  ϕ− , if t ≤ − 1  2 − + − 1 1 1 ϕ(t) := ϕ + ϕ − ϕ (t + 2 ) if − 2 < t < 2  + 1 ϕ if t ≥ 2 , and ϕk(s) := ϕ(ks) for k large enough. Note that the curve t ∈ (−1/2, 1/2) 7→ exp(ιϕ(t)) 1 − + parametrizes the geodesic arc γ± ⊂ S which connects u to u , consistently with the choice done in formula (3.29). Then we put uk(x) := exp ιϕk(νH · (x − p)) for x ∈ U. 26 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

2 ∞ 2 We prove that Guk *Gu in D2(U×R ). Let us ﬁx φ ∈ Cc (U×R ). Since uk → u in measure, we have that Z Z

Guk (φ(x, y) dx) = φ(x, uk(x)) dx → φ(x, u(x)) dx = Gu(φ(x, y) dx) . U U 0 0 Writing x ∈ U as x = x + sνH with x ∈ H , s ∈ R, for l = 1, 2 we further obtain that Z G (φ(x, y) dxl∧dy1) = (−1)2−l φ(x, u (x))∂ u1 (x) dx uk b k xl k U Z 3−l 0 l = (−1) φ(x, uk(x)) sin(ϕk(νH · (x − p)))ϕk(νH · x)νH dx U 1/2k Z Z 3−l 0 0 l 1 0 = (−1) φ x + sνH , exp ιϕk(s) sin ϕk(s) ϕk(s) ds νH dH (x ) H −1/2k 1/2 Z Z 3−l 0 t 0 l 1 0 = (−1) φ x + k νH , exp ιϕ(t) sin ϕ(t) ϕ (t) dt νH dH (x ) H −1/2 Z Z 2−l 0 t 1 l 1 0 = (−1) φ x + k νH , y dy νH dH (x )

H γ± Z Z 2−l 0 1 l 1 0 l 1 → (−1) φ x , y dy νH dH (x ) = Gu(φ(x, y) dxb∧dy ) ,

H γ± 1 − + where γ± ⊂ S is the geodesic arc connecting u to u . With analogous computations one proves G (φ(x, y) dxl∧dy2) → G (φ(x, y) dxl∧dy2). uk b u b Hence, due to Stokes’ Theorem we have that

0 = ∂Guk (η) = Guk (dη) → Gu(dη) = ∂Gu(η) , which proves (3.34). 2 2 Case 3: supp(η) ⊂ Bρ(p)×R , where p = εi + εe1 + εe2 for some i ∈ Z . In this case we will approximate the current Gu with graphs of a sequence of functions uk which are Lipschitz outside the point p. For notation simplicity we let ϕb1, ϕb2, ϕb3, ϕb4 ∈ [0, 2π) be the phases deﬁned by the relations

u(εi + εe1 + εe2) =: u1 = exp(ιϕ1) , u(εi + εe2) =: u2 = exp(ιϕ2) , b b (3.35) u(εi) =: u3 = exp(ιϕb3) , u(εi + εe1) =: u4 = exp(ιϕb4) . We deﬁne the auxiliary angles

ϕσ(h+1) := ϕbσ(h) + Ψ(ϕbσ(h+1) − ϕbσ(h)) , (3.36) for h = 1, 2, 3, 4, where σ(h) ∈ {1, 2, 3, 4} is such that σ(h) ≡ h mod 4 (ϕσ(h+1) is the oriented angle in [−π, π] between the two vectors uσ(h) and uσ(h+1) ). We introduce the 2π -periodic function ϕk : R → R  ϕ , if − π + h π < ϑ ≤ h π − 1 ,  bσ(h) 4 2 2 2k ϕ (ϑ) := π 1 π 1 π 1 k ϕbσ(h) + k ϕσ(h+1) − ϕbσ(h) (ϑ − h 2 + 2k ) , if h 2 − 2k < ϑ < h 2 + 2k  π 1 π π ϕσ(h+1) , if h 2 + 2k ≤ ϑ < 4 + h 2 . π π π π for ϑ ∈ − 4 + h 2 , 4 + h 2 ), h ∈ Z. The function ϕk might have jumps at the points π π 4 + h 2 , h ∈ Z; note, however, that the amplitude of the jump is given by ϕbσ(h+1) − ϕσ(h+1) = ϕbσ(h+1) − ϕbσ(h) − Ψ(ϕbσ(h+1) − ϕbσ(h)) = Q(ϕbσ(h+1) − ϕbσ(h)) ∈ 2πZ , according to (2.4). FROM THE N -CLOCK MODEL TO THE XY MODEL 27

ϕ1

ϕ4 2π

π π π − 4 0 4 2 ϕ 1 1 k

Figure 6. Example for the deﬁnition of ϕk for h = 0 .

1 1 1 We now define a map vk : S → S . Given y ∈ S , let ϑ(y) ∈ [0, 2π) be the angle such that y = exp(ιϑ(y)) and set vk(y) := exp ιϕk(ϑ(y)) . The definition actually does not depend on the choice of the phase ϑ(y), due to the 2π - periodicity of ϕk . Thus we could also choose ϑ(y) ∈ [2πh, 2π(h + 1)) for any h ∈ Z. Note that vk is continuous: indeed the possible jumps of ϕk have amplitude in 2πZ, and thus are not seen by vk . In particular, we can compute the degree of the map vk via the formula Z Z 3 # X deg(v )2π = deg(v ) ω 1 = v ω 1 = ϕ − ϕ k k S k S σ(h+1) bσ(h) 1 1 h=0 S S 3 X = Ψ(ϕbσ(h+1) − ϕbσ(h)) = du(εi)2π , h=0 1 1 where ωS is the volume form on S and du(εi) is the discrete vorticity defined in (2.6). 1 We now define the map uk : Bρ(p) → S by x−p uk(x) := vk |x−p| . Note that, if (r, ϑ) are polar coordinates for the point x − p, then the polar coordinates of uk(x) are (1, ϕk(ϑ)).

u2 u1 u2 u1 1 ∼ k u3 u4 u3 u4

Figure 7. Example of the approximation uk (on the left) of the function u (on the right). The jump set of the function u is expanded and a transition between the jumps of u is 1 constructed using the geodesic arcs in S between the traces. If u has a nontrivial discrete vorticity as in the picture, then the graph Guk of the function uk has a hole in the center, x as it happens for the graph of the map x 7→ |x| . The hole is then preserved in the passage to limit to Gu , see formula (3.37).

By [31, 3.2.2, Example 2] we get that 1 1 1 2 2 ∂Guk |Bρ(p)× = − deg(vk)δp× S = −du(εi)δp× S = −µu× S |Bρ(p)× . (3.37) R J K J K J K R 2 Therefore, to conclude the proof it suﬃces to show the convergence Guk *Gu in D2(Ω×R ), so that 1 −µu× S (η) = ∂Guk (η) → ∂Gu(η) . ∞ J K2 To do so, let us ﬁx φ ∈ Cc (Bρ(p)×R ). Since uk → u in measure, we have that Z Z

Guk (φ(x, y) dx) = φ(x, uk(x)) dx → φ(x, u(x)) dx = Gu(φ(x, y) dx) .

Bρ(p) Bρ(p) 28 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

l m To compute the limit on forms of the type φ(x, y) dxb∧dy , observe that uk is not constant only in the 4 sectors of Bρ(p) given in polar coordinates by

h π 1 π 1 Ak := (r, ϑ): r ∈ (0, ρ), ϑ ∈ h 2 − 2k , h 2 + 2k , h ∈ {0, 1, 2, 3} , thus, for l, m = 1, 2, Z 2−l l m m (−1) G (φ(x, y) dx ∧dy ) = φ(x, u (x))∂ l u (x) dx uk b k x k

Bρ(p) 3 Z X m = φ(x, uk(x))∂xl uk (x) dx .

h=0 h Ak

h The integrals on the sets Ak can be computed in polar coordinates. We show the computations for h = 0 and m = 1, the other cases being analogous. Changing variables in the integral on the interval (−1/2k, 1/2k) we obtain Z 1 φ(x, uk(x))∂x2 uk(x) dx

0 Ak ρ 1/2k Z Z 0 = − φ r exp ιϑ , exp ιϕk(ϑ) sin(ϕk(ϑ))ϕk(ϑ) cos(ϑ) dϑ dr 0 −1/2k ρ 1/2 Z Z t 0 t = − φ r exp ι k , γ41(t) sin(γ41(t))γ41(t) cos k dt dr 0 −1/2 ρ 1/2 Z Z Z Z 0 1 2 1 → − φ (r, 0), γ41(t) sin(γ41(t))γ41(t) dt dr = φ(x, y) dy ν dH (x) ,

0 −1/2 J41 γ41 and

ρ 1/2k Z Z Z 1 0 φ(x, uk(x))∂x1 uk(x) dx = φ r exp ιϑ , exp ιϕk(ϑ) sin(ϕk(ϑ))ϕk(ϑ) sin(ϑ) dϑ dr 0 0 Ak −1/2k ρ 1/2 Z Z t 0 t = φ r exp ι k , γ41(t) sin(γ41(t))γ41(t) sin k dt dr → 0 , 0 −1/2

1 where t ∈ (−1/2, 1/2) 7→ γ41(t) := exp ι(ϕb4 + (ϕ1 − ϕb4)(t + 2 )) is a parametrization of the 1 geodesic arc γ41 ∈ S which connects u4 to u1 (cf. the deﬁnition of ϕb4 in (3.35) and of ϕ1 in (3.36)) and J41 is the subset of Ju ∩ Bρ(p) where u jumps from u4 to u1 , oriented with normal ν = (0, 1). This concludes the proof.

f 1 1 Lemma 3.12. Let µε, µ ∈ Mb(Ω) and assume that µε * µ in Ω. Then µε× S * µ× S 2 J K J K in D1(Ω×R ).

∞ 2 Proof. Let us ﬁx φ ∈ Cc (Ω×R ). For l = 1, 2, by the very deﬁnition of the product of a 0-current and a 1-current we infer

1 l 1 l µε× S (φ(x, y) dx ) = 0 = µ× S (φ(x, y) dx ) . J K J K FROM THE N -CLOCK MODEL TO THE XY MODEL 29

Hence we only need to compute for m = 1, 2 Z Z Z 1 m 1 m m µε× S (φ(x, y) dy ) = S (φ(x, y) dy ) dµε(x) = φ(x, y) dy dµε(x) J K J K 1 Ω Ω S Z Z m m = ψ (x) dµε(x) → ψ (x) dµ(x) Ω Ω m R m 0,1 since ψ (x) := 1 φ(x, y) dy is a function in C (Ω). S c

4. Proofs in the regime ε θε ε| log ε| We now start with the proof of Theorems 1.2 and 1.3, which share some similarities (actually the ﬁrst one is a special case of the second one). We split the arguments in several intermediate results, which we will also use to analyze the other regimes. Due to that fact, we state in each result the assumptions on θε separately. In what follows, for A ⊂ R2 we shall use the localized energy given by 1 X E (u; A) := ε2|u(εi) − u(εj)|2. ε 2 hi,ji εi,εj∈A 4.1. Compactness and lower bound in absence of vortices. In this section we consider 2 1 a generic sequence uε : ε → Sε such that Eε(uε) is bounded. First we prove that such Z εθε sequences are compact in L1(Ω) with limits in BV (Ω; S1). 1 Proposition 4.1 (Compactness in BV ). Assume that θε 1 and Eε(uε) ≤ C . Then εθε 1 there exists a subsequence (not relabeled) and a function u ∈ BV (Ω; S ) such that uε → u 1 2 in L (Ω) and uε * u in BVloc(Ω; R ).

1 Proof. Let us ﬁx A ⊂⊂ Ω. Observe that dS (u(εi), u(εj)) ≥ θε whenever u(εi) 6= u(εj). Thus by (2.1) we infer that

1 1 X 2 2 C ≥ Eε(uε) = ε |u(εi) − u(εj)| εθε 2εθε hi,ji

1 X 2 1 1 = ε 2 sin 2 dS u(εi), u(εj) |u(εi) − u(εj)| 2εθε (4.1) hi,ji

θε θε sin( 2 ) X 2 sin( 2 ) ≥ ε|u(εi) − u(εj)| ≥ |Duε|(A) . θε θε hi,ji 1 Hence uε is bounded in BV (A; S ) and we conclude that (up to a subsequence) uε → u 1 2 1 in L (A) and uε * u in BV (A; R ) for some u ∈ BV (A; S ) with |Du|(A) ≤ C . Since A ⊂⊂ Ω was arbitrary and the constant C does not depend on A, the claim follows from a diagonal argument and the equiintegrability of uε . In the next lemma we prove lower bound for the energy still at the discrete level.

Lemma 4.2. Assume that θε 1 and let σ ∈ (0, 1). Then for ε small enough we have 2 1 |uε(εi) − uε(εj)| ≥ (1 − σ)θεdS uε(εi), uε(εj) . (4.2) In particular 1 1 X 1 Eε(uε) ≥ (1 − σ) εdS uε(εi), uε(εj) . (4.3) εθε 2 hi,ji

1 1 Proof. By (2.1) we have that |uε(εi) − uε(εj)| = 2 sin 2 dS (uε(εi), uε(εj) . Since uε takes 1 values in Sε , the geodesic distance dS (uε(εi), uε(εj)) is an integer multiple of θε , i.e., there 1 exists a k ∈ N (depending on i, j , and ε) such that dS (uε(εi), uε(εj)) = kθε . We can assume k 6= 0, otherwise inequality (4.2) is trivial. Moreover, note that kθε ≤ π . 30 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

θε Due to Taylor’s formula there exists a ζ ∈ [0, k 2 ] such that 3 3 θε θε 1 θε θε 1 θε sin k 2 = k 2 − 6 cos(ζ) k 2 ≥ k 2 − 6 k 2 √ θε Dividing by k 2 we get that

θε sin k √ h 2i √ 2 ≥ k 1 − 1 k θε (4.4) θε 6 2 k 2

If k ≥ 9, using the fact that kθε ≤ π ≤ 4 we obtain that sin k θε √ √ 2 ≥ k 1 ≥ 1 . (4.5) θε 3 k 2 Otherwise, if k ≤ 8, (4.4) directly implies

sin k θε √ 2 ≥ 1 − 3θ2 ≥ 1 − σ , (4.6) θε ε k 2 for ε small enough. Squaring both sides in (4.5) and (4.6) (notice that kθε ∈ [0, π] implies θε sin k 2 ≥ 0) we have that 2 θε 2 4 sin k 2 ≥ (1 − σ)kθε .

1 We conclude the proof of (4.2) by replacing kθε = dS (uε(εi), uε(εj)) in the last inequality and by (2.1).

We now recast the energy as a parametric integral of the currents Guε . To do so, we 2 2 deﬁne the convex and positively 1-homogeneous function Φ: Λ2(R ×R ) 7→ R by Φ(ξ) := p(ξ21)2 + (ξ22)2 + p(ξ11)2 + (ξ12)2 (4.7) for every

00 21 22 11 12 00 2 2 ξ = ξ e1 ∧ e2 + ξ e1 ∧ e¯1 + ξ e1 ∧ e¯2 + ξ e2 ∧ e¯1 + ξ e2 ∧ e¯2 + ξ e¯1 ∧ e¯2 ∈ Λ2(R ×R ) . Lemma 4.3. For every open set A ⊂⊂ Ω and ε small enough we have Z 1 X εd 1 u (εi), u (εj) ≥ Φ(G~ ) d|G | . 2 S ε ε uε uε hi,ji 2 A×R

Proof. By the explicit formulas (3.32)–(3.33) for the orientation of Guε we infer that h i Φ(G~ )(x, y) = 1 (x) |ν2 (x)|p(y2)2 + (y1)2 + |ν1 (x)|p(y2)2 + (y1)2 uε Juε uε uε = 1 (x)|ν (x)| . Juε uε 1 2 Moreover, we recall that |Guε | = H Mε , where (a) (j) ε Mε = Mε ∪ Mε = {(x, uε(x)) : x ∈ Ω \ Juε } ∪ {(x, y): x ∈ Juε , y ∈ supp(γx)} , ε − + γx being the geodesic arc that connects uε (x) to uε (x). Therefore Z Z Z Z ~ 2 (j) 1 1 Φ(Guε ) d|Guε | = |νuε |1 dH Mε = dH (y) |νuε (x)|1 dH (x)

2 2 ε A×R A×R Juε ∩A supp(γx) Z − + 1 1 = dS uε (x), uε (x) |νuε (x)|1 dH (x)

Juε ∩A 1 X ≤ εd 1 u (εi), u (εj) . 2 S ε ε hi,ji

Next we show that energy bounds also yield compactness for the associated currents Guε . FROM THE N -CLOCK MODEL TO THE XY MODEL 31

1 Proposition 4.4 (Compactness in cart(Ω×S )). Assume that θε ε| log ε| as well as 1 2 Eε(uε) ≤ C . Let Gu ∈ D2(Ω× ) be the currents associated to uε deﬁned as in (3.28)– εθε ε R 2 (3.30). Then there exists a subsequence (not relabeled) and a current T ∈ D2(Ω×R ) such 2 1 that Guε *T in D2(Ω×R ). Moreover, T ∈ cart(Ω×S ) and uT = u a.e. in Ω, where 1 uT is the BV function associated to T given by Theorem 3.5 and u ∈ BV (Ω; S ) is the function given by Proposition 4.1. Proof. Let us ﬁx an open set A ⊂⊂ Ω. Since Φ(ξ) ≥ p(ξ21)2 + (ξ22)2 + (ξ11)2 + (ξ12)2 , we deduce the estimate 2 (a) 2 (j) 2 |Guε |(A×R ) = |Guε |(M ∩ A×R ) + |Guε |(M ∩ A×R ) Z 1 X ≤ |A| + Φ(G~ u ) d|Gu | ≤ |Ω| + εd 1 uε(εi), uε(εj) ε ε 2 S (4.8) 2 hi,ji A×R 2 ≤ |Ω| + Eε(uε) ≤ C, εθε where in the last inequality we employed (4.3) with σ = 1/2. By the Compactness Theorem for currents [31, 2.2.3, Proposition 2 and Theorem 1-(i)] we deduce that there exists a 2 subsequence (not relabeled) and a current T ∈ D2(Ω×R ) with |T | < ∞ such that Guε *T 2 in D2(Ω×R ). Let us prove that T ∈ cart(Ω×S1):

• T is an i.m. rectifiable current: each Guε is an i.m. rectifiable current (see Proposi- 2 2 tion 3.10). Moreover, |Guε |(A×R ) and |∂Guε |(A×R ) are equibounded for every A ⊂⊂ Ω (by (4.8) and the argument for the next point). Thus by the Closure Theorem [31, 2.2.4, Theorem 1] the limit T is an i.m. rectifiable current too. 1 2 2 • ∂T |Ω×R = 0: by Proposition 3.11 we have ∂Guε |Ω×R = −µuε × S . By Re- 2 J K mark 2.5, Lemma 3.12, and since ∂Guε * ∂T in D1(Ω×R ), we conclude that 2 ∂T |Ω×R = 0. Ω 2 • π#T = Ω : let us fix ω ∈ D (Ω), i.e., a 2-form of the type ω(x) = φ(x) dx with ∞ J K R Ω φ ∈ Cc (Ω). Then Guε (ω) = Ω φ(x) dx. Thus π#Guε = Ω . Passing to the limit as ε → 0 we get the desired condition. J K 2 • T | dx ≥ 0: let ω ∈ D2(Ω×R ) be of the form ω(x, y) = φ(x, y) dx with φ ∈ ∞ 2 R Cc (Ω×R ) with φ ≥ 0. Then Guε (ω) = Ω φ(x, uε(x)) dx ≥ 0. Passing to the limit as ε → 0 we get T (ω) ≥ 0. •| T | < ∞: this is a consequence of the Compactness Theorem for currents (see above). R •k T k1 < ∞: note that, by (3.5), kGuε k1 = Ω |uε| dx = |Ω|. By the lower semi- 2 continuity of k · k1 with respect to the convergence in D2(Ω×R ) we deduce that kT k1 ≤ |Ω|. • supp(T ) ⊂ Ω×S1 : let us fix ω ∈ D2(Ω×R2) with supp(ω) ⊂⊂ (Ω×R2) \ (Ω×S1).

Then Guε (ω) = 0. Passing to the limit as ε → 0, we conclude that T (ω) = 0.

To prove that uT = u a.e. we observe that uε → u implies Z Z

Guε (φ(x, y) dx) = φ(x, uε(x)) dx → φ(x, u(x)) dx Ω Ω ∞ 2 for every φ ∈ Cc (Ω×R ). On the other hand, due to Theorem 3.5 Z

Guε (φ(x, y) dx) → T (φ(x, y) dx) = φ(x, uT (x)) dx . Ω By the arbitrariness of φ, this concludes the proof.

Proposition 4.5 (Lower bound for the parametric integral). Assume that θε ε| log ε| 1 2 and that Eε(uε) ≤ C . Let Gu ∈ D2(Ω× ) be the currents associated to uε deﬁned as εθε ε R 1 in (3.28)–(3.30) and assume that Guε *T , where T ∈ cart(Ω×S ) is a current given by 32 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

Proposition 4.4, represented as T = T~|T |. Then Z Z Φ(T~) d|T | ≤ lim inf Φ(G~ u ) d|Gu | . (4.9) ε→0 ε ε 2 2 A×R A×R for every open set A ⊂⊂ Ω. Proof. The statement is a direct consequence of the lower semicontinuity of parametric integrals with respect to the mass bounded weak convergence of currents, [32, 1.3.1, Theo- rem 1]. We can write explicitly the parametric integral in the left-hand side of (4.9) in terms of the BV function u, limit of the sequence uε . We recall that by (3.17) the jump-concentration part of T is given by Z Z (jc) l m 2−l m l 1 T (φ(x, y) dxb∧dy ) = (−1) φ(x, y) dy νT (x) dH (x) . T JT γx 1 For H -a.e. x ∈ JT we deﬁne the number Z T 1 `T (x) := length(γx ) = |m(x, y)| dH (y) , (4.10)

T supp(γx ) T where m(x, y) is the integer deﬁned in (3.18). Notice that by length(γx ) we mean the T 1 length of the curve γx counted with its multiplicity and not the H Hausdorﬀ measure of − + 1 its support. Observe that, in particular, `T (x) = dS u (x), u (x) if x ∈ Ju \L, whilst `T (x) = 2π|k(x)| if x ∈ L \ Ju . The full form of the parametric integral is contained in the lemma below.

Lemma 4.6. Let T ∈ cart(Ω×S1) and u ∈ BV (Ω; S1) be as in Proposition 4.4, and let Φ be the parametric integrand deﬁned in (4.7). Then Z Z Z (c) 1 Φ(T~) d|T | = |∇u|2,1 dx + |D u|2,1(Ω) + `T (x)|νT (x)|1 dH (x) .

2 Ω×R Ω JT Proof. We employ the mutually singular decomposition given by Theorem 3.5 and Propo- sition 3.8, so that |T | = H2 M(a) + H2 M(c) + |T (jc)|. First of all note that by (3.20) ∞ and (3.11) for every ψ ∈ Cc (Ω) we have Z 1 Z ψ(x) dH2(x, y) = ψ(x) dx , (4.11) p1 + |∇u(x)|2 M(a) Ω since both integrals are equal to T (a)(ψ(x) dx). By approximation, the above equality is true for every ψ :Ω → R such that (x, y) 7→ ψ(x) is H2 M(a) -measurable and x 7→ ψ(x) 2 is L -measurable. Note that x 7→ |∇u(x)|2,1 satisﬁes these measurability properties thanks to (3.20). In particular, we deduce that Z Z 2 (a) 1 2 Φ(T~(x, y)) dH M (x, y) = |∇u(x)|2,1 dH (x, y) p1 + |∇u(x)|2 2 Ω×R M(a) Z = |∇u(x)|2,1 dx . Ω

Next note that, by (3.13), for every function ψ ∈ Cc(Ω) it holds true (c) (c) sup T (ωe) = sup T (ω) . (4.12) |ωe(x,y)|≤ψ(x) |ω(x)|≤ψ(x) 2 2 (c) ωe∈D (Ω×R ) ω |D u|−measurable 2 2 (c) Indeed given ωe ∈ D (Ω×R ) such that |ω(x, y)| ≤ ψ(x), one can deﬁne the |D u|- measurable 2-form ω(x) := ωe(x, ueT (x)) to prove that the left-hand side is greater than or equal to the right-hand side. For the reverse inequality, given a |D(c)u|-measurable FROM THE N -CLOCK MODEL TO THE XY MODEL 33

2-form ω such that |ω(x)| ≤ ψ(x), one can regularize it and then deﬁne the 2-form ∞ ωe(x, y) := ω(x)ζ(y), where ζ ∈ Cc (B2) is such that ζ(y) = 1 for |y| ≤ 1 (note that ζ does not aﬀect the value of T (c)(ω) thanks to (3.13)). Since |T (c)| = H2 M(c) and by (3.13), equality (4.12) implies that Z Z ψ(x) dH2(x, y) = ψ(x) d|D(c)u|(x) , (4.13)

M(c) Ω for every function ψ ∈ Cc(Ω). By approximation, (4.13) holds true for every ψ :Ω → R such 2 (c) (c) dD(c)u that ψ is H M -measurable and |D u|-measurable. The function x 7→ d|D(c)u| (x) 2,1 satisﬁes these measurability properties, cf. (3.21), thus (3.21) implies

Z Z dD(c)u Φ(T~(x, y)) dH2 M(c)(x, y) = (x) dH2(x, y) d|D(c)u| 2,1 2 Ω×R M(c) (c) = |D u|2,1(Ω) .

Finally, by (3.22) we get that Z Z (jc) 2 (jc) Φ(T~(x, y)) d|T |(x, y) = |m(x, y)||νT (x)|1 dH M (x, y)

2 2 Ω×R Ω×R Z Z 1 1 1 = M(jc) (x, y)|m(x, y)||νT (x)|1 dH (y) dH (x)

1 JT S Z Z 1 1 = |m(x, y)| dH (y) |νT (x)|1 dH (x)

T JT supp(γx ) Z 1 = `T (x)|νT (x)|1 dH (x) .

In the second equality we employed the coarea formula for rectiﬁable sets [29, Theorem 3.2.22] 1 1 (applied with W = JT ×S , Z = JT , f given by the projection JT ×S → JT , and 1 g = M(jc) |m||νT |1 ) and in the third equality we used (3.27).

Remark 4.7. In presence of vortices, we will work with cartesian currents on punctured open PN sets. Given a measure µ = h=1 dhδxh and an open set A, we adopt the notation

Aµ := A \ supp(µ) = A \{x1, . . . , xN }

ρ SN and Aµ := A \ h=1 Bρ(xh). 1 2 We observe that a current T ∈ cart(Ωµ×S ) can be extended to a current T ∈ D2(Ω×R ). 1 Indeed, since T ∈ cart(Ωµ×S ), it can be represented as Z 2 2 2 T (ω) = hω, ξiθ dH M , for ω ∈ D (Ωµ×R ) ,

2 Ωµ×R

1 2 according to the notation in (3.2), where M ⊂ Ωµ×S H -a.e. (cf. the proof of Propo- sition 3.8 for the last fact). The integral above can be extended to a linear functional on forms ω ∈ D2(Ω×R2), namely Z 2 2 2 T (ω) = hω, ξiθ dH M , for ω ∈ D (Ω×R ) .

2 Ω×R 34 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

2 2 To prove the continuity of this functional, let us ﬁx a form ω ∈ D (Ω×R ) with supx |ω(x)| ≤ 1. We have the bound Z 2 |T (ω)| ≤ |T ((1 − ζ)ω)| + ζhω, ξiθ dH M 2 Ω×R (4.14) N Z 2 X 2 ≤ |T |(Ωµ×R ) + |θ| dH M

h=1 2 Bρ(xh)×R

∞ SN where ζ ∈ Cc (Ω) is such that 0 ≤ ζ ≤ 1, supp(ζ) ⊂ h=1 Bρ(xh), and ζ ≡ 1 on Bρ/2(xh) 2 for every h = 1,...,N . Letting ρ → 0 in the inequality above, we get |T (ω)| ≤ |T |(Ωµ×R ) 2 2 2 1 2 since H M ∩ ({xh}×R ) ≤ H {xh}×S = 0 for h = 1,...,N and θ is H M- 2 summable. This shows that T ∈ D2(Ω×R ). 2 2 Moreover, by the arbitrariness of ω in (4.14) we deduce that |T |(Ω×R ) = |T |(Ωµ×R ) and, in particular, Z Z Z Z (c) 1 Φ(T~) d|T | = Φ(T~) d|T | = |∇u|2,1 dx + |D u|2,1(Ω) + `T (x)|νT (x)|1 dH (x) .

2 2 Ω×R Ωµ×R Ω JT

To state the ﬁnal lower bound result, for every u ∈ BV (Ω; S1) we introduce the energy Z 1 1 J (u; Ω) := inf `T (x)|νT (x)|1 dH (x): T ∈ cart(Ω×S ), uT = u a.e. in Ω . (4.15)

1 Proposition 4.8 (Lower bound). Assume that θε ε| log ε| and Eε(uε) ≤ C . If εθε 1 1 uε → u in L (Ω), where u ∈ BV (Ω; S ) is as in Proposition 4.1, then Z (c) 1 |∇u|2,1 dx + |D u|2,1(Ω) + J (u; Ω) ≤ lim inf Eε(uε) . (4.16) ε→0 εθε Ω Proof. Fix σ ∈ (0, 1) and A ⊂⊂ Ω an open set. By Lemma 4.2 and Lemma 4.3 we deduce that Z ~ 1 (1 − σ) Φ(Guε ) d|Guε | ≤ E(uε) . εθε 2 A×R Passing to the limit as ε → 0, Proposition 4.5 implies Z 1 (1 − σ) Φ(T~) d|T | ≤ lim inf E(uε) . ε→0 εθε 2 A×R Letting σ → 1 and A → Ω, by Lemma 4.6 we conclude that (4.16) holds true. Remark 4.9. The lower bound (4.16) is greater or equal than the anisotropic total variation, namely Z Z Z (c) (c) − + 1 1 |∇u|2,1 dx+|D u|2,1(Ω)+J (u; Ω) ≥ |∇u|2,1 dx+|D u|2,1(Ω)+ dS (u , u )|νu|1 dH

Ω Ω Ju

1 for all u ∈ BV (Ω; S ). This can be seen using the deﬁnition of `T (x) for a given T ∈ 1 1 − + 1 cart(Ω×S ) with uT = u. Indeed, for H -a.e. x ∈ Ju ∩ L we have dS (u (x), u (x)) ≤ T T − + 1 length(γx ) = `T (x), since γx is a curve connecting u (x) and u (x) in S . 4.2. Compactness and lower bound in presence of vortices. Next we extend the results of the previous subsection to the case of M vortices, which corresponds to a certain 2 rate of blow-up of the energy. Again we consider a general sequence uε : εZ → Sε and the associated current Guε . FROM THE N -CLOCK MODEL TO THE XY MODEL 35

Proposition 4.10 (M vortices – Compactness). Assume that ε θε ε| log ε| and that there exist M ∈ N and C > 0 such that 1 ε Eε(uε) − 2πM| log ε| ≤ C. (4.17) εθε θε PN f Then there exists µ = h=1 dhδxh with dh ∈ Z such that µuε * µ (up to a subsequence) and |µ|(Ω) ≤ M . If, in addition, |µ|(Ω) = M , then there exist u ∈ BV (Ω; S1) and 2 T ∈ D2(Ω×R ) such that 1 2 ∗ 2 (i) uε → u in L (Ω; R ) and uε * u weakly* in BVloc(Ωµ; R ); 1 (ii) T ∈ cart(Ωµ×S ) and uT = u a.e. in Ω; 2 (iii) Guε *T in D2(Ωµ×R ) (up to a subsequence); 1 (iv) ∂T |Ω× 2 = −µ× S . R J K Proof. From (4.17) it follows that 1 θ E (u ) ≤ 2πM + C ε , ε2| log ε| ε ε ε| log ε| f PN so that by Proposition 2.4 we get that (up to a subsequence) µuε * µ = h=1 dhδxh and |µ|(Ω) ≤ M . From now on we assume that the equality |µ|(Ω) = M holds true, namely PN h=1 |dh| = M . Let ρ > 0 small enough such that the balls Bρ(xh) are pairwise disjoint and their closure is contained in Ω. We recall the localized lower bound for the XY -model in [8, Theorem 3.1], which states 1 ρ lim inf Eε(uε; Bρ(xh)) − 2π|dh| log ≥ Ce for some Ce ∈ R . (4.18) ε→0 ε2 ε

From this inequality and the fact that ε θε we deduce that 1 ε lim inf Eε(uε; Bρ(xh)) − 2π|dh|| log ε| ε→0 εθε θε 1 ε ε = lim inf Eε(uε; Bρ(xh)) − 2π|dh|| log ε| − 2π|dh| log ρ (4.19) ε→0 εθε θε θε ε 1 ρ = lim inf 2 Eε(uε; Bρ(xh)) − 2π|dh| log ≥ 0 . ε→0 θε ε ε Summing over h = 1,...,N , the superadditivity of the lim inf yields N X 1 ε lim inf Eε(uε; Bρ(xh)) − 2πM| log ε| ≥ 0 . (4.20) ε→0 εθε θε h=1 Therefore the bound (4.17) implies N 1 ρ X 1 ε lim sup Eε(uε;Ωµ) ≤ C − lim inf Eε(uε; Bρ(xh)) − 2πM| log ε| ≤ C, ε→0 εθε ε→0 εθε θε h=1 so that, for ε small enough, 1 ρ Eε(uε;Ωµ) ≤ 2C, (4.21) εθε where C is independent of ρ. By Proposition 4.1 and Proposition 4.4, with a diagonal 1 1 ∗ argument we obtain that there exist u ∈ BV (Ω; S ) and T ∈ cart(Ωµ×S ) such that uε * u 1 2 weakly* in BVloc(Ωµ; S ), Guε *T in D2(Ωµ×R ) up to a subsequence, and uT = u a.e. in Ω. Since uε is equiintegrable, the local weak* BV -convergence further implies strong convergence in L1(Ω; R2). Thus (i)–(iii) hold true. 2 By Remark 4.7, the current T can be extended to a current T ∈ D2(Ω×R ). Thus, it only remains to prove (iv). The argument is local and we can work close to a single atom xh of µ. Without loss of generality, we can assume that xh = 0 and Ω = B , where B is the unit ball centered at 0. First of all let us notice that supp(∂T ) ⊂ {0}×S1 . Indeed, on the one hand if ω ∈ D1(B×R2) is such that supp(ω) ⊂ (B×R2) \ ({0}×R2), 36 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

1 1 2 then ∂T (ω) = 0, since T ∈ cart (B \{0})×S ; on the other hand, if ω ∈ D (B×R ) is such that supp(ω) ⊂ (B×R2) \ (B×S1), then supp(dω) ⊂ (B×R2) \ (B×S1) and thus ∂T (ω) = T (dω) = 0, since supp(T ) ⊂ B×S1 . In conclusion supp(∂T ) ⊂ ({0}×R2) ∩ (B×S1) = {0}×S1 . Being ∂T a boundaryless 1-current with support in a 1-dimensional manifold, the Constancy Theorem [31, 5.3.1, Theorem 2] yields the existence of a number c ∈ R such that 1 2 ∂T |B×R = −c δ0× S . ∞ J K 1 Let us now fix a function ζ ∈ Cc (B) with ζ ≡ 1 in the ball B1/2 of radius 2 centered 1 1 at 0. Let us define the 1-form ω = ζωS , ωS being the 0-homogeneous extension of the 1 2 2 2 volume form of S to R \{0}. Observing that dω ∈ D (B \{0})×R , the convergence f in (ii), Proposition 3.11, and the flat convergence µuε * µ yield

−c 2π = ∂T (ω) = T (dω) = lim Gu (dω) = lim ∂Gu (ω) = lim −hµu , ζi2π = −hµ, ζi2π ε→0 ε ε→0 ε ε→0 ε and thus c = µ({0}). This concludes the proof. We are now in a position to prove the lower bound with M vortices. To state the result, we introduce the set of admissible currents 2 1 Adm(µ, u; Ω) := T ∈ D2(Ω×R ): T ∈ cart(Ωµ×S ) , (4.22) 1 ∂T |Ω× 2 = −µ× S , uT = u a.e. in Ω R J K and, similarly to (4.15), the energy Z 1 J (µ, u; Ω) := inf `T (x)|νT (x)|1 dH (x): T ∈ Adm(µ, u; Ω) (4.23)

PN 1 for every µ = h=1 dhδxh and u ∈ BV (Ω; S ) with `T (x) deﬁned in (4.10). One can prove that Adm(µ, u; Ω) is non-empty.9

Proposition 4.11 (M vortices – Lower bound ). Assume that θε ε| log ε| and (4.17) f PN 1 2 holds. Assume further that µuε * µ = h=1 dhδxh with |µ|(Ω) = M , uε → u in L (Ω; R ) with u ∈ BV (Ω; S1) as in Proposition 4.10. Then Z (c) 1 ε |∇u|2,1 dx + |D u|2,1(Ω) + J (µ, u; Ω) ≤ lim inf Eε(uε) − 2πM| log ε| . (4.24) ε→0 εθε θε Ω ρ Proof. Let us ﬁx A ⊂⊂ Ωµ and σ ∈ (0, 1). Then there exists ρ > 0 such that A ⊂⊂ Ωµ . Thanks to (4.3) and Lemma 4.3, for ε small enough we infer that Z ~ 1 ρ (1 − σ) Φ(Guε ) d|Guε | ≤ Eε(uε;Ωµ) . εθε 2 A×R 2 Passing to a subsequence if necessary, we have that Guε *T in D2(Ωµ × R ) for some 2 T ∈ D2(Ω × R ) given by Proposition 4.10. As in Proposition 4.5 and from the lower bound (4.20) we deduce that Z Z (1 − σ) Φ(T~) d|T | ≤ lim inf (1 − σ) Φ(G~ u ) d|Gu | ε→0 ε ε 2 2 A×R A×R 1 ρ 1 ε ≤ lim inf Eε(uε;Ωµ) ≤ lim inf Eε(uε) − 2πM| log ε| . ε→0 εθε ε→0 εθε θε

9 1 By Proposition 3.9 there exists a current T ∈ cart(Ω×S ) such that uT = u. Let γ1, . . . , γN be pairwise disjoint unit speed Lipschitz curves such that γh connects xh to ∂Ω. Deﬁne Lh to be the 1- PN 1 current τ(supp(γh), −dh, γ˙h), so that ∂Lh = dhδxh . Then T + h=1 Lh× S ∈ Adm(µ, u; Ω). J K FROM THE N -CLOCK MODEL TO THE XY MODEL 37

Letting A → Ωµ and σ → 0 we conclude that Z 1 ε Φ(T~) d|T | ≤ lim inf Eε(uε) − 2πM| log ε| . ε→0 εθε θε Ωµ By Proposition 4.10 (ii) & (iv) the current T is admissible in the infimum problem defining J (µ, u; Ω), so that (4.24) is a direct consequence of Lemma 4.6 and Remark 4.7. 4.3. Upper bound in absence of vortices. To reduce notation, for u ∈ BV (Ω; S1) we set Z (c) E(u) := |∇u|2,1 dx + |D u|2,1(Ω) + J (u; Ω) Ω with J (u; Ω) given by (4.15). The proof of the Γ-limsup inequality is done in several steps which gradually simplify the map u ∈ BV (Ω; S1) that we want to approximate. The main reason is that we are able to do precise estimates for the upper bound only in the case where 2 the recovery sequence uε : εZ ∩ Ω → Sε approximates a piecewise constant map. Specifically, the simplifications of the map u ∈ BV (Ω; S1) can be summarized as follows: 1 (1) We find a cartesian current T ∈ cart(Ω×S ) such that uT = u and whose parametric integral R Φ(T~) d|T | is approximately equal to the energy E(u) of the BV map u, cf. (4.25) in Proposition 4.12. (2) Using the Approximation Theorem for cartesian currents, we find a map ue ∈ ∞ 1 1,1 1 1 2 C (Ω;e S )∩W (Ω;e S )(Ωe ⊃⊃ Ω) such that ue is close to u in L (Ω; R ) and whose 1,1 R R ~ anisotropic W norm E(ue) = |∇ue|2,1 dx is approximately equal to Φ(T ) d|T | (and thus to E(u)). We need regularity slightly outside Ω in the discretization step (3). To this end, we apply Lemma 3.4 to extend the current T to the set Ωe and then we apply the Approximation Theorem to the extended current, cf. Propo- sition 4.12. 2 −n (3) We consider a lattice λnZ with λn = 2 (λn ε) and we sample ue in the squares of this lattice. This allows us to define a piecewise constant map un on 2 1 2 the lattice λnZ such that un is close to ue (and thus to u) in L (Ω; R ) and R + − 1 1 whose anisotropic BV norm J dS (un , un )|νun |1 dH is approximately equal to R un R |∇ue|2,1 dx (and thus to E(u)), cf. Lemma 4.13. To recover the energy |∇ue|2,1 dx 2 in the whole Ω, we may need to sample ue also in squares of the lattice λnZ which intersect the complement of Ω: for this reason we need regularity up to a set Ωe ⊃⊃ Ω in the regularization step (2). (4) To define uε , we construct transitions in Sε between the values of un in two adjacent squares, cf. (4.35). This construction needs some care and is presented in detail in Proposition 4.16. In the next proposition we approximate the map u with a sequence of smooth maps.

Proposition 4.12. Let u ∈ BV (Ω; S1). Then there exist an open set Ωe ⊃⊃ Ω and a ∞ 1 1,1 1 sequence of smooth maps uh ∈ C (Ω;e S ) ∩ W (Ω;e S ) such that uh → u strongly in L1(Ω; R2) and Z Z (c) lim sup |∇uh|2,1 dx ≤ |∇u|2,1 dx + |D u|2,1(Ω) + J (u; Ω) . h→+∞ Ω Ω 1 Proof. Let η > 0 and let T ∈ cart(Ω×S ) with uT = u a.e. in Ω be such that Z 1 `T (x)|νT (x)|1 dH (x) ≤ J (u; Ω) + η . (4.25)

JT Note that by Lemma 4.6 Z Z Z (c) 1 |∇u|2,1 dx + |D u|2,1(Ω) + `T (x)|νT (x)|1 dH (x) = Φ(T~) d|T | ,

2 Ω JT Ω×R 38 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF where Φ is the parametric integrand deﬁned in (4.7). By Lemma 3.4 we can extend the current T to Ωe×S1 for some Ωe ⊃⊃ Ω without charging the boundary of Ω, i.e., we can assume T ∈ cart(Ωe×S1) and |T |(∂Ω×R2) = 0. Thanks to the Approximation Theorem 3.3 we ﬁnd a sequence of smooth maps uh ∈ ∞ 1 2 2 2 C (Ω;e S ) such that Guh *T in D2(Ωe×R ) and |Guh |(Ωe×R ) → |T |(Ωe×R ). In par- 2 2 2 ticular, since |T | does not charge ∂Ω × R , we have |Guh |(Ω×R ) → |T |(Ω×R ) and the 1 2 convergence uh → uT = u strongly in L (Ω; R ). Therefore, by Reshetnyak’s Continuity Theorem [12, Theorem 2.39] we have Z Z ~ ~ Φ(Guh ) d|Guh | → Φ(T ) d|T | .

2 2 Ω×R Ω×R By (3.4) and the Area Formula we can write Z Z ~ Φ(Guh ) d|Guh | = |∇uh|2,1(x) dx .

2 Ω×R Ω This implies that Z Z (c) lim sup |∇uh|2,1(x) dx ≤ |∇u|2,1 dx + |D u|2,1(Ω) + J (u; Ω) + η . h→+∞ Ω Ω Since η > 0 was arbitrary, we conclude the proof. 2 The next lemma states that it is possible to discretize on a lattice λnZ a smooth map with values in S1 in such a way that the anisotropic BV norm does not increase. The discretized maps un satisfy in addition an ’almost continuity property’, cf. (4.27), which states that if λn is chosen small enough, then the constant values of un in two neighboring cubes are close. 1 −n Lemma 4.13 (Discretization of smooth S -valued maps). Let λn := 2 , n ∈ N and let O, Oe be bounded, open sets such that O ⊂⊂ Oe . Assume that u ∈ C∞(Oe; S1) ∩ W 1,1(Oe; S1). 1 Then there exist a sequence of piecewise constant maps un ∈ PCλn (S ) such that un → u strongly in L1(O; R2) as n → +∞ and Z Z + − 1 1 lim sup dS (un , un )|νun |1 dH ≤ |∇u|2,1 dx , (4.26) n→+∞ λn Jun ∩O O where Oλn is the union of half-open squares given by

λn [ 2 O := {Iλn (λnz): z ∈ Z such that Iλn (λnz) ∩ O 6= Ø} . Moreover , for every δ > 0 there exists n = n(u, δ, Oe) such that for every n ≥ n and for 2 every z1, z2 ∈ Z with Iλn (λnz1) ∩ Iλn (λnz2) 6= Ø and Iλn (λnzi) ∩ O 6= Ø we have 1 dS un(λn(z1)), un(λn(z2)) ≤ δ . (4.27) Proof. Let O0 be an open set such that O ⊂⊂ O0 ⊂⊂ Oe and let n be so large that for every n ≥ n Oλn ⊂ O0 . 2 For every z ∈ Z such that Iλn (λnz) ∩ O 6= Ø we define 1 1 un(λnz) := u λn(z + 2 e1 + 2 e2) , 1 1 λn(z + 2 e1 + 2 e2) being the center of the square Iλn (λnz). The definition is well-posed, 1 1 2 since λn(z + 2 e1 + 2 e2) ∈ Oe . Then we extend un to λnZ by choosing an arbitrary value 1 1 in S . This defines a piecewise constant map un ∈ PCλn (S ). 0 Since u is continuous on O , it follows that un → u pointwise on O and therefore also strongly in L1(O; R2) by dominated convergence. Next we show (4.26). For i ∈ {1, 2} we define 2 Zi(λn) := {z ∈ Z : Iλn (λnz) ∩ O 6= Ø and Iλn (λn(z + ei)) ∩ O 6= Ø} . FROM THE N -CLOCK MODEL TO THE XY MODEL 39

∞ Let z ∈ Zi(λn). Since u is C in the interior of the rectangle Iλn (λnz)∪Iλn (λn(z +ei)), it ∞ admits a C lifting ϕ such that u = exp(ιϕ) in the interior of Iλn (λnz) ∪ Iλn (λn(z + ei)). Then, by the fundamental theorem of calculus,

1 1 1 1 1 1 dS un(λn(z + ei)), un(λnz) = dS u λn(z + ei + 2 e1 + 2 e2) , u λn(z + 2 e1 + 2 e2)

≤ ϕλ (z + e + 1 e + 1 e ) − ϕλ (z + 1 e + 1 e ) n i 2 1 2 2 n 2 1 2 2 Z 1 1 1 ≤ λn ∂iϕ λn(z + tei + 2 e1 + 2 e2) dt 0 Z 1 1 1 = λn ∂iu λn(z + tei + 2 e1 + 2 e2) dt . 0 (4.28)

We notice, in addition, that for every t ∈ [0, 1] and z ∈ Zi(λn) Z 2 1 1 ∂iu(x) dx − λ ∂iu λn(z + tei + e1 + e2) n 2 2 Iλn (λnz) Z 1 1 (4.29) ≤ ∂iu(x) − ∂iu λn(z + tei + 2 e1 + 2 e2) dx

Iλn (λnz) 3 2 3 ≤ 2λnk∇ ukL∞(O0) =: C(u)λn . From (4.28)–(4.29) it follows that Z 2 + − 1 X X 1 1 dS (un , un )|νun |1 dH ≤ λndS un(λn(z + ei)), un(λnz) i=1 λn z∈Zi(λn) Jun ∩O 2 Z 1 X X 2 1 1 ≤ λn ∂iu λn(z + tei + 2 e1 + 2 e2) dt 0 i=1 z∈Zi(λn) 2 Z 1 Z X X 3 ≤ ∂iu(x) dx + C(u)λn dt 0 i=1 z∈Zi(λn) Iλn (λnz) Z 0 ≤ |∇u|2,1 dx + C(u)|O |λn . O0 We conclude the proof of (4.26) letting n → +∞ and then O0 & O. Finally, in order to prove (4.27), observe that the condition I (λ z ) ∩ I (λ z ) 6= Ø √ λn n 1 λn n 2 1 1 1 implies that |λn(z1 + 2 e1 + e2) − λn(z2 + 2 e1 + 2 e2)| ≤ 2λn , so the claim follows from the Lipschitz continuity of u on the larger set Oe , which (for n large enough) contains the straight line between the centers of the cubes . We are now ready to deal with the main part of the upper bound, namely the construction 2 of a recovery sequence uε : εZ → Sε . Thanks to the previous simpliﬁcations, it will be 2 enough to approximate the energy of piecewise constant maps on the lattice λnZ which come from Lemma 4.13. To deﬁne the recovery sequence we shall construct a minimal transition (in S1 ) between two constant values of S1 . For this reason it is convenient to introduce some notation about geodesics in S1 and to recall an elementary stability property that we prove for the sake of completeness.

1 2 1 1 2 1 2 1 1 Deﬁnition 4.14. For u , u ∈ S denote by Geo[u , u ]: [0, dS (u , u )] → S the (in case of non-uniqueness counterclockwise rotating) unit speed geodesic between u1 and u2 which 1 2 1 1 2 2 1 2 1 we extend by Geo[u , u ](t) = u for t < 0 and Geo[u , u ](t) = u for t > dS (u , u ). As such the geodesics are 1-Lipschitz continuous functions on R. We further set mid(u1, u2) = 1 1 2 1 Geo( 2 dS (u , u )) as the midpoint on that geodesic. 40 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

Lemma 4.15. There exists a constant c > 0 such that whenever u1, u2, b ∈ S1 are such 1 2 that u , u ∈ Bc(b), then for all t ∈ R 1 2 1 2 1 |Geo[u , b](t) − Geo[u , b](t)| ≤ dS (u , u ) . Proof. First note that for c < 2 both geodesics are unique. Up to a rotation we may also r assume that b = e1 . Let ϕ1, ϕ2 ∈ (−π, π] be the unique angles such that u = exp(ιϕr) for r 1 r = 1, 2. Then dS (u , b) = |ϕr| and we distinguish two cases: if ϕ1 · ϕ2 ≤ 0, then |Geo[u1, b](t) − Geo[u2, b](t)| ≤|Geo[u1, b](t) − b| + |b − Geo[u2, b](t)| 1 2 1 1 ≤dS (Geo[u , b](t), b) + dS (Geo[u , b](t), b) ≤ |ϕ1| + |ϕ2| 1 2 1 =|ϕ1 − ϕ2| = dS (u , u ) , where in the last equality we used that ϕ1 − ϕ2 ∈ (−π, π] for c small enough. If instead 2 1 ϕ1 · ϕ2 > 0, assume without loss of generality that 0 < ϕ1 < ϕ2 . Then Geo[b, u ](ϕ1) = u , 2 1 or more suited for our purposes, Geo[u , b](ϕ2 − ϕ1) = u . Hence for c small enough ( 1 1 u if t < 0 , Geo[u , b](t) = 2 Geo[u , b](t + ϕ2 − ϕ1) if t ≥ 0 , so that for t ≥ 0 we have the estimate 1 2 2 2 |Geo[u , b](t) − Geo[u , b](t)| =|Geo[u , b](t + ϕ2 − ϕ1) − Geo[u , b](t)| 1 2 1 ≤|ϕ2 − ϕ1| = dS (u , u ) , 1 2 1 2 1 2 1 while for t < 0 we have by deﬁnition |Geo[u , b](t) − Geo[u , b](t)| = |u − u | ≤ dS (u , u ). 1 1 We introduce a map which will be used to project vectors of S on Sε . Given u ∈ S we 1 let ϕu ∈ [0, 2π) be the unique angle such that u = exp(ιϕu). We deﬁne Pε : S → Sε by j k ϕu Pε(u) = exp ιθε . (4.30) θε Combined with Propositions 4.1 and 4.8 the next result completes the proof of Theorem 1.2.

1 Proposition 4.16. Assume that ε θε 1 and let u ∈ BV (Ω; S ). Then there exists a 1 2 sequence uε ∈ PCε(Sε) such that uε → u strongly in L (Ω; R ) and Z 1 (c) lim sup Eε(uε) ≤ |∇u|2,1 dx + |D u|2,1(Ω) + J (u; Ω) . ε→0 εθε Ω Proof. By Proposition 4.12, Lemma 4.13 and by the lower semicontinuity of the the Γ- 1 1 limsup with respect to the L convergence, it is enough to prove that for un ∈ PCλn (S ) we have Z 1 + − 1 1 Γ- lim sup Eε(un) ≤ dS (un , un )|νun |1 dH . (4.31) ε→0 εθε λn Jun ∩Ω Since un is fixed in the following discussion, to simplify the notation we denote un by u and λn by λ, always assuming that λ 1. 2 We will define a recovery sequence locally on each half-open cube Iλ(λz) for z ∈ Z . 0 First, we define a boundary condition on ∂Iλ(λz). For a side S = {λz + tei : t ∈ [0, λ]} 0 2 1 2 3 1 3 ε 1 with z ∈ Z and i ∈ {1, 2}, and three values v = (v , v , v ) ∈ (S ) , we set bS[v]: S → S as  1 ε v if t ∈ c0 [0, 1) ,  θε  1 2  1 2 d 1 (v ,v )θε ε ε Geo[v , v ] S (t − c0 ) if t ∈ c0 [1, 2) ,  c0ε θε θε ε 0 2 ε ε b [v](λz + tei) = v if t ∈ [2c0 , λ − 2c0 ) , S θε θε 2 3  2 3 d 1 (v ,v )θε ε ε Geo[v , v ] S (t − (λ − 2c0 ) if t ∈ λ − c0 (1, 2] ,  c0ε θε θε  3 ε v if t ∈ λ − c0 [0, 1] . θε (4.32) FROM THE N -CLOCK MODEL TO THE XY MODEL 41

The particular choice of the constant c0 is not important. For this proof we only need that c0 > 2π . This condition will be clear only after (4.40). (However, to apply this construction later also in the proof of Proposition 4.22 we need to choose a larger constant, namely ε c0 = 777.) Since → 0 by assumption, the function bS[v] can be interpreted as follows: in θε a small neighborhood of the two endpoints of S we set the two values v1 and v3 , while in a contiguous small neighborhood we use the geodesic for a transition to the value v2 , which is taken on most of the side. 1 Next, given u ∈ PCλ(S ) and a side S as above we specify the values 0 − + 0 vS(u) = (u(λz ), mid(uS , uS ), u(λ(z + ei))) , (4.33) − + where uS and uS denote the (constant) traces of u along the side S and the midpoint is 1 given by Deﬁnition 4.14. The boundary values bz,ε : ∂Iλ(λz) → S are then deﬁned by ε 0 0 2 bz,ε(x) = bS[vS(u)](x) if x = λz + tei ∈ S for some z ∈ Z and t ∈ [0, λ] .

Note that this function is well-defined also in the corners with bz,ε(λz0) = u(λz0) for all 2 z0 ∈ Z . Moreover, since we have chosen unit speed geodesics and c0 > 2π , on each side S ε the function bS[vS(u)] satisfies a Lipschitz-estimate of the form 1 θ |bε [v (u)](x) − bε [v (u)](y)| ≤ ε |x − y| x, y ∈ S. (4.34) S S S S 2 ε Repeating the construction on every half-open cube we obtain a continuous function on the S skeleton 2 ∂Iλ(λz). z∈Z We are now in a position to define the recovery sequence of u. We will interpolate between the constant u(λz) and the boundary value bz,ε in Iλ(λz). This will be done on a mesoscale towards the boundary ∂Iλ(λz). Let P : Iλ(λz) → ∂Iλ(λz) be a function satisfying |P (x) − x| = dist(x, ∂Iλ(λz)) for all x ∈ Iλ(λz) (such a function can be defined 2 1 globally by periodicity). To reduce notation, let uz = u(λz). Setu ¯ε : εZ ∩ Iλ(λz) → S as −1 u¯ε(εi) = Geo [bz,ε(P (εi)), uz] θεε dist εi, ∂Iλ(λz) , (4.35) with the extended geodesics given by Definition 4.14. Note that in generalu ¯ε(εi) ∈/ Sε . Hence we define uε ∈ PCε(Sε) by uε := Pε(¯uε) with the operator Pε given by (4.30). 1 2 We claim that uε converges to u in L (Ω; R ). Indeed, for all εi ∈ Iλ(λz) we have by Definition 4.14 ε uε(εi) = Pε(uz) if dist(εi, ∂Iλ(λz)) ≥ π , (4.36) θε ε so that the assumptions → 0 and θε → 0 yield that uε → uz in measure on Iλ(λz). θε Here we used that |Pε − I| ≤ θε . Vitali’s convergence theorem then implies uε → u in L1(Ω; R2). 2 Next we bound the differences uε(εi) − uε(εj) for all i, j ∈ Z with |i − j| = 1. Step 1 (interactions within one cube) 2 We start with εi, εj ∈ εZ ∩ Iλ(λz) for the same z and |i − j| = 1. Let us write I = Iλ(λz) for short. One has to distinguish several cases:

−1 −1 Case 1: If dist(εi, ∂I) ≥ πεθε and dist(εj, ∂I) ≥ πεθε , then (4.36) yields

|uε(εi) − uε(εj)| = 0 . Since for neighboring lattice points it holds that θ ε |dist(εi, ∂I) − dist(εj, ∂I)| ≤ θ , (4.37) ε ε for the remaining cases we can assume that −1 max{dist(εi, ∂I), dist(εj, ∂I)} < (π + 1)εθε . (4.38) Case 2: We next analyze when P (εi) and P (εj) lie on diﬀerent 1-dimensional boundary segments Si 6= Sj of I . We claim that P (εi) and P (εj) are then close to a node of the 42 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

2 lattice λZ . Indeed, denote by ΠSi and ΠSj the projections onto the subspaces spanned by the segments Si and Sj , respectively. Since by (4.38) −1 |P (εi) − P (εj)| ≤ ε|i − j| + dist(εi, ∂I) + dist(εj, ∂I) ≤ (2π + 2)εθε + ε , for ε small enough the sides Si and Sj cannot be parallel. Hence the point λzi,j := 2 ΠSi (ΠSj (εi)) belongs to Si ∩ Sj ⊂ λZ and therefore the 1-Lipschitz continuity of ΠSi and

ΠSj combined with (4.38) implies 2 2 dist(P (εi), λZ ) = dist(ΠSi (εi), λZ ) ≤ |εi − ΠSj (εi)| −1 ≤ |εi − εj| + |εj − ΠSj (εj)| + |ΠSj (εj) − ΠSj (εi)| ≤ 2ε + (π + 1)εθε . (4.39) Exchanging the roles of i and j we derive by the same argument the bound 2 −1 dist(P (εj), λZ ) ≤ 2ε + (π + 1)εθε . (4.40) −1 For ε small enough both terms can be bounded by 2πεθε . In particular, the distance to 2 λZ of both P (εi) and P (εj) is realized by the point λzi,j , which is an endpoint of both the sides Si and Sj . Hence from the deﬁnition of the boundary condition bz,ε in (4.32) and (4.33), and the fact that c0 > 2π we deduce that

bz,ε(P (εi)) = bz,ε(P (εj)) = uzi,j .

Equation (4.37), the 1-Lipschitz continuity of Geo[uzi,j , uz] and the construction ofu ¯ε then yield

|u¯ε(εi) − u¯ε(εj)| ≤ θε .

From the deﬁnition of the function Pε and the above inequality it follows that

|uε(εi) − uε(εj)| ≤ θε . (4.41) Moreover, note that by (4.38), (4.39), and (4.40), for ε small enough, 2 2 −1 −1 dist(εj, λZ ) ≤ |P (εj) − εj| + dist(P (εj), λZ ) ≤ (2π + 2)εθε + 2ε < 2c0εθε , 2 2 −1 −1 (4.42) dist(εi, λZ ) ≤ |P (εi) − εi| + dist(P (εi), λZ ) ≤ (2π + 2)εθε + 2ε < 2c0εθε . These inequalities will be used in Step 3 to count how many interactions fall into Case 2.

Case 3: Now consider points i and j such that P (εi) = ΠSi (εi) and P (εj) = ΠSi (εj) 2 −1 and assume additionally that dist(P (εj), λZ ) ≥ 3c0εθε . Since ΠSi is 1-Lipschitz, this 2 −1 implies that dist(P (εi), λZ ) ≥ 2c0εθε for ε small enough. Hence by the definition of the boundary condition (cf. (4.32) and (4.33)) b (P (εi)) = b (P (εj)) = mid(u− , u+ ) . z,ε z,ε Si Si Using again the 1-Lipschitz-continuity of the geodesic Geo[mid(u− , u+ ), u ], similar to Si Si z (4.41) we obtain that |uε(εi) − uε(εj)| ≤ θε . (4.43) However, we need to analyze more accurately which points yield a non-zero difference. On the one hand, the projection property of P and the definition ofu ¯ε yield the implication − + −1 if dist(εj, S ) = dist(εj, ∂I) ≥ d 1 (mid(u , u ), u )εθ then u¯ (εj) = u . (4.44) i S Si Si z ε ε z The same conclusion holds true for εi. Hence for Case 3 the estimate (4.43) needs to be taken into account only for (i, j) such that one of them violates the condition in (4.44), while for other couples εi, εj the difference vanishes as in Case 1.

On the other hand, using that P (εi) = ΠSi (εi) and P (εj) = ΠSi (εj), one can show the following implication (where k means parallel):

(εi−εj) k Si = 0 =⇒ dist(εi, ∂I) = dist(εj, ∂I) =⇒ |uε(εi)−uε(εj)| = 0. (4.45)

Case 4: It remains to treat the case of points i and j such that P (εi) = ΠSi (εi) and 2 −1 P (εj) = ΠSi (εj), but dist(P (εj), λZ ) < 3c0εθε . Here we use the Lipschitz-continuity of bz,ε on Si and the stability estimate of Lemma 4.15. For the latter, we need that bz,ε(P (εi)) and bz,ε(P (εj)) are suﬃciently close to uz . Since on Si the boundary condition FROM THE N -CLOCK MODEL TO THE XY MODEL 43

1 3 bz,ε is deﬁned by geodesic interpolation between the elements of the vector vSi (u) ∈ (S ) deﬁned in (4.33) and u ∈ {u− , u+ }, we know that z Si Si

1 |bz,ε(P (εi)) − uz| ≤ dS (bz,ε(P (εi)), uz) − + − + ≤ max d 1 ((vS (u), er), mid(u , u )) + d 1 (mid(u , u ), uz) r=1,3 S i Si Si S Si Si

− + 1 − + = max d 1 ((vS (u), er), mid(u , u )) + d 1 (u , u ) r=1,3 S i Si Si 2 S Si Si − + ≤ max d 1 ((vS (u), er), uz) + d 1 (u , u ) . r=1,3 S i S Si Si

Recall that the ﬁrst and third component of vSi [u] are given by the evaluation of u at the endpoints of Si . Hence by the almost continuity estimate (4.27) we deduce for λ 1 that

|bz,ε(P (εi)) − uz| < c , where c is the constant given by Lemma 4.15. Repeating the argument one proves the −1 analogue estimate for P (εj). To reduce notation, we set dε,i = θεε dist(εi, ∂I) and dε,j = −1 θεε dist(εj, ∂I). Then by the triangle inequality, (4.34), and the applicable Lemma 4.15 we have

|u¯ε(εi) − u¯ε(εj)| ≤ Geo[bz,ε(P (εi)), uz](dε,i) − Geo[bz,ε(P (εi)), uz](dε,j)

+ Geo[bz,ε(P (εi)), uz](dε,j) − Geo[bz,ε(P (εj)), uz](dε,j)

1 ≤ |dε,i − dε,j| + dS (bz,ε(P (εi)), bz,ε(P (εj))) π ≤ θ + θ ε−1|Π (εi) − Π (εj)| ≤ 2θ . ε 4 ε Si Si ε Hence in Case 4 we deduce the slightly weaker bound

|uε(εi) − uε(εj)| ≤ 2θε . (4.46) Finally the location condition on j and (4.38) imply that 2 2 −1 dist(εj, λZ ) ≤ |P (εj) − εj| + dist(P (εj), λZ ) < 4c0εθε . (4.47) Step 2 (interactions between diﬀerent cubes) Now we consider points εi ∈ Iλ(λzi) and εj ∈ Iλ(λzj) with zi 6= zj and |i − j| = 1. By the deﬁnition ofu ¯ε via geodesics and by the 1-Lipschitz continuity of the latter we have θ |u¯ (εi) − b (P (εi))| = |u¯ (εi) − Geo[b (P (εi)), u¯ ](0)| ≤ ε dist(εi, ∂I (λz )) , ε zi,ε ε zi,ε zi ε λ i θ |u¯ (εj) − b (P (εj))| = |u¯ (εj) − Geo[b (P (εj)), u¯ ](0)| ≤ ε dist(εj, ∂I (λz )) . ε zj ,ε ε zj ,ε zj ε λ j

Note that there exists a boundary segment Sij of ∂Iλ(λzj) such that the line segment [εi, εj] intersects Sij orthogonally and moreover Sij ⊂ ∂Iλ(λzi). In particular,

dist(εi, ∂Iλ(λzi)) + dist(εj, ∂Iλ(λzj)) ≤ ε . Summing the previous two estimates then yields

|u¯ε(εi) − bzi,ε(P (εi))| + |u¯ε(εj) − bzj ,ε(P (εj))| ≤ θε . (4.48)

We claim that either P (εi) ∈ Sij and P (εj) ∈ Sij or that both P (εi) and P (εj) are close to 2 λZ . Indeed, ﬁrst assume that P (εj) ∈/ Sij . Then there exists another facet Sj of Iλ(λzj) such that P (εj) ∈ Sj . Since dist(εj, Sj) ≤ ε and dist(εj, Sij) ≤ ε, the sides Sj and Sij cannot be parallel. Denoting by ΠS the projection onto the subspace spanned by a segment 2 S , we deduce that ΠSj (ΠSij (εj)) ∈ Sj ∩ Sij ⊂ λZ . Hence 2 2 dist(P (εj), λZ ) = dist(ΠSj (εj), λZ ) ≤ |εj − ΠSij (εj)| ≤ ε .

For P (εi) we check two possibilities. First consider P (εi) ∈ Sij . Then we may assume that

P (εj) ∈/ Sij as above. From the Lipschitz-continuity of ΠSij we infer 2 2 dist(P (εi), λZ ) = dist(ΠSij (εi), λZ ) ≤ |ΠSij (εi) − ΠSij (ΠSj (εj))|

≤ |εi − εj| + |εj − ΠSj (εj)| ≤ 2ε . 44 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

On the contrary, if P (εi) ∈/ Sij , then there exists a facet Si 6= Sij of Iλ(λzi) such that P (εi) ∈ Si . Since Si and Sij are both sides of the cube Iλ(λzi) which cannot be 2 parallel, we deduce that ΠSi (ΠSij (εi)) ∈ span(Si) ∩ span(Sij) ⊂ λZ and thus the deﬁning property of Sij implies that 2 2 dist(P (εi), λZ ) = dist(ΠSi (εi), λZ ) ≤ |εi − ΠSij (εi)| ≤ ε . 2 It remains to establish an estimate for dist(P (εj), λZ ) when P (εi) ∈/ Sij and P (εj) ∈ Sij . In this case we have 2 2 dist(P (εj), λZ ) = dist(ΠSij (εj), λZ ) ≤ |ΠSij (εj) − ΠSij (ΠSi (εi)|

≤ |εj − εi| + |εi − ΠSi (εi)| ≤ 2ε . To sum up, we have proved the following two alternatives:

(i) P (εi),P (εj) ∈ Sij ; (ii) max{dist(P (εi), λZ2), dist(P (εj), λZ2)} ≤ 2ε. Again we treat the two cases separately. 2 Case 5: Note that the conditions in (ii) above imply that the unique points λz¯i, λz¯j ∈ λZ realizing the minimal distance satisfy

|λz¯i − λz¯j| ≤ |λz¯i − P (εi)| + |P (εi) − εi| + |εi − εj| + |εj − P (εj)| + |P (εj) − λz¯j| ≤ 7ε, so that necessarilyz ¯i =z ¯j for ε small enough. In particular, the construction of the boundary condition forces bzi,ε(P (εi)) = bzj ,ε(P (εj)) = uz¯i for ε small enough. From (4.48) we infer

|uε(εi) − uε(εj)| ≤ θε . (4.49) Furthermore we know that 2 2 dist(εj, λZ ) ≤ dist(P (εj), λZ ) + dist(εj, ∂Iλ(λz)) ≤ 3ε . (4.50)

Case 6: Let us now analyze the case P (εi),P (εj) ∈ Sij . By the symmetric deﬁnition bzi,ε and bzj ,ε coincide on Sij . Since by assumption the segment [εi, εj] is orthogonal to Sij and

Sij ⊂ ∂Iλ(λzi) ∩ ∂Iλ(λzj), we have P (εi) = ΠSij (εi) = ΠSij (εj) = P (εj). Hence estimate (4.48) yields

|uε(εi) − uε(εj)| ≤ θε . (4.51) Moreover, we can give an estimate for the location of εj by

dist(εj, Sij) = dist(εj, ∂Iλ(λzj)) ≤ ε. (4.52)

Step 3 (energy estimates) Let us ﬁrst sum up our analysis hitherto. The interactions of couples (εi, εj) with |i−j| = 1 and at least one point in an half-open cube Iλ(λz) can be grouped as follows:

(1) In Case 1 it holds that |uε(εi) − uε(εj)| = 0. 2 −1 (2) In the Cases 2, 4, and 5 we have for ε small enough dist(εj, λZ ) ≤ 4c0εθε (see (4.42), (4.47), and (4.50)) and by (4.41), (4.46), and (4.49) the continuity estimate

|uε(εi) − uε(εj)| ≤ 2θε . (3) In Cases 3 and 6, according to (4.43)–(4.45) respectively (4.51)–(4.52), there exists a side S of Iλ(λz) such that ( ε θε if dist(εj, S) < κS(z) θ + ε and (i − j) ⊥ S, |uε(εi) − uε(εj)| ≤ ε 0 otherwise,

1 − + − + 1 1 where κS(z) := 2 dS (uS , uS ) = dS (mid(uS , uS ), uz). FROM THE N -CLOCK MODEL TO THE XY MODEL 45

Now we are in a position to estimate the discrete energy. Due to (1)-(3) above we can bound it via

1 2 −1 Eε(uε) ≤ Cεθε# Ωε ∩ dist(·, λZ ) ≤ Cεθε εθε X X 2 n ε o + εθε# ε ∩ Iλ(λz) ∩ dist(·,S) < κS(z) + 2ε , Z θε Iλ(λz)∩Ω6=Ø S⊂∂Iλ(λz) where we used that each point in Z2 has four neighbors, but for (3) we have to count only half of the interactions. We claim that the ﬁrst right hand side term vanishes when ε → 0. To this end, ﬁx a large cube Q such that Ω ⊂⊂ Q. For ε small enough we have

2 2 −1 X −2 2 −2 ε # Ω ∩ dist(·, λ ) ≤ Cεθ ≤ |B −1 (x)| ≤ C|Q|λ ε θ . ε Z ε 2Cεθε ε 2 x∈λZ ∩Q Inserting this estimate into the ﬁrst term, we obtain

2 −1 −2 ε εθε# Ωε ∩ dist(·, λZ ) ≤ Cεθε ≤ C|Q|λ , (4.53) θε which vanishes when ε → 0 due to the assumption ε θε . Now we treat the second term. Since each S is a segment of length λ, for any ﬁxed κ > 0 it holds that

2 n ε o θε 2ε ε εθε# εZ ∩ Iλ(λz) ∩ dist(·,S) < θ κ + 2ε ≤ λ + κ + 6ε κ + 4ε . ε ε θε θε

For only finitely many cubes Iλ(λz) intersecting Ω, we can insert this estimate with κ = κS(z), pass to the limit in ε and obtain by (4.53) that Z 1 X X λ − + − + 1 1 1 lim sup Eε(uε) ≤ dS (uS , uS ) = dS (u , u )|νu|1 dH . ε→0 εθε 2 Iλ(λz)∩Ω6=Ø S⊂∂Iλ(λz) λ Ju∩Ω (4.54) This estimate agrees with (4.31) and hence concludes the proof. 4.4. Upper bound in presence of vortices. Also in the case of vortices the construction of the recovery sequence is done by gradually simplifying the map u ∈ BV (Ω; S1), following the main idea of Section 4.3. However, due to the presence of the vortex measure PN µ = h=1 dhδxh , in general the map u cannot be approximated by smooth maps with values in S1 . This requires some additional steps in the simplification of u. For notational convenience, set Z (c) E(µ, u) = |∇u|2,1 dx + |D u|2,1(Ω) + J (µ, u; Ω) Ω with J (µ, u; Ω) given by (4.23). Then our approximation procedure is as follows: 1 (1) We find a current T such that ∂T = −µ× S , uT = u, and the parametric integral R Φ(T~) d|T | is approximately equal to theJ energyK E(µ, u) of the BV map u and the measure µ. (2) Using the Approximation Theorem of cartesian currents we find a map with finitely ∞ 1 1,1 1 many singularities ue ∈ C (Ωe\{x1, . . . , xN }; S )∩W (Ω;e S ), Ωe ⊃⊃ Ω, such that ue 1 2 1,1 R is close to u in L (Ω; R ), and the anisotropic W norm E(µ, ue) = |∇ue|2,1 dx is approximately equal to R Φ(T~) d|T | (and thus to E(µ, u)), cf. Lemma 4.17. We need regularity in a larger set Ωe ⊃⊃ Ω for the same reasons as in Section 4.3. (3) The function ue has topological degree around xh equal to dh . If |dh| > 1, we modify the map ue around xh by splitting the singularity in |dh| singularities of degree ±1 without essentially increasing the W 1,1 energy, cf. Lemma 4.18. This is needed since it is energetically more convenient to approximate two singularities of degree 1 than one singularity of degree 2. In Lemma 4.20 we further move the singularities on a lattice committing a further negligible energetic error. 46 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

(4) After the modification in (3), we can assume that ue has topological degree ±1 around the singularities {x1, . . . , xN }. Then we modify u in such a way that it ±1 e x−xh 1,1 coincides with near xh , without notably increasing the W energy, cf. |x−xh| Lemma 4.21. This will simplify the construction of the recovery sequence and the estimation of the energy close to vortices. 2 −n (5) As in Section 4.3, we consider a lattice λnZ , λn = 2 , and we construct a 2 piecewise constant map un by suitably sampling ue on the lattice λnZ , applying Lemma 4.13 far from the singularities. (6) The recovery sequence uε is defined as follows: outside small cubes which contain the singularities of u we construct uε that approximates un as in Section 4.3; in an even e ±1 x−xh smaller cube close to a singularity xh we define uε by projecting on Sε ; |x−xh| |ε 2 in the intermediate layer one has to interpolate between the two constructions.Z This step is very delicate and requires a decomposition of said layer into squares which become finer and finer when we approach the inner cube, cf. Proposition 4.22. We start with the approximation result for currents T with boundary ∂T = −µ× S1 which appear in the definition of the limit functional. J K PN Lemma 4.17 (Approximations creating finitely many singularities). Let µ = h=1 dhδxh 2 1 1 2 and let T ∈ D2(Ω×R ) be such that T ∈ cart(Ωµ×S ) and ∂T |Ω×R = −µ× S . Then ∞ 1 J 1K,1 1 there exist an open set Ωe ⊃⊃ Ω and a sequence of functions uk ∈ C (Ωeµ; S ) ∩ W (Ω;e S ) such that 1 2 uk → uT in L (Ω; R ) , (4.55) 2 Guk *T in D2(Ωµ×R ) , (4.56) 2 2 |Guk |(Ω×R ) → |T |(Ω×R ) . (4.57) Moreover deg(uk)(xh) = dh , for h = 1,...,N. (4.58) Proof. Let Ω0 and Ω00 be open sets with Lipschitz boundary such that Ω0 ⊂⊂ Ω00 ⊂⊂ Ω 0 0 2 and {x1, . . . , xN } ⊂ Ω and let us define the open set O := Ω \ Ω . Since ∂T |O×R = 0, we have T ∈ cart(O×S1). By Lemma 3.4 there exist an open set Oe ⊃⊃ O and a current 1 2 2 2 Te ∈ cart(Oe×S ) such that Te|O×R = T |O×R and |Te|(∂O×R ) = 0. In particular,

T | 00 0 2 = T | 00 0 2 . (4.59) e (Ω \Ω )×R (Ω \Ω )×R This allows us to glue together the currents T and Te. To do so, we define the set Ωe := Ω∪Oe 2 ∞ 00 and the current S ∈ D2(Ωe×R ) as follows. Fix a cut-off function ζ ∈ Cc (Ω ) such that 0 0 ≤ ζ ≤ 1 and ζ ≡ 1 on a neighborhood of Ω . For every ω ∈ D2(Ωe×R2) we put S(ω) := T (ζω) + Te (1 − ζ)ω . 2 Then by (4.59) it follows that S|Ω× 2 = T , S| 0 2 = T | 0 2 , and |S|(∂Ω× ) = R (Ωe\Ω )×R e (Ωe\Ω )×R R 0. In particular, using the product rule for the exterior derivative, for any 1-form ω ∈ D1(Ωe×S1) we find that ∂S(ω) = T (ζ dω) + Te((1 − ζ) dω) 1 = ∂T (ζω) − T ( dζ ∧ ω) + ∂Te((1 − ζ)ω) + Te( dζ ∧ ω) = −µ× S (ω) , 0 J K where we used that ζ ≡ 1 on supp(µ) and dζ ∧ ω ∈ D2(Ω00 \ Ω ×R2). Hence S ∈ 1 1 cart(Ωµ× ) and ∂S| 2 = −µ× . e S Ωe×R S 1 J K Since S ∈ cart(Ωeµ×S ), by the Approximation Theorem for cartesian currents (The- ∞ 1 1,1 1 orem 3.3) there exists a sequence uk ∈ C (Ωeµ; S ) ∩ W (Ω;e S ) such that Guk *S in 2 2 2 D2(Ωeµ×R ) and |Guk |(Ωeµ×R ) → |S|(Ωeµ×R ). In particular, we get (4.56) and thus (4.55), see the proof of Proposition 4.4. Moreover 2 2 |Guk |(Ωe×R ) → |S|(Ωe×R ) , (4.60) FROM THE N -CLOCK MODEL TO THE XY MODEL 47

2 since Guk and S do not charge the sets {xh}×R (being i.m. rectifiable 2-currents concentrated on a subset of R2×S1 , see also Remark 4.7). Thanks to the convergence in (4.56) we can prove (4.58). Indeed, let h = 1,...,N . For ρ > 0 small enough we have (e.g., by [33, Section 6, Proposition 1]) 1 2 ∂Guk |Bρ(xh)× = − deg(uk)(xh)δxh × S . R J K ∞ Let us fix a cut-off function ζ ∈ Cc (Bρ(xh)) such that ζ ≡ 1 on Bρ/2(xh) and define 1 1 1 the 1-form ω = ζωS , ωS being the 0-homogeneous extension of the volume form of S to 2 2 2 R \{0}. Observing that dω ∈ D (Bρ(xh) \{xh})×R , the convergence in (4.56) implies that

− deg(uk)(xh) = ∂Guk (ω) = Guk (dω) → T (dω) = ∂T (ω) = −dh .

Then deg(uk)(xh) is a sequence of integer numbers which converges to the integer number dh . Thus for k large enough deg(uk)(xh) = dh . 2 To conclude, we observe that (4.60) and |S|(∂Ω×R ) = 0 imply (4.57). The next result shows how to reduce the analysis to singularities with degree ±1.

∞ Lemma 4.18 (Splitting of the degree). Let V := {x1, . . . , xN } ⊂ Ω and let u ∈ C (Ω \ 1 1,1 1 V ; S ) ∩ W (Ω; S ) be such that deg(u)(xh) 6= 0 for h = 1,...,N . Then for 0 < τ 1 there exist a set V = {xτ , . . . , xτ } ⊂ Ω and a function uτ ∈ C∞(Ω \ V ; 1) ∩ W 1,1(Ω; 1) τ 1 Nτ τ S S such that uτ → u strongly in L1(Ω; R2), τ τ | deg(u )(xh)| = 1 , for h = 1,...,Nτ , and Z Z τ lim |∇u |2,1 dx = |∇u|2,1 dx , (4.61) τ→0 Ω Ω

Nτ N X τ τ X Nτ = | deg(u )(xh)| = | deg(u)(xh)| . (4.62) h=1 h=1

τ Nτ τ τ Moreover, deﬁning the measures µ = P deg(u )(x )δ τ , we have that h=1 h xh N τ f X µ * deg(u)(xh)δxh as τ → 0 . h=1

Finally, if u ∈ C∞(Ωe \ V ; S1) ∩ W 1,1(Ω;e S1) for some Ωe ⊃⊃ Ω, then one can additionally choose uτ ∈ C∞(Ωe \ V ; S1) ∩ W 1,1(Ω;e S1). Remark 4.19. In this section we shall apply Lemma 4.18 to functions given by Lemma 4.17, cf. (4.58). In Section 6 we have to consider u ∈ C∞(Ω \ V ; S1) ∩ W 1,1(Ω; S1) without assuming that deg(u)(xh) 6= 0 for every h = 1,...,N . In that case, the statement of the lemma holds true, but (4.62) needs to be adapted to

N X Nτ = | deg(u)(xh)| + 2#{xh : deg(u)(xh) = 0} . h=1 The argument in the proof remains unchanged.

Proof of Lemma 4.18. Via an iterative construction we create | deg(u)(xh)| diﬀerent singularities out of one singularity whenever | deg(u)(xh)| > 1 in such a way that the new function is close in energy. To reduce notation, we assume that x1 = 0 and deg(u)(x1) > 1 (the case of a negative degree less than −1 can be treated similarly). We equip R2 with the complex τ ∞ 1 product, which we denote by . Given 0 < τ 1 we set u ∈ C (Ω \ (V ∪ {τe1}); S ) as x −1 (x − τe ) uτ (x) = u(x) 1 . |x| |x − τe1| 48 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

Deﬁned as above, it follows that uτ → u in L1(Ω; R2) by dominated convergence. Next, we estimate its anisotropic gradient norm. Using the product rule for the complex product, for i = 1, 2 we obtain −1 τ x (x − τe1) ∂iu (x) = ∂iu(x) |x| |x − τe1| ( −1 −1 ) x (x − τe1) x (x − τe1) + u(x) ∂i + ∂i . |x| |x − τe1| |x| |x − τe1| A straightforward computation shows that, for a.e. x ∈ Ω, we have ( −1 −1 ) τ x x x x lim ∂iu (x) = ∂iu(x) + u ∂i + ∂i τ→0 |x| |x| |x| |x| ( ) x −1 x = ∂ u(x) + u ∂ = ∂ u(x) . i i |x| |x| i

x −1 1 In order to use dominated convergence, we observe that |x| = |x| (x1, −x2), so that

x −1 x − τe 2 2 τ 1 |∂iu (x)| ≤ |∂iu(x)| + ∂i + ∂i ≤ |∂iu(x)| + + . |x| |x − τe1| |x| |x − τ| The right-hand side is equi-integrable on Ω ⊂ R2 , so that we conclude Z Z τ lim |∇u |2,1 dx = |∇u|2,1 dx . τ→0 Ω Ω Finally, we need to compute the degree of uτ . To reduce notation, we introduce the complex- valued functions 1 1 ue(x) = u1(x) + ιu2(x), f(x) = (x1 − ιx2), g(x) = (x1 − τ) + ιx2 . |x| |x − τe1| In the planar setting, the degree around a point x can be expressed via the winding number, that means Z d(ufg) Z (du)fg + u(df)g + uf(dg) (2πι) deg(uτ )(x) = e = e e e ufge ufge ∂Br (x) ∂Br (x)

=2πι (deg(u)(x) − δ0(x) + δτe1 (x)) . τ We deduce that the degree of u is of the form (recalling that x1 = 0)  deg(u)(x1) − 1 if x = x1 , τ  deg(u )(x) = 1 if x = x1 + τe1 , deg(u)(x) otherwise , where for the second equality we used that deg(u)(x1 +τe1) = 0 due to the local smoothness of u around x1 + τe1 (see [21, Corollary 8]). Repeating this construction (with τ/2, τ/4, and so on) we find a finite set V = {xτ , . . . , xτ } and a sequence uτ ∈ C∞(Ω \ V ; 1) ∩ τ 1 Nτ τ S 1,1 1 τ 1 2 τ τ W (Ω; S ) such that u → u in L (Ω; R ), deg(u )(xh) ∈ {±1} and (4.61)–(4.62) hold true. The claim on the flat convergence follows by the construction. 2 In the next lemma we move the singularities onto a lattice λnZ which makes them 1 compatible with a piecewise constant approximation un ∈ PCλn (S ).

Lemma 4.20 (Moving singularities on a lattice). Let V := {x1, . . . , xN } ⊂ Ω, let Ωe ⊃⊃ Ω, ∞ 1 1,1 1 and let u ∈ C (Ωe \ V ; S ) ∩ W (Ω;e S ). Then for every λ > 0 there exist a set Vλ = λ λ 2 λ ∞ 1 1,1 1 λ {x1 , . . . , xN } ⊂ λZ ∩ Ω and a map u ∈ C (Ωe \ Vλ; S ) ∩ W (Ω;e S ) such that u → u 1,1 1 λ λ strongly in W (Ω;e S ) as λ → 0. Moreover, deg(u )(xh) = deg(u)(xh) for h = 1,...,N FROM THE N -CLOCK MODEL TO THE XY MODEL 49

λ PN λ λ for λ small enough and, deﬁning the measures µ = deg(u )(x )δ λ , it holds that h=1 h xh λ f PN µ * h=1 deg(u)(xh)δxh .

λ 2 λ Proof. For every λ > 0 and h = 1,...,N we choose xh ∈ λZ ∩ Ω such that xh → xh as λ λ → 0. For every λ > 0 there exists a diffeomorphism ψλ : Ωe → Ωe such that ψλ(xh) = xh for h = 1,...,N (see, e.g., [36, p. 210] for an explicit construction). We remark that it is −1 possible to construct ψλ in such a way that kψλ − idkC1 and kψλ − idkC1 are controlled λ −1 by maxh |xh − xh| for every λ > 0. In particular, kψλ − idkC1 , kψλ − idkC1 → 0 for λ ∞ 1 1,1 1 λ λ → 0. We define u := u ◦ ψλ ∈ C (Ωe \ Vλ; S ) ∩ W (Ω;e S ). Then u → u strongly in 1,1 1 W (Ω;e S ) as λ → 0. Let us fix ρ > 0 such that the balls Bρ(xh) are pairwise disjoint λ and contained in Ω.e For λ small enough, we have that xh ∈ Bρ/4(xh) for h = 1,...,N . ∞ Let ζ ∈ Cc (Bρ(xh)) such that ζ ≡ 1 on Bρ/2(xh). By [20, Theorem B.1] we have that Z Z λ λ λ λ ⊥ λ→0 ⊥ 2π deg(u )(xh) = − (u ×∇u ) · ∇ζ dx → − (u×∇u) · ∇ζ dx = 2π deg(u)(xh)

Bρ(xh) Bρ(xh)

⊥ where (u×∇u) = (u1∂2u2 − u2∂2u1, u2∂1u1 − u1∂1u2).

Before we come to the constructions starting from the discrete environment, we modify the target sequence one last time to give it a more explicit structure close to the singularities.

∞ 1 Lemma 4.21 (Modiﬁcation near a singularity). Let ρ ≤ 1 and let u ∈ C (Bρ \{0}; S ) ∩ 1,1 1 ∞ 1 W (Bρ; S ) with | deg(u)(0)| = 1. Then for every σ > 0 there exist ue ∈ C (Bρ \{0}; S )∩ 1,1 1 2 W (Bρ; S ) and a radius η0 ∈ (0, ρ ) such that i) u(x) = (x1,±x2) for every x ∈ B \{0}; e |x| η0 ii) u(x) = u(x) for every x ∈ B \ B√ ; e ρ η0 iii) it holds that Z Z |∇ue|2,1 dx ≤ |∇u|2,1 dx + σ .

Bρ Bρ

Proof. We give a proof in the case deg(u)(0) = 1, the case deg(u)(0) = −1 being completely analogous. We also assume, without loss of generality, that ρ = 1 and we denote Bρ simply by B . Let us consider the set Σ := {(x1, 0) : 0 ≤ x1 ≤ 1}. To modify the map u we will actually modify its lifting ϕ. We start by discussing some useful properties of ϕ. Since B \ Σ is simply connected, the map u: B \ Σ → S1 admits a lifting ϕ: B \ Σ → R, i.e., a function satisfying u = exp(ιϕ). The function ϕ is unique up to integer multiples of 2π and has the same regularity of u, namely ϕ ∈ C∞(B \ Σ; R) ∩ W 1,1(B \ Σ; R). The fact that u ∈ C∞(B \{0}; S1) can be translated in terms of the regularity of ϕ as follows: for every x ∈ Σ ∩ (B \{0}) and r > 0 such that Br(x) ⊂⊂ B \{0}, we have

∞ 1 + ϕ + 1 − (ϕ − 2π) ∈ C (Br(x)) , (4.63) Br (x) Br (x)

± ∞ 1 where Br (x) = {x ∈ Br(x): ± x2 > 0}. To show this, we observe that u ∈ C (Br(x); S ), ∞ thus it admits a lifting ϕx ∈ C (Br(x)). Up to adding an integer multiple of 2π to ϕx , by + the uniqueness of the lifting up to integer multiples of 2π we have ϕ = ϕx in Br (x) and − 1 1 there exists a kx ∈ Z such that ϕ = ϕx +2πkx in Br (x), thus + ϕ+ − (ϕ−2πkx) = Br (x) Br (x) ∞ ϕx ∈ C (Br(x)). To prove that kx = 1, we observe that the proven regularity implies, in ± particular, that the restrictions ϕ| ± admit traces ϕ on Σ in the classical sense. To Br (x) + − compute the jump ϕ −ϕ = −2πkx at a point x ∈ Σ∩(B \{0}), we parametrize the circle ∂B|x| counterclockwise with the closed path γ(t) = |x| exp(ι2πt), t ∈ [0, 1]. Observing that 50 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

∇ϕ = u1∇u2 − u2∇u1 in B \ Σ, we infer that

1 1 Z d Z ϕ+(x) − ϕ−(x) = ϕ(γ(0+)) − ϕ(γ(1−)) = − ϕ(γ(t)) dt = − ∇ϕ(γ(t)) · γ˙ (t) dt dt 0 0 Z 1 = − (u1∇u2 − u2∇u1) · τ dH = −2π deg(u)(0) = −2π . γ

This proves kx = 1, and in turn (4.63). Finally, ϕ ∈ L2(B). Indeed, by the Sobolev Embedding Theorem, ϕ ∈ W 1,1(B±) =⇒ ∗ ϕ ∈ L1 (B±) = L2(B±). We are now in a position to define a modification ϕe of ϕ. Let us fix σ > 0. By a classical ∞ √ capacity argument, we find η0 > 0 small enough and a cut-off function ζ ∈ Cc (B η0 ),

0 ≤ ζ ≤ 1, ζ ≡ 1 on Bη0 satisfying

2 C C 2 √ k∇ζkL2(B) ≤ η ≤ < σ . log 0 | log η0| η0 We use the cut-off ζ to interpolate between ϕ and the principal argument arg. The x function arg is defined in polar coordinates (ρ, ϑ) by arg(ρ, ϑ) = ϑ and satisfies |x| = exp(ι arg(x)) in B \{0}. In particular, also arg ∈ C∞(B \ Σ; R) ∩ W 1,1(B \ Σ; R) and it satisfies the regularity property as in (4.63). Let us define

ϕe := ζ arg +(1 − ζ)ϕ , ue := exp(ιϕe) . ∞ Since ϕe ∈ C (B \ Σ; R) and for every x ∈ Σ ∩ (B \{0}) and r > 0 such that Br(x) ⊂⊂ ∞ ∞ B \{0}, we have 1 + ϕ + 1 − (ϕ − 2π) ∈ C (Br(x); R), we deduce that u ∈ C (B \ Br (x) e Br (x) e e 1 {0}; S ). By definition ue satisfies (i) and (ii). To prove (iii), let us compute Z Z Z Z |∇ue|2,1 dx = |∇ϕe|1 dx ≤ |∇ϕe − ∇ϕ|1 dx + |∇ϕ|1 dx B B\Σ B\Σ B\Σ Z Z = |∇ϕe − ∇ϕ|1 dx + |∇u|2,1 dx , B\Σ B thus it only remains to estimate the first integral in the right-hand side. We have Z √ Z √ Z |∇ϕe − ∇ϕ|1 dx ≤ 2 |∇ζ|| arg −ϕ| dx + 2 ζ|∇ arg −∇ϕ| dx B\Σ B\Σ B\Σ Z √ √ ≤ 2k∇ζkL2(B) k arg kL2(B) + kϕkL2(B) + 2 |∇ arg | + |∇ϕ| dx

√ B η0 \Σ √ √ ≤ 2k arg kL2(B) + 2kϕkL2(B) + 1 σ , if η0 > 0 is also chosen small enough such that √ Z 2 |∇ arg | + |∇ϕ| dx < σ .

√ B η0 \Σ

This concludes the proof. Now we are in a position to construct the discrete recovery sequence.

−n Proposition 4.22. Assume that ε θε ε| log ε|. Let λn := 2 , n ∈ N and let 2 ∞ 1 1,1 1 V := {x1, . . . , xN } ⊂ λnZ ∩ Ω. Assume that u ∈ C (Ωe \ V ; S ) ∩ W (Ω;e S ) with FROM THE N -CLOCK MODEL TO THE XY MODEL 51

Ωe ⊃⊃ Ω has the following structure: | deg(u)(xh)| = 1 for all 1 ≤ h ≤ N and there exists η0 > 0 such that 1 0 x − xh u(x) = in Bη0 (xh) . 0 deg(u)(xh) |x − xh| PN Set moreover µ = h=1 deg(u)(xh)δxh . Then there exists a sequence uε :Ωε → Sε such f 1 2 that µuε * µ, uε → u in L (Ω; R ), and 1 ε Z lim sup Eε(uε) − 2π|µ|(Ω)| log ε| ≤ |∇u|2,1 dx . ε→0 εθε θε Ω Remark 4.23. We emphasize that the construction presented below also works under the sole assumption ε θε . The scaling θε ε| log ε| will be used only from (4.130) on, where we identify the flat limit of the discrete vorticity measure. This observation will be useful to study the regimes θε ∼ ε| log ε| in Section 5 and ε| log ε| θε in Section 6. Proof of Proposition 4.22. We will divide the proof into several steps. First we define a good approximation very close to the singularity. Then we define an interpolation between this construction and the piecewise constant approximations provided by Lemma 4.13 far from the singularities. In a third step we estimate the energetic contribution of this interpolation. In the final step we conclude by bounding the energy and identifying the flat limit of the discrete vorticity measures. Step 1 Local discrete approximation of (degree ±1)-singularities. We define a local recovery sequence close to the first singularity x1 . In the whole proof we will assume that deg(u)(x1) = 1, so that

x − x1 u(x) = in Bη0 (x1) . (4.64) |x − x1|

1 0 x−x1 (If, instead, deg(u)(x1) = −1, we have u(x) = and the construction below 0 −1 |x−x1| is adapted accordingly.) For simpliﬁcation, we specify u(x1) := e1 . Next we partition R2 \{0} according to the value of the angle in polar coordinates. More precisely, for k = 0, 1,...,Nε − 1 we set 2 Sk,ε := {x = r exp(ιϕ) ∈ R : r > 0 , ϕ ∈ [kθε, (k + 1)θε)} . (4.65)

Based on this partition, we approximate the functions u with sequences vε ∈ PCε(Sε) 2 deﬁned on εZ \{x1} by

vε(εi) = exp(ιkθε) if εi − x1 ∈ Sk,ε (4.66) while vε(εi) = e1 if εi = x1 . Note that vε = Pε(u) by deﬁnition of Pε , see (4.30). Then, ε ε ε writing εi − x1 = |εi − x1| exp(ι(ki θε + φi )) with φi ∈ [0, θε), it holds that ε ε ε |u(εi) − vε(εi)| ≤ | exp(ι(ki θε + φi )) − exp(ιki θε)| ≤ θε , (4.67) −1 for εi ∈ Bη0 (x1). We deﬁne the radius rε = 4εθε (its role will become clear below). We start estimating the energy of vε in B2rε (x1). By a change of variables we may assume that 2 x1 = 0 in order to reduce the notation. Observe that for any two vectors a, b ∈ R we have |a|2 − |b|2 ≤ |a − b|(|a| + |b|) ≤ 2|b||a − b| + |a − b|2 . (4.68) Hence, using also (4.67), 1 1 Eε(vε; B2rε ) ≤ Eε(u; B2rε ) εθε εθε 1 X 2 2 2 + ε |vε(εi) − vε(εj)| − |u(εi) − u(εj)| 2εθε hi,ji εi,εj∈B2rε

1 C X 2 2 ≤ Eε(u; B2rε ) + ε θε|u(εi) − u(εj)| + θε . (4.69) εθε εθε hi,ji εi,εj∈B2rε 52 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

As we shall prove next, the last sum vanishes, while the first right hand side term scales exactly like 2π| log ε| ε in the sense that the difference vanishes. To this end, we derive an θε estimate for the finite differences of u away from the singularity. Since for any t ∈ [0, 1] and i, j ∈ Z2 with |i − j| = 1 we have |(1 − t)εi + tεj| ≥ |εi| − ε ,

2 for any εi, εj ∈ εZ \ B2ε with |i − j| = 1 the regularity of u in Bη0 \{0} implies Z 1 |u(εi) − u(εj)| ≤ |∇u(tεi + (1 − t)εj)(εi − εj)| dt . 0

Since i − j ∈ {±e1, ±e2}, a direct computation yields the two cases Z 1 |i · e |  2 dt if (i − j) k e ,  2 1  0 |ti + (1 − t)j| |u(εi) − u(εj)| ≤ (4.70) Z 1 |i · e |  1  2 dt if (i − j) k e2 , 0 |ti + (1 − t)j|

(−,+) (+,+) εQ3 B2rε εQ3

2rε ∼ ε

Q6ε

2rε (−,−) (+,−) εQ3 εQ3

Figure 8. The decomposition of B2rε used to bound the energy in (4.72) makes use of the s trimmed quadrants εQ3 .

To further simplify the energy, given a sign s = (s1, s2) ∈ {(+, +), (−, +), (−, −), (+, −)} s and n ∈ N we deﬁne the trimmed quadrants Qn as s 2 Qn := {x ∈ R : s1 x · e1 ≥ n, s2 x · e2 ≥ n} . (4.71) Then applying Jensen’s inequality in (4.70) and specifying n = 3 for the trimming, we can bound the energy via

d2rε/εe 2 1 X 1 s 1 C X 2 k Eε(u; B2rε ) ≤ Eε(u, εQ3 ∩B2rε )+ Eε(u; Q6ε)+ ε 4 , (4.72) εθε εθε εθε εθε (k − 1) s k=2 see Figure 8. The last sum is converging with respect to k , so that the second and third term can be estimated by

d2rε/εe 2 1 C X 2 k ε Eε(u; Q6ε) + ε 4 ≤ C , (4.73) εθε εθε (k − 1) θε k=2 where for the ﬁrst term we used the trivial estimate |u(εi) − u(εj)|2 ≤ 4. On every trimmed s quadrant Q3 we can use again Jensen’s inequality in (4.70) and a monotonicity argument FROM THE N -CLOCK MODEL TO THE XY MODEL 53 to deduce s X 2 2 2 Eε(u; εQ3 ∩ B2rε ) = ε |u(ε(i + s1e1)) − u(εi)| + |u(ε(i + s2e2)) − u(εi)| 2 εi∈εZ ∩B2rε s i∈Q3 X ε2 Z ε2 ≤ ε2 ≤ dx . (4.74) |εi|2 |x|2 2 s εi∈εZ ∩B2rε εQ ∩B2r s 2 ε i∈Q3 Note that we shifted the trimming in the last inequality to pass from discrete to continuum. We sum (4.74) over all possible s and, after multiplying with 1 , we infer that εθε X 1 1 Z ε2 ε E (u; εQs ∩ B ) ≤ dx ≤ 2π| log ε| , (4.75) εθ ε 3 2rε εθ |x|2 θ s ε ε ε B2rε \Bε −1 since rε = 4εθε < 1. The combination of (4.72), (4.73), and (4.75) yields 1 ε ε Eε(u; B2rε ) − 2π| log ε| ≤ C . (4.76) εθε θε θε −1 In order to control the remaining sum in (4.69), ﬁrst note that since rε = 4εθε , on the one hand we have that 1 X 2 2 θε 2 ε ε θε ≤ C (2rε + 2ε) ≤ C . (4.77) εθε ε θε hi,ji εi,εj∈B2rε s On the other hand, using the trimmed quadrants Q3 to split the sum as in (4.72) yields √ Z d2rε/εe 1 X 2 2 ε C X 2 k ε θε|u(εi) − u(εj)| ≤ θε dx + Cε + ε θε 2 . εθε εθε |x| εθε (k − 1) hi,ji k=2 B2rε \Bε εi,εj∈B2rε √ Note that due to the non-quadratic structure we have the additional constant 2 in front of the integral. Moreover, the last sum diverges logarithmically, but we have an additional factor θε which compensates this growth. We conclude that

1 X 2 2 ε θε|u(εi) − u(εj)| ≤ C rε + ε + ε| log θε| . εθε hi,ji εi,εj∈B2rε

Since ε θε 1 the right hand side vanishes when ε → 0. Thus this estimate, (4.69), (4.76), and (4.77) imply that 1 ε lim sup Eε(vε; B2rε ) − 2π| log ε| ≤ 0 . (4.78) ε→0 εθε θε

Next we control the energy in Bη(x1) \ Brε (x1) for 0 < η < η0 , where η0 is given by the assumptions. To this end, we need to examine the precise behavior of the sequence vε for 2 i, j ∈ Z satisfying |i − j| = 1 and |εj − x1|, |εi − x1| ≥ rε . The basic idea is that for many such pairs the energy contribution vanishes. Indeed, write such points as ε ε ε ε ε ε εi − x1 = ri exp(ι(ki θε + φi )), εj − x1 = rj exp(ι(kj θε + φj )) (4.79) ε ε ε ε with ki , kj ∈ {0,...,Nε − 1} and φi , φj ∈ [0, θε). By (2.2) we obtain ε ε ε ε ε ε ε = ri exp(ι(ki θε + φi )) − rj exp(ι(kj θε + φj )) ε ε ε ε ε ε ≥ rε| exp(ι(ki θε + φi )) − exp(ι(kj θε + φj ))| − |ri − rj |

2rε ε ε ε ε ≥ min |(ki − kj )θε + φi − φj + 2πn| − ε n∈{0,±1} π

2rεθε ε ε ≥ min (|ki − kj + Nεn| − 1) − ε . n∈{0,±1} π 54 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

−1 Inserting rε = 4εθε , the above estimate can be rearranged into π ε ε ≥ min (|ki − kj + Nεn| − 1) . 4 n∈{0,±1} ε ε Since ki − kj + Nεn is an integer, we get the following two possibilities: ε ε ε ε (i) ki − kj − Nεn = 0, which is possible only for n = 0, since ki , kj ∈ {0,...,Nε − 1}. This yields vε(εi) = vε(εj) since εi and εj belong to the same sector; ε ε (ii) ki − kj = ±1 mod (Nε), which implies

|vε(εi) − vε(εj)| ≤ θε . ε ε Moreover, since k 6= k we infer that dist(εi − x , ∂S ε ) ≤ ε. i j 1 ki ,ε With this information at hand, we can estimate the energy by bounding the number of all 2 SNε−1 points in εZ ∩ Bη(x1) which are ε-close to one of the lines in k=0 ∂Sk,ε + x1 . Since −1 Nε ≤ Cθε , this leads to

Nε−1 2 X 2 2 Eε(vε; Bη(x1) \ Brε (x1)) ≤ Cθε ε #{εi ∈ εZ ∩ Bη(x1) : dist(εi, ∂Sk,ε + x1) ≤ ε} k=0

≤ Cθε(η + 2ε)ε .

Dividing the inequality by εθε we obtain for ε small enough that 1 1 1 Eε(vε; Bη(x1)) ≤ Eε(vε; B2rε (x1)) + Eε(vε; Bη(x1) \ Brε (x1)) εθε εθε εθε 1 ≤ Eε(vε; B2rε (x1)) + Cη , εθε where we used that rε ε to split the energy via changing the inner radius from rε to 2rε . Subtracting the term 2π| log ε| ε and using (4.78), we proved that θε 1 ε lim sup Eε(vε; Bη(x1)) − 2π| log ε| ≤ Cη (4.80) ε→0 εθε θε for some uniform constant C < +∞. Step 2 An interpolation between singular and piecewise constant approximations. We do the construction in the case where the singularity lies in the origin. The case of singularities contained in λZ2 will be treated with a translation argument. Consider a cube m(λ) m(λ) 2 Q(λ) = [−2 λ, 2 λ] , where λ = λk with k ≥ n will be small, but fixed in this step, and 1 m(λ) ∈ N is chosen maximal such that Q(λ) ⊂ Bη/2 with fixed 0 < η < η0 . Note that the corners of Q(λ) belong to λZ2 . Define then a sequence of dyadically shrinking m(λ) −k m(λ) −k 2 cubes by Qk = [(−2 + (2 − 2 ))λ, (2 − (2 − 2 ))λ] for k ≥ 0. Here the factor −k Pk −l 2 − 2 is chosen as the value of the geometric sum l=0 2 . For notational reasons we m(λ) m(λ) also set Q−1 := Q(λ) and Q−2 := [−(2 + 1)λ, (2 + 1)λ]. Then for k ≥ 0 the layer Lk = Qk−1 \ Qk can be decomposed into finitely many closed cubes with disjoint interior and side lengths 2−kλ. Indeed, those cubes are given by the closures of the half-open cubes belonging to the family z −k −k 2 2 z Qk := qk = 2 λz + [0, 2 λ) : z ∈ Z , qk ⊂ Lk . With a slight abuse of notation we also set z 2 2 z Q−1 := q−1 = λz + [0, λ) : z ∈ Z , q−1 ⊂ L−1 .

For k ≥ −1, a generic element of Qk is of the form z − max{k,0} − max{k,0} 2 qk = 2 λz + [0, 2 λ) ⊂ Lk .

(See Figure 9.) We introduced the square Q−2 and the family of cubes Q−1 since they will be useful later to glue in the layer L−1 = Q−2 \ Q−1 the construction of the recovery sequence uε inside Q(λ) and outside Q(λ), respectively. The construction of uε outside Q(λ) will be based, as in Proposition 4.16, on a piecewise constant approximation of u on the λZ2 lattice and its boundary value on ∂Q(λ) will agree with that of the construction FROM THE N -CLOCK MODEL TO THE XY MODEL 55

2 from the inside. For this reason the cubes in Q−1 have volume λ , as those of Q0 , instead of the notationally more consistent volume (2λ)2 . We further choose kε ∈ N as the unique number such that −kε −kε+1 2 ≤ θε < 2 . (4.81) Note that, in particular, we have that

Qkε ⊃ B(2m(λ)−2)λ . (4.82)

Q−1 = Q(λ) Q−2 λ

z qk ∈ Qk

2−kλ

0 2−kλz

2m(λ)λ

η/2

Figure 9. Dyadic decomposition of Q(λ) and example of a square belonging to the family Qk (in the picture, k = 1 ). The ball contained in all squares Qk is given by (4.82).

z − max{k,0} − max{k,0} 2 To each (non-empty, half-open) cube qk = 2 λz + [0, 2 λ) ∩ Lk ∈ Qk , we associate the value z − max{k,0} 1 1 uk,ε = u(2 λ(z + 2 e1 + 2 e2)) , − max{k,0} 1 1 z z where 2 λ(z + 2 e1 + 2 e2) is the midpoint of the cube qk . We use these values uk,ε to deﬁne an interpolation similar to the one in the proof of Proposition 4.16, but on a family of shrinking cubes. In order to obtain quantitative energy estimates, we need a bound on z the diﬀerences of the values uk,ε between cubes which touch at their boundaries. A key ingredient will be the estimate

x y x|y| − x|x| + x|x| − y|x| |x − y| − ≤ ≤ 2 , (4.83) |x| |y| |x||y| |y| 2 z which is valid for x, y ∈ R \{0}. Due to (4.82) it holds that 0 ∈/ qk for −1 ≤ k ≤ kε . Hence, for two touching cubes qz1 and qz2 with −1 ≤ k , k ≤ k (i.e., qz1 ∩ qz2 6= Ø), the k1 k2 1 2 ε k1 k2 estimate (4.83) implies the bound √ 2λ(2−k1 + 2−k2 ) |uz1 − uz2 | ≤ 2 , (4.84) k1,ε k2,ε − max{kl,0} 1 1 min |2 λ(zl + e1 + e2)| l=1,2 2 2 where we used that the distance between midpoints is bounded by the sum of the diameters of the cubes. Assuming that m(λ) ≥ 2, the inclusion (4.82) implies that the denominator can be estimated from below via

2− max{kl,0}λ(z + 1 e + 1 e ) ≥ (2m(λ) − 2)λ ≥ 2m(λ)−1λ . l 2 1 2 2

−kε In combination with (4.84) and the bound 2 ≤ θε (cf. (4.81)) we obtain

z1 z2 3−m(λ) −k1 −k2 kε−kl−m(λ) |uk ,ε − uk ,ε| ≤ 2 (2 + 2 ) ≤ 16 max 2 θε . (4.85) 1 2 l=1,2

Next, we deﬁne the piecewise constant functionw ¯ε : Q−2 \ Qkε → Sε via z − max{k,0} − max{k,0} 2 w¯ε(x) = uk,ε if x ∈ 2 λz + [0, 2 λ) ∩ Lk , −1 ≤ k ≤ kε . (4.86)

Note that this function is pointwise well-deﬁned except on parts of ∂Q−2 since we consider half-open cubes. In order to deﬁne an interpolation between cubes which approximates the z piecewise constant functionw ¯ε , we introduce again boundary conditions. In each cube qk ∈ 56 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

Qk , we define the boundary conditions only on those sides which are not contained in ∂Qk (recall that ∂Qk is the inner part of the boundary of the layer Lk ). On the side contained in ∂Qk (if there is any) we define the boundary condition via the cubes in Lk+1 (cf. Figure 10). To fix ideas, in what follows one can use an iterative definition starting with k = kε , for which we neglect the inner boundary.

∂Qk ∂Qk−1

z qk

Lk+1 Lk Lk−1

z Figure 10. Sides of a cube qk contained in the layer Lk where we deﬁne the boundary conditions.

−k 0 −k 0 2 For a generic side S = {2 λz + tei : t ∈ [0, 2 λ]} with z ∈ Z , k ≥ 0 and i ∈ {1, 2}, 1 2 3 1 3 ε 1 and three values w = (w , w , w ) ∈ (S ) , we set bS,k[w]: S → S as ε ε k bS,k[w](x) = b2kS[w](2 x) , ε 0 0 where bS0 [w] is defined in (4.32) for every side S = {λz + tei : t ∈ [0, λ]}. In this proof we work with the constant c0 := 393 in (4.32); this choice will be clear only after formula (4.102). Given a side S as above satisfying additionally S ⊂ Lk with k ≥ 0 and ε S * ∂Qk (recall that the layer Lk is closed), we specify the three values w = wS on S by ε −k 0 − + −k 0 wS = w¯ε(2 λz ), mid (w ¯ε)S , (w ¯ε)S ), w¯ε(2 λ(z + ei) , (4.87) − + where (w ¯ε)S and (w ¯ε)S denote the (constant) traces along the side S of the functionw ¯ε defined in (4.86) (note that on sides in ∂Qk the trace from outside Lk may be non-constant because the cubes shrink). It is only here where we have to use the values in the layer L−1 . z z 1 Fix a cube qk ∈ Qk with k ≥ 0 and define the boundary values bk,ε[z]: ∂qk \ ∂Qk → S via the formula ε ε −k 0 0 2 −k bk,ε[z](x) = bS,k[wS](x) if x = 2 λz + tei ∈ S for some z ∈ Z and t ∈ [0, 2 λ] .

Having in mind the deﬁnition (4.32), on each side S the function bk,ε[z] satisﬁes the Lipschitz-estimate ε ε k ε ε k bk,ε[z](x) − bk,ε[z](y) = b2kS[wS](2 x) − b2kS[wS](2 y) k ε i ε 22 θε 1 ≤ max dS (wS) , (wS) |x − y| i=1,3 c0ε k−1 ε i ε 2 π2 θε ≤ max (wS) − (wS) |x − y| , (4.88) i=1,3 c0 ε where we used (2.2) in the last inequality. We continue with estimating the right hand side of (4.88). On the one hand, equation (4.85) implies

ε 2 ± + − ± 1 + − |(w ) − (w ¯ ) | ≤ d 1 mid((w ¯ ) , (w ¯ ) ), (w ¯ ) = d 1 (w ¯ ) , (w ¯ ) S ε S S ε S ε S ε S 2 S ε S ε S π ≤ |(w ¯ )+ − (w ¯ )−| ≤ C 2kε−k−m(λ)θ , (4.89) 4 ε S ε S 1 ε with C1 = 32, where in the last inequality we also used that due to the deﬁnition (4.86) and the fact that S ⊂ ∂qz we have (w ¯ )± = uz± for some z ∈ 2 and −1 ≤ k ≤ k with k ε S k±,ε ± Z ± ε ε |k − k±| ≤ 1. On the other hand, observe that in the deﬁnition of wS in (4.87) the points FROM THE N -CLOCK MODEL TO THE XY MODEL 57

2−kλz0 and 2−kλ(z0 + e ) belong to S . This implies that the cubes qz1 and qz3 used in i k1 k3 ε 1 −k 0 ε 3 −k 0 the deﬁnition (4.86) for (wS) =w ¯ε(2 λz ) and (wS) =w ¯ε(2 λ(z + ei)), respectively, must touch both the cubes qz± used in the deﬁnition (4.86) for (w ¯ )± . Hence, again due k± ε S to (4.85),

ε i ± kε−k−m(λ) |(wS) − (w ¯ε)S | ≤ C12 θε , for i = 1, 3 . (4.90)

kε Combining the last two estimates with (4.88) and the bound 2 θε ≤ 2 yields θ |b [z](x) − b [z](y)| ≤ ε |x − y| , (4.91) k,ε k,ε ε k−1 kε−k−m(λ) π2 where we used that 2C12 θε ≤ 1. Next observe that the locally deﬁned c0 boundary values yield a function

[ [ z 1 z bε : ∂qk \ ∂Qkε → S , x 7→ bε(x) := bk,ε[z](x) if x ∈ ∂qk \ ∂Qk . (4.92) z 0≤k≤kε qk∈Qk

We briefly explain the idea how to construct the recovery sequence. In Qkε we put the value of the function vε used in Step 1 and defined in (4.66), namely, we approximate directly x |x| close to its singularity. In the first layer Lkε we keep this construction and then we start z z an interpolation scheme with respect to the cubes qk , where we put the value uk,ε in most part of a cube. The boundary conditions bk,ε[z] help to control interactions between different cubes. In the estimates we can allow for multiplicative constants since the total contribution will be proportional to 2m(λ)λ, which scales like η . However, a precise dependence on the energy with respect to the layer number k will be crucial since at the end we have to sum over all layers. z Now let us start with the details. For the moment fix 0 ≤ k < kε . Given a cube qk ∈ Qk , z z z z let Pk,z : qk → ∂qk be any function such that |Pk,z(x) − x| = dist(x, ∂qk) for all x ∈ qk . 2 z 1 Setu ¯ε : εZ ∩ qk → S as z −1 z u¯ε(εi) = Geo bε(Pk,z(εi)), uk,ε θεε dist εi, ∂qk , with the extended geodesics given by Definition 4.14 and bε given by (4.92). Since in general u¯ε(εi) ∈/ Sε , we project it. The function uε in the square Q(λ) is then given by ( vε(εi) if εi ∈ Qkε−1 , uε(εi) := z z (4.93) Pε(¯uε(εi)) if εi ∈ qk for some qk ∈ Qk with 0 ≤ k < kε , with the operator Pε defined in (4.30). In this step we are interested in the energy restricted to Q(λ) and for this reason we defined uε only in Q(λ). The sequence uε will be defined later in Step 4 outside Q(λ), that means, far from the singularity, as in Proposition 4.16. 1 z First let us identify the L (Q(λ))-limit of uε . To this end, observe that for all εi ∈ qk with 0 ≤ k < kε we have by Definition 4.14 z −1 z z 1 uε(εi) = Pε(uk,ε) if θεε dist(εi, ∂qk) ≥ dS (bε(Pk,z(εi)), uk,ε) . (4.94) z We need to quantify the dependence on k in the right hand side. Let S ⊂ ∂qk be a side such that Pk,z(εi) ∈ S . Since the boundary datum bε restricted to S interpolates ε via geodesic arcs between the three elements of the vector wS defined in (4.87) and by z − + construction uk,ε ∈ {(w ¯ε)S , (w ¯ε)S }, it follows from (4.89) and (4.90) (with k + 1 in place of k if S ⊂ ∂Qk , which improves the estimate) that

z + − kε−k−m(λ) 1 1 dS (bε(Pk,z(εi)), uk,ε) ≤ dS bε(Pk,z(εi)), mid((w ¯ε)S , (w ¯ε)S ) + C12 θε ε i + − kε−k−m(λ) ≤ max d 1 (wS) , mid((w ¯ε) , (wε) ) + C12 θε i=1,3 S S S

kε−k−m(λ) ≤ C22 θε , (4.95) π for C2 = 2 + 1 C1 ≤ 96. In particular, the condition (4.94) implies that

z z kε−k uε(εi) = Pε(uk,ε) if dist(εi, ∂qk) ≥ C2 2 ε . (4.96) 58 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

kε kε z Since 2 θε ≤ 2 (cf. (4.81)), the term 2 ε vanishes when ε → 0. As the measure of each qk is 2−kλ, we deduce from (4.67) that pointwise a.e. in Q(λ) (and thus in L1(Q(λ))) it holds that

 x m(λ) m(λ) 2  on Q∞ := [(−2 + 2)λ, (2 − 2)λ] , |x| λ  uε → u0 = (4.97)  2−k−1λ(2z + e + e )  1 2 z  −k−1 if x ∈ qk ∩ Lk for some k ∈ N ∪ {0} . |2 λ(2z + e1 + e2)| T∞ λ x Notice that Q∞ = k=0 Qk and that u0 = |x| except in the layer Q(λ) \ Q∞ , whose thickness is 2λ, thus infinitesimal when λ → 0. 2 Below we bound the differences uε(εi)−uε(εj) for all εi, εj ∈ εZ ∩Q(λ) with |i−j| = 1. Substep 2.1 (interactions within a single cube) 2 z Consider first εi, εj ∈ εZ ∩ qk with 0 ≤ k < kε and |i − j| = 1. We distinguish several cases:

z kε−k z kε−k Case 1: If dist(εi, ∂qk) ≥ C2 2 ε and dist(εj, ∂qk) ≥ C2 2 ε, then from (4.96) we infer that z z |uε(εi) − uε(εj)| = |Pε(uk,ε) − Pε(uk,ε)| = 0 . z By the Lipschitz continuity of dist(·, ∂qk), we can from now on assume that

z z kε−k max{dist(εi, ∂qk), dist(εj, ∂qk)} < (C2 + 1)2 ε . (4.98)

Case 2: We first analyze when Pk,z(εi) and Pk,z(εj) lie on different 1-dimensional boundary z segments Si 6= Sj of qk . We claim that Pk,z(εi) and Pk,z(εj) are then close to a node of −k 2 the lattice 2 λZ . Indeed, denote by ΠSi and ΠSj the projections onto the subspaces spanned by the segments Si and Sj , respectively. Assumption (4.98) and the defining property of Pk,z imply that

z z kε−k |Pk,z(εi) − Pk,z(εj)| ≤ ε|i − j| + dist(εi, ∂qk) + dist(εj, ∂qk) ≤ 2(C2 + 1)2 ε + ε .

kε Hence for ε small enough the sides Si and Sj cannot be parallel because 2 ε λ. −k 2 Therefore the point ΠSi (ΠSj (εi)) belongs to Si ∩ Sj ⊂ 2 λZ . We claim that

bε(Pk,z(εi)) = bε(Pk,z(εj)) =w ¯ε(ΠSi (ΠSj (εi))) . (4.99) ∗ ∗ ∗ ∗ Indeed, let us denote by 0 ≤ ki , kj ≤ kε the layer numbers and by Si ⊂ Si and Sj ⊂ Sj the sides satisfying

∗ ∗ ε ε ki ε ε kj bε(Pk,z(εi)) = b k∗ [w ∗ ](2 Pk,z(εi)) , bε(Pk,z(εj)) = b k∗ [w ∗ ](2 Pk,z(εj)) . 2 i S∗ Si j ∗ Sj i 2 Sj ∗ ∗ (The sides Si and Sj are needed due to the fact that Si or Sj may be contained in ∂Qk , where bε is defined using the cubes which decompose the layer Lk+1 ; if, for instance, Si is ∗ ∗ ∗ not contained in ∂Qk , then ki = k and Si = Si .) Since by the dyadic construction Si either agrees with Si or is exactly one half of the side Si , it follows that ΠSi (ΠSj (εi)) is an ∗ ∗ endpoint of Si . By the same reasoning it is also an endpoint of Sj . Since ΠSi (ΠSj (εi)) = ∗ ∗ ki kj ΠSj (ΠSi (εj)), it then suffices to show that 2 Pk,z(εi) and 2 Pk,z(εj) are sufficiently close ∗ ∗ ki kj to 2 ΠSi (ΠSj (εi)) and 2 ΠSj (ΠSi (εj)), respectively, since by construction the boundary datum is constant in a neighborhood of the endpoints of a side. The 1-Lipschitz continuity of ΠSi and ΠSj combined with (4.98) yields

|Pk,z(εi) − ΠSi (ΠSj (εi))| ≤ |εi − ΠSj (εi)| ≤ |εi − εj| + |εj − ΠSj (εj)| + |ΠSj (εi) − ΠSj (εj)|

kε−k ≤ 2ε + (C2 + 1)2 ε . (4.100) Similarly we can derive the estimate

kε−k |Pk,z(εj) − ΠSj (ΠSi (εj))| ≤ 2ε + (C2 + 1)2 ε . (4.101) FROM THE N -CLOCK MODEL TO THE XY MODEL 59

−k+1 −1 By (4.81), for ε small enough both terms can be bounded by 2 (C2 + 2)εθε . Since ∗ ∗ ∗ ∗ ki kj k ≤ ki , kj ≤ k + 1, multiplying (4.100) by 2 and (4.101) by 2 yields n ∗ ∗ ∗ ∗ o ki ki kj kj −1 max |2 Pk,z(εi) − 2 ΠSi (ΠSj (εi))|, |2 Pk,z(εj) − 2 ΠSj (ΠSi (εj))| ≤ C3εθε , (4.102) where C3 = 4(C2 + 2) = 392 < c0 , c0 being the constant in the deﬁnition (4.32) (thus explaining the choice c0 = 393). The estimate (4.102) thus implies (4.99). Having in mind that θ ε |dist(εi, ∂I) − dist(εj, ∂I)| ≤ θ , ε ε z the 1-Lipschitz continuity of Geo[w ¯ε(ΠSi (ΠSj (εi))), uk,ε] and the formula foru ¯ε then yield that

|u¯ε(εi) − u¯ε(εj)| ≤ θε .

From the deﬁnition of the function Pε and the previous estimate we infer that

|uε(εi) − uε(εj)| ≤ θε . (4.103)

Case 3: It remains to treat the case of points i and j such that Pk,z(εi) = ΠSi (εi) and

Pk,z(εj) = ΠSi (εj). Here we use the Lipschitz-continuity of bε on Si . Note that bε might be deﬁned separately on two smaller sides contained in Si , but nevertheless the Lipschitz condition on sides (cf. (4.91)) holds on the whole Si because of the convexity of sides. Moreover, we want to apply the stability estimate of Lemma 4.15. To this end, observe that by (4.95) and (4.81) we have

z kε−k−m(λ) −m(λ) |bε(Pk,z(εi)) − uk,ε| ≤ C22 θε ≤ 2C22 −m(λ) and the right hand side can be made arbitrarily small (speciﬁcally, 2C22 < c, where c is the constant given in Lemma 4.15) since m(λ) 1 for small λ. The same estimate −1 z holds with i replaced by j . To reduce notation, we set dε,i = θεε dist(εi, ∂qk) and −1 z dε,j = θεε dist(εj, ∂qk). Then by the triangle inequality, (4.91), and Lemma 4.15 we have z z |u¯ε(εi) − u¯ε(εj)| ≤ Geo[bε(Pk,z(εi)), uk,ε](dε,i) − Geo[bε(Pk,z(εi)), uk,ε](dε,j) z z + Geo[bε(Pk,z(εi)), uk,ε](dε,j) − Geo[bε(Pk,z(εj)), uk,ε](dε,j)

1 ≤ |dε,i − dε,j| + dS (bε(Pk,z(εi)), bε(Pk,z(εj))) π −1 π ≤ θε + 2 θεε |ΠSi (εi) − ΠSi (εj)| ≤ (1 + 2 )θε . Hence we deduce the weaker but still suﬃcient bound

|uε(εi) − uε(εj)| ≤ 3θε . (4.104) Substep 2.2 (interactions between diﬀerent cubes) z z Now we consider lattice points εi ∈ qzi and εj ∈ q j with qzi 6= q j and |i − j| = 1. In ki kj ki kj this substep we assume that 0 ≤ ki, kj ≤ kε − 1, that means, we consider only the layers where we interpolate. We assume without loss of generality that ki ≤ kj . Apart from the ∗ ∗ layer numbers ki and kj we have to focus on the numbers ki and kj characterized by the property P (εi) ∈ L ∗ \ ∂Q ∗ ,P (εj) ∈ L ∗ \ ∂Q ∗ , ki,zi ki ki kj ,zj kj kj that means, those values which determine the rescaling of the boundary conditions. Note that from the deﬁnition of Pk,z it follows that ∗ ∗ ki ≤ ki ≤ ki + 1, kj ≤ kj ≤ kj + 1 . (4.105) z Since all cubes qk are half-open and oriented along the coordinate axes, there exists a side S of ∂qzj such that the segment [εi, εj] intersects S orthogonally and additionally ij kj ij z S ⊂ ∂qzi ∩ ∂q j , (4.106) ij ki kj where we used that kj ≥ ki to ensure the inclusion. In particular, z dist(εi, ∂qzi ) + dist(εj, ∂q j ) ≤ ε (4.107) ki kj 60 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF which implies that

|u¯ε(εi) − bε(Pki,zi (εi))| + |u¯ε(εj) − bε(Pkj ,zj (εj))| ≤ θε . (4.109)

Note that the above estimate does not give information on |u¯ε(εi) − u¯ε(εj)| since, a priori, bε(Pki,zi (εi)) might diﬀer from bε(Pkj ,zj (εj)). We will show that this is not the case. To this end, we shall prove two alternatives:

(i) Pki,zi (εi) ∈ Sij and Pkj ,zj (εj) ∈ Sij ; ∗ ∗ (4.110) ki 2 ki+3 kj 2 kj +2 (ii) dist(2 Pki,zi (εi), λZ ) ≤ 2 ε and dist(2 Pkj ,zj (εj), λZ ) ≤ 2 ε . Indeed, ﬁrst assume that P (εj) ∈/ S . Then there exists another facet S of qzj , kj ,zj ij j kj

Sj 6= Sij , such that Pkj ,zj (εj) ∈ Sj . Since dist(εj, Sij) ≤ ε and dist(εj, Sj) ≤ ε, the sides S and S cannot be parallel since the distance between parallel sides of ∂qzj is given by ij j kj −kj 1 −kj 2 2 λ ≥ 2 θελ ε. Hence ΠSj (ΠSij (εj)) ∈ Sj ∩ Sij ⊂ 2 λZ , so that

−kj 2 −kj 2 dist(Pkj ,zj (εj), 2 λZ ) = dist(ΠSj (εj), 2 λZ ) ≤ |εj − ΠSij (εj)| ≤ ε . (4.111) In particular, applying (4.105) we deduce from the above estimate that

∗ ∗ ∗ kj 2 kj −kj 2 dist(2 Pkj ,zj (εj), λZ ) = 2 dist(Pkj ,zj (εj), 2 λZ ) (4.112) kj +1 −kj 2 kj +1 ≤ 2 dist(Pkj ,zj (εj), 2 λZ ) ≤ 2 ε ,

∗ −kj 2 −kj 2 where we used that 2 λZ ⊂ 2 λZ . For the point Pki,zi (εi) consider ﬁrst the case

Pki,zi (εi) ∈ Sij . Due to what we aim to prove we then assume that Pkj ,zj (εj) ∈ Sj \ Sij as above, so that (4.112) holds. Hence (4.108) and (4.111) imply

−kj 2 −kj 2 2 dist(Pki,zi (εi), 2 λZ ) ≤ dist(Pkj ,zj (εj), 2 λZ ) + 2ε ≤ 3ε < 2 ε .

In order to conclude the claimed estimate, observe that the condition Pki,zi (εi) ∈ Sij ⊂ ∂qzj ⊂ L forces k∗ ≥ k so that kj kj i j

∗ ∗ ∗ ki 2 ki −ki 2 dist(2 Pki,zi (εi), λZ ) = 2 dist(Pki,zi (εi), 2 λZ )

ki+1 −kj 2 ki+3 ≤ 2 dist(Pki,zi (εi), 2 λZ ) ≤ 2 ε , (4.113)

∗ −kj 2 −ki 2 where we used that 2 λZ ⊂ 2 λZ . On the contrary, if Pki,zi (εi) ∈/ Sij , denote by S a facet of qzi such P (εi) ∈ S , S 6= S . Then S and S do not lie on the same i ki ki,zi i i ij i ij straight line. To show this, we argue by contradiction. Assume that Si ⊂ span(Sij). Since the segment [εi, εj] is orthogonal to Sij , this would imply the false statement

ΠSi (εi) = ΠSij (εi) = ΠSij (εj) ∈ Sij , where the last inclusion holds since εj ∈ qzj and S is a side of the cube qzj . Since S kj ij kj i and Sij can neither be parallel for ε small enough, we conclude that ΠSi (ΠSij (εi)) ∈ span(S )∩span(S ). Since, by (4.106), S ⊂ ∂qzi , we know that Π (Π (εi)) ∈ 2−ki λ 2 . i ij ij ki Si Sij Z Thus the deﬁning property of Sij yields

−ki 2 −ki 2 dist(Pki,zi (εi), 2 λZ ) = dist(ΠSi (εi), 2 λZ ) ≤ |ΠSij (εi) − εi| ≤ |εi − εj| = ε . Again in combination with (4.105) this inequality implies the estimate

∗ ki 2 ki+1 −ki 2 ki+1 dist(2 Pki,zi (εi), λZ ) ≤ 2 dist(Pki,zi (εi), 2 λZ ) ≤ 2 ε . (4.114) ∗ kj 2 It remains to establish an estimate for dist(2 Pkj ,zj (εj), λZ ) when Pki,zi (εi) ∈/ Sij and

Pkj ,zj (εj) ∈ Sij . In this case we have

−kj 2 −ki 2 dist(Pkj ,zj (εj), 2 λZ ) = dist(ΠSij (εj), 2 λZ ) ≤ |ΠSij (εj) − ΠSij (ΠSi (εi))|

≤ |εj − εi| + |εi − ΠSi (εi)| ≤ 2ε , FROM THE N -CLOCK MODEL TO THE XY MODEL 61

−ki 2 −kj 2 where we used the inclusion 2 λZ ⊂ 2 λZ (recall the assumption ki ≤ kj at the beginning of Substep 2.2). From the above inequality we deduce the estimate

∗ kj 2 kj +1 −kj 2 kj +2 dist(2 Pkj ,zj (εj), λZ ) ≤ 2 dist(Pkj ,zj (εj), 2 λZ ) ≤ 2 ε . (4.115) Combining the estimates (4.112), (4.113), (4.114), and (4.115) we proved the claimed alternatives (i) or (ii) in (4.110). We analyze them separately below.

∗ ∗ ki 2 ki+3 kj 2 kj +2 Case 5: Assume dist(2 Pki,zi (εi), λZ ) ≤ 2 ε and dist(2 Pkj ,zj (εj), λZ ) ≤ 2 ε 2 (that means, alternative (ii)) and denote by λz¯i, λz¯j ∈ λZ points realizing the minimal −k∗ −k∗ distance. We start by observing that 2 i λz¯i = 2 j λz¯j . Indeed, on the one hand we use (4.108) to estimate

∗ ∗ ∗ ∗ −ki −kj −ki −kj |2 λz¯i − 2 λz¯j| ≤ |2 λz¯i − Pki,zi (εi)| + |Pkj ,zj (εj) − 2 λz¯j| + 2ε ≤ 14ε .

− max{k∗,k∗} On the other hand, since both points on the left hand side belong to 2 i j λ 2 ⊂ ∗ ∗ Z −kj −1 2 −kj −1 1 −ki −kj 2 λZ and 2 λ ≥ 2 θελ ε by (4.81), we deduce that 2 λz¯i = 2 λz¯j . We −k∗ −k∗ set pij := 2 i λz¯i = 2 j λz¯j . Let now S and S be the sides of the cubes in Q ∗ and Q ∗ , respectively, such that i j ki kj

Pki,zi (εi) ∈ Si , Pkj ,zj (εj) ∈ Sj , and

∗ ∗ ε ε ki ε ε kj b (P (εi)) = b k∗ [w ](2 P (εi)) , b (P (εi)) = b k∗ [w ](2 P (εj)) . ε ki,zi i Si ki,zi ε kj ,zj j Sj kj ,zj 2 Si 2 Sj (4.116) We claim that pij ∈ Si ∩ Sj . Indeed, since by assumption

|pij − Pki,zi (εi)| ≤ 8ε , |pij − Pkj ,zj (εj)| ≤ 4ε

z 2 −k 2 and for a given side S ⊂ ∂qk with 0 ≤ k < kε and z ∈ Z it holds that dist(S, 2 λZ \S) ≥ 2−kλ ε, the claim follows by a triangle inequality argument. Moreover, recalling that ki+3 −1 −1 kj +2 −1 ki, kj ≤ kε−1, property (4.81) yields 2 ε ≤ 4εθε < c0εθε and 2 ε ≤ 2εθε < −1 c0εθε , c0 being the constant used in the deﬁnition (4.32). Thus we conclude from (4.116) and the deﬁnition of the boundary condition that

bε(Pki,zi (εi)) = bε(pij) = bε(Pkj ,zj (εj)) , where we used that pij has to be an endpoint of Si and of Sj . Combined with (4.109) we infer

|uε(εi) − uε(εj)| ≤ θε . (4.117)

Case 6: Finally we analyze the case Pki,zi (εi) ∈ Sij and Pkj ,zi (εj) ∈ Sij (that means, alternative (i)). Since by assumption the line segment [εi, εj] intersects Sij orthogonally z and S is a side of both cubes qzi and q j , we know that P (εi) = P (εj). In ij ki kj ki,zi kj ,zj combination with estimate (4.109) we therefore obtain

|uε(εi) − uε(εj)| ≤ θε . (4.118)

Substep 2.3 (interactions between Qkε−1 and the layers) 2 2 In this step we consider the case where εi ∈ εZ ∩ Qkε−1 but εj ∈ εZ \ Qkε−1 . Since

|εi − εj| = ε, it follows that εi ∈ Lkε and εj ∈ Lkε−1 , that means, the last and the last but −kε 1 one layers. Indeed, the thickness of the last layer is 2 λ ≥ 2 θελ ε, so that the claim 62 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF follows by a triangle inequality argument. Let z ∈ 2 be such that εj ∈ qzj . From the j Z kε−1 deﬁnition of uε in (4.93) and (4.67) we infer

|uε(εi) − uε(εj)| = |vε(εi) − Pε(¯uε(εj))| ≤ |vε(εi) − u¯ε(εj)| + θε

≤ |vε(εi) − bε(Pzj ,kε−1(εj))| + |bε(Pzj ,kε−1(εj)) − u¯ε(εj)| + θε

≤ |vε(εi) − bε(Pzj ,kε−1(εj))| + 2θε , where the last inequality can be proven as the estimates (4.48) and (4.109). Using the general estimate (4.95) with k = kε − 1 we can further estimate the right hand side to conclude that z |u (εi) − u (εj)| ≤ |v (εi) − u j | + C θ = |v (εi) − u(2−kε+1λ(z + 1 e + 1 e ))| + C θ . ε ε ε kε−1,ε ε ε j 2 1 2 2 ε (4.119) z Recall that 2−kε+1λ(z + 1 e + 1 e ) is the midpoint of the cube q j , so that by (4.81) j 2 1 2 2 kε−1

−kε+1 ≤ ε + 2 λ ≤ ε + 2θελ . We insert this bound with the estimates (4.67) and (4.83) in (4.119) to obtain

−kε+1 1 1 |uε(εi) − uε(εj)| ≤ |u(εi) − u(2 λ(zj + 2 e1 + 2 e2))| + C θε

−kε+1 1 1 |εi − 2 λ(zj + 2 e1 + 2 e2)| ε + 2θελ ≤ 2 + C θε ≤ C + C θε, −kε+1 1 1 m(λ) |2 λ(zj + 2 e1 + 2 e2)| (2 − 2)λ where for the last inequality we used the set inclusion (4.82). Since ε θε , for λ > 0 ﬁxed we can assume that ε ≤ θελ, so that the last estimate turns into the bound

|uε(εi) − uε(εj)| ≤ C θε . (4.120)

Step 3 (energy estimates in Q(λ)) Let us ﬁrst summarize what we have proven so far. By our choice of m(λ) at the beginning of Step 2 we have Q(λ) ⊂ Bη/2 and (4.82). Hence we can use the bound (4.80) of Step 1 to control the energy due to interactions with both points in Qkε−1 , where uε = vε , cf. (4.93).

For the interactions with at least one point in Q(λ) \ Qkε−1 , we showed in Substeps 2.1–2.3 (cf. (4.103), (4.104), (4.117), (4.118), and (4.120)) that the bound

|uε(εi) − uε(εj)| ≤ C θε (4.121) holds with a uniform constant C < +∞. In order to obtain precise estimates on the energy due to interactions with at least one point in Q(λ) \ Qkε−1 , we have to count the number of lattice points εi, εj satisfying uε(εi) 6= uε(εj). For such points (4.121) will suﬃce. z 2 Let us ﬁx such εi, εj . Then there exists a cube qk ∈ Qk with 0 ≤ k < kε and z ∈ Z with z kε−k dist(εi, ∂qk) ≤ C 2 ε . (4.122)

Indeed, if εi, εj ∈ Q(λ)\Qkε−1 and they belong to the same cube of Qk (Substep 2.1), then this is a consequence of (4.98). If εi, εj ∈ Q(λ) \ Qkε−1 , but they belong to two diﬀerent cubes (Substep 2.2), then this follows from (4.107). Finally, if, for instance, εi ∈ Qkε−1 and

εj ∈ Q(λ) \ Qkε−1 (Substep 2.3), then this is a consequence of the fact that εi ∈ Lkε and

εj ∈ Lkε−1 (see also (4.107)). Therefore it suﬃces to count lattice points that satisfy (4.122). From a covering argument with cubes of volume ε2 and (4.81) we infer that

2 2 z kε−k −k kε−k kε−k ε #{εi ∈ εZ : dist(εi, ∂qk) ≤ C2 ε} ≤ 4(2 λ + 2(C2 ε + ε))(2C2 ε + 2ε) −k −k −1 −k −1 ≤ C (2 λ + ε(2 θε + 1))(2 θε + 1)ε −2k −1 ≤ C 2 λεθε , −1 where in the last inequality we also used that εθε λ for ε small enough. Next recall z that the number of cubes qk in the layer Lk can be roughly bounded by m(λ) k #Qk ≤ C 2 2 . FROM THE N -CLOCK MODEL TO THE XY MODEL 63

Combining the previous two estimates with (4.121) we infer that the energy of uε can be estimated from above via

2 kε 1 1 Cθε m(λ) X −k −1 Eε(uε; Q(λ)) ≤ Eε(vε; Bη) + 2 λ 2 εθε εθε εθε εθε k=0

1 m(λ) ≤ Eε(vε; Bη) + C2 λ . εθε Due to the choice of m(λ) it holds that 2m(λ)λ ≤ η . Subtracting the term 2π| log ε| ε and θε inserting the upper bound (4.80) we conclude that 1 ε lim sup Eε(uε; Q(λ)) − 2π| log ε| ≤ C η . (4.123) ε→0 εθε θε We emphasize that Q(λ) implicitly depends on η through the quantity 2m(λ)λ. Step 4 (from local to global constructions) We are now in a position to define uε globally. In this step we stress again the dependence on n of λn . We start by repeating the construction presented in Step 2 around each singularity xh of u, by defining uε as in (4.93) (combined with a reflection if deg(u)(xh) = −1) in the squares Q(λn, xh) = Q(λn) + xh . To define uε outside the squares Q(λn, xh), we start by observing that the square xh + m(λn)+1 m(λn)+1 [−2 λn, 2 λn] is not contained in Bη/2(xh), since m(λn) has been chosen as m(λn) m(λn) the maximal integer such that Q(λn, xh) = xh + [−2 λn, 2 λn] ⊂ Bη/2(xh). This m(λn)+1 yields η/4 ≤ 2 λn and thus, by (4.82),

λn Qk (xh) = xh + Qk ⊃ B(2m(λn)−2)λ(xh) ⊃ Bη/16(xh) . (4.124)

λn Note that here we stress the dependence of Qk (xh) on λn , in contrast to the notation λn adopted for Qk in Step 2. We recall that Q−1(xh) = Q(λn, xh). SN SN Applying Lemma 4.13 with O = Ω \ h=1 Bη/16(xh) and Oe = Ωe \ h=1 Bη/32(xh) ∞ SN 1 to u ∈ C (Ωe \ h=1 Bη/32(xh); S ), we get a sequence of piecewise constant functions 1 1 SN 2 un ∈ PCλn (S ) such that un → u strongly in L (Ω \ h=1 Bη/16(xh); R ) and, by (4.124), Z Z − + 1 1 lim sup dS (un , un )|νun |1 dH ≤ |∇u|2,1 dx . (4.125) n→+∞ SN λn λn Ω Jun ∩(Ω\ h=1 Q0 (xh))

λn 2 Notice that the squares Q0 (xh) have vertices on the lattice λnZ . 2 SN λn 0 Let us fix n large enough. For εi ∈ εZ \ h=1 Q0 (xh) we define uε(εi) as the recovery sequence given in the proof of Proposition 4.16 for the piecewise constant function un ∈ 1 0 PCλn (S ) with the constant c0 = 393 in (4.32). Then we define uε(εi) := uε(εi) for εi ∈ 2 SN 2 SN λn 2 εZ \ h=1 Q(λn, xh) = εZ \ h=1 Q−1(xh). This completes the definition of uε in εZ . We claim that

2 λn λn 0 εi ∈ εZ ∩ Q−1(xh) and dist(εi, ∂Q−1(xh)) ≤ ε =⇒ uε(εi) = uε(εi) , (4.126) that means, the two constructions given by Step 2 and Proposition 4.16 are identical. Indeed, z0 ﬁrst note that the assumptions on εi above imply that εi ∈ L0 . Hence we ﬁnd q0 ∈ Q0 z0 z0 λn such that εi ∈ q0 . We now consider the two cases P0,z0 (εi) ∈ ∂q0 \ ∂Q−1(xh) and λn z0 λn z0 P0,z0 (εi) ∈ ∂Q−1(xh). If P0,z0 (εi) ∈ ∂q0 \ ∂Q−1(xh), let Si ⊂ ∂q0 be the side such λn that P0,z0 (εi) ∈ Si . By the assumption in (4.126), Si is not contained in ∂Q0 (xh) and thus z0 λn 2 it intersects a side S0 of q0 such that S0 ⊂ ∂Q−1(xh). In particular, ΠSi (ΠS0 (εi)) ∈ λnZ is an endpoint of Si and, by (4.126), ε |P0,z0 (εi) − ΠSi (ΠS0 (εi))| = |ΠSi (εi) − ΠSi (ΠS0 (εi))| ≤ |εi − ΠS0 (εi)| ≤ ε c0 , θε

λn where we used that S0 is the side such that dist(εi, S0) = dist(εi, ∂Q−1(xh)) ≤ ε. Since 2 P0,z0 (εi) is close enough to the corner pi,0 := ΠSi (ΠS0 (εi)) ∈ λnZ , the boundary condition 64 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

used for the deﬁnition of uε at P0,z0 (εi) agrees with its value at the corner, cf. (4.32). Therefore h i z0 −1 z0 uε(εi) = Geo bε(pi,0), u0,ε θεε dist(εi, ∂q0 ) h i 1 1 1 1 −1 z0 = Geo u pi,0 + λn( 2 e1 + 2 e2) , u λn(z0 + 2 e1 + 2 e2) θεε dist(εi, ∂q0 ) .

0 z0 The same holds true for uε . This concludes the proof of (4.126) when P0,z0 (εi) ∈ ∂q0 \ λn λn z0 ∂Q−1(xh). If, instead, P0,z0 (εi) ∈ ∂Q−1(xh), let Si be the side of q0 such that P0,z0 (εi) ∈ S . Then the two 3-tuples of values v (u) and wε defined in (4.33) and (4.87), respectively, i Si Si coincide (note that, by definition, both cubes in Q0 and Q−1 have size λ). Then (4.126) follows in this case too. Taking into account (4.97) on each Q(λn, xh), the function uε ∈ PCε(Sε) converges in 1 2 1 1 L (Ω; R ) to the function ˚un ∈ L (Ω; S ) defined by  SN un(x) if x ∈ Ω \ Q(λn, xh) ,  h=1  ˚un(x) := ! 1 0  λn  u0 (x − xh) if x ∈ Q(λn, xh) for some 1 ≤ h ≤ N,  0 deg(u)(xh)

We remark that the precise structure for ﬁxed λn is not important. Just note that due to the fact that Q(λn, xh) ⊂ Bη/2(xh) and (4.97), the layer in each Q(λn, xh) where ˚un diﬀers ±1 x−xh from , and thus from u, is of thickness 2λn . Consequently |x−xh| 1 2 ˚un → u in L (Ω; R ) as n → +∞ . (4.127)

It remains to estimate the energy of uε in terms of λn and η . In particular, we need to estimate the interactions between the square Q(λn, xh) and the exterior. Thanks to (4.126) we can split the energy as N 1 ε 1 0 [ λn Eε(uε; Ω) − 2πN| log ε| ≤ Eε uε;Ω \ Q0 (xh) εθε θε εθε h=1 N X 1 ε + Eε(uε; Q(λn, xh)) − 2π| log ε| . εθε θε h=1 By (4.123) and (4.54) we can pass to the limit in ε and conclude that Z 1 ε − + 1 1 lim sup Eε(uε; Ω) − 2πN| log ε| ≤ dS (un , un )|νun |1dH ε→0 εθε θε SN λn λn Jun ∩(Ω\ h=1 Q0 (xh)) + C η . (4.128)

Before we can conclude, we have to identify the flat limit of the vorticity measures µuε associated with the sequence uε . Since the right hand side in (4.128) is finite, Proposi- f PN tion 4.10 implies that (up to a subsequence) µuε * µ¯ for someµ ¯ = k=1 dkδyk with dk ∈ Z and |µ¯|(Ω) ≤ N (we allow dk = 0 in order to sum from 1 to N ). We claim that µ¯ = µ with µ defined in the statement of the proposition. Here comes the argument. Fix ε x0 ∈ Ω \{x1, . . . , xN }. Since the singular part 2πN| log ε| of the estimate (4.128) is con- θε SN centrated in the set h=1 B2rε (xh) (cf. (4.78)) and rε → 0, we deduce that for 0 < ρ λ small enough 1 lim sup Eε(uε; Bρ(x0)) < +∞ . (4.129) ε→0 εθε Since we assume here that θε ε| log ε|, Remark 2.5 yields that f µuε Bρ(x0) * 0 . (4.130) 0,1 Testing this convergence with a Lipschitz-function ϕ ∈ Cc (Bρ(x0)) such that ϕ(x0) = 1 we obtain that x0 ∈/ {y1, . . . , yN } (or x0 = yk for some k with dk = 0). Since x0 ∈ PN Ω \{x1, . . . , xN } was arbitrary, we can writeµ ¯ = h=1 dhδxh . FROM THE N -CLOCK MODEL TO THE XY MODEL 65

It remains to prove that dh = deg(u)(xh) for all 1 ≤ h ≤ N . Note that for ρ λ it holds that uε = vε on each Bρ(xh), where vε is defined in (4.66). Due to (4.80) we have for ε small enough 1 θ E (v ; B (x )) ≤ Cη ε + 2π| log ε| ≤ C| log ε| . (4.131) ε2 ε ε ρ h ε Hence we can apply [7, Proposotion 5.2], which states that in dimension 2 the flat convergence of µvε Bρ(xh) is equivalent to the flat convergence of the (normalized) Jacobians of the piecewise affine interpolation of vε on Bρ(xh). Denote this piecewise affine interpolation and the one associated to the function u on Bρ(xh) by vbε and ub(ε), respectively. Inserting the estimate (4.67) in the definition of the piecewise affine interpolation one can show that

|vbε(x) − ub(ε)(x)| ≤ Cθε for all x ∈ Bρ(xh) . (4.132) Taking into account one more time the estimate (4.131), we conclude that kv − u(ε)k 2 k∇v k 2 + k∇u(ε)k 2 bε b L (Bρ/2(xh)) bε L (Bρ/2(xh)) b L (Bρ/2(xh)) 1 2 1 1 1 ≤ Cθ E (v ; B (x )) + E (u; B (x )) ≤ Cθ | log ε| 2 , (4.133) ε ε2 ε ε ρ h ε2 ε ρ h ε 1 where the bound ε2 Eε(u; Bρ(xh)) ≤ C| log ε| can be proven with similar arguments used to show (4.76). The above right hand side vanishes when ε → 0. Thus [7, Lemma 3.1] implies ±1 f x−xh that the Jacobians fulﬁll Jvε − Ju(ε) * 0. Recalling that u = on Bρ(xh), it b b |x−xh| follows from Step 1 of the proof of [4, Theorem 5.1 (ii)] that 1 Ju(ε) *f deg(u)(x )δ . π b h xh 0,1 Fixing again ϕ ∈ Cc (Bρ(xh)) such that ϕ(xh) = 1, the above arguments imply 1 1 dh = hµ, ϕi = limhµu , ϕi = limh Jvε, ϕi = limh Ju(ε), ϕi = deg(u)(xh) ε→0 ε ε→0 π b ε→0 π b as claimed. Since the limit measure equals µ for all λ, we deduce from the L1(Ω)-lower semicontinuity of the Γ- lim sup 10, (4.127), and (4.125) that 1 ε Z Γ- lim sup Eε − 2πN| log ε| (u, µ) ≤ C η + |∇u|2,1 dx . ε→0 εθε θε Ω The claim then follows by the arbitrariness of 0 < η < η0 (recall that |µ|(Ω) = N ). In the next proof we make rigorous the steps described at the beginning of Section 4.4. Together with Propositions 4.10 and 4.11 we thereby prove Theorem 1.3.

Proposition 4.24 (Γ-lim sup inequality). Assume that ε θε ε| log ε|. Let µ = PN 1 h=1 dhδxh with |µ|(Ω) = M ∈ N and let u ∈ BV (Ω; S ). Then Z 1 ε (c) Γ- lim sup Eε − 2πM| log ε| (u, µ) ≤ |∇u|2,1 dx + |D u|2,1(Ω) + J (µ, u; Ω) . ε→0 εθε θε Ω 2 Proof. Let us ﬁx σ > 0. By the deﬁnition (4.23) of J there exists a T ∈ D2(Ω×R ), with 1 1 2 T ∈ cart(Ωµ×S ), ∂T |Ω×R = −µ× S , and uT = u such that Z Z J K (c) Φ(T~) d|T | ≤ |∇u|2,1 dx + |D u|2,1(Ω) + J (µ, u; Ω) + σ . (4.134)

2 Ω×R Ω

In the previous inequality we applied Lemma 4.6 with Ωµ in place of Ω (cf. Remark 4.7). Due to Lemma 4.17 we ﬁnd an open set Ωe ⊃⊃ Ω and a sequence of functions uk ∈ ∞ 1 1,1 1 1 2 2 2 C (Ωeµ; S ) ∩ W (Ω;e S ) such that uk → u in L (Ω; R ), |Guk |(Ω×R ) → |T |(Ω×R ),

10We stated our result without specifying the metric space. To state Theorem 1.3 in the framework of 1 2 Γ-convergence, one can do as follows: for u ∈ L (Ω; R ) and µ ∈ Mb(Ω) with kµkﬂat ∈ [M −1/2,M +1/2], 1 ε set Eε(u, µ) < +∞ if and only if u ∈ PCε(Sε) and µ = µu , where Eε(u, µ) = Eε(u) − 2πM| log ε| . e e εθε θε 1 The result in Theorem 1.3 corresponds to computing the Γ-limit of Eeε with respect to the L -convergence of u and the ﬂat convergence of µ. 66 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

and deg(uk)(xh) = dh for h = 1,...,N . From Reshetnyak’s Continuity Theorem we infer that Z Z Z ~ ~ |∇uk|2,1 dx = Φ(Guk ) d|Guk | ≤ Φ(T ) d|T | + σ , (4.135)

2 2 Ω Ω×R Ω×R for k large enough. In the ﬁrst equality we applied Lemma 4.6 & Remark 4.7 to uk in Ωµ . Applying Lemmata 4.18, 4.20, and 4.21 we reduce to the assumptions in Proposition 4.22. By the lower semicontinuity of the Γ-lim sup with respect to the strong L1 -convergence of u and the ﬂat convergence of µ, we conclude the proof.

5. Proofs in the regime θε ∼ ε| log ε| In this regime a lot of arguments can be transferred from Section 4, so we start immedi- ately with the proof of Theorem 1.5 and just point out the differences. Proof of Theorem 1.5. We first show the compactness statement. Hence consider a sequence 2 1 uε : εZ → Sε with ε2| log ε| Eε(uε) ≤ C . Then Proposition 2.4 implies that (up to a sub- f PN sequence) µuε * µ for some measure µ = h=1 dhδxh with xh ∈ Ω and dh ∈ Z. Since θε = ε| log ε| we can also apply Proposition 4.1 and deduce that (again up to subsequences) 1 1 uε → u in L (Ω) for some u ∈ BV (Ω; S ). 1 PN Next we proof the lower bound. Fix u ∈ BV (Ω; S ) and µ = h=1 dhδxh for some xh ∈ Ω and dh ∈ Z. Passing to a subsequence it suffices to consider a sequence uε ∈ PCε(Sε) such 1 2 f that uε → u in L (Ω; R ), µuε * µ, and 1 lim Eε(uε) = C < +∞ . ε→0 ε2| log ε| In order to obtain the claimed lower bound, we need to investigate the asymptotic behavior of the currents Guε . Due to the estimate (4.8), which does not depend on the scaling of θε , there exists a further subsequence (not relabeled) such that Guε *T for some 2 T ∈ D2(Ω × R ) with |T | < +∞. Moreover, the combination of Proposition 3.11 and 1 2 Lemma 3.12 yields that ∂T |Ω×R = −µ× S . We still need to show that uT = u and 1 T ∈ cart(Ω × ). Since we already knowJ thatK ∂T | 2 = 0, these two properties follow µ S Ωµ×R exactly as in the proof of Proposition 4.4. Now we can conclude the lower bound. Fix ρ A ⊂⊂ Ωµ and σ > 0. Then there exists ρ > 0 such that A ⊂⊂ Ωµ . From the Lemmata 4.2 and 4.3 we infer that for ε small enough N Z 1 X 1 ~ Eε(uε) ≥ Eε(uε; Bρ(xh)) + (1 − σ) Φ(Guε ) d|Guε | . εθε εθε 2 h=1 A×R Along the chosen subsequence we now insert (4.19) to bound the first term, while we use the lower semicontinuity of parametric integrals with respect to the mass bounded weak convergence of currents (cf. [32, 1.3.1, Theorem 1]) for the second one. Then

N 1 X Z lim inf Eε(uε) ≥ 2π |dh| + Φ(T~) d|T | , ε→0 εθε 2 h=1 A×R where we already used that σ > 0 was arbitrary. Letting A ↑ Ωµ we conclude from Lemma 4.6 & Remark 4.7 that 1 Z lim inf Eε(uε) ≥ 2π|µ|(Ω) + Φ(T~) d|T | ε→0 εθ 2 ε Ωµ×R Z (c) ≥ 2π|µ|(Ω) + |∇u|2,1 dx + |D u|2,1(Ω) + J (µ, u; Ω) , Ω which is exactly the claimed lower bound. The arguments for the upper bound are identical to the ones used to prove Theorem 1.3 except for the identiﬁcation of the ﬂat limit in the proof of the upper bound (cf. Step 4 in the proof of Proposition 4.22), since the conclusion (4.130) is not valid in the scaling regime FROM THE N -CLOCK MODEL TO THE XY MODEL 67

θε = ε| log ε|. Hence, taking the same sequence uε as for (4.128), we argue slightly diﬀerent via a diagonal argument. Since θε = ε| log ε|, the energy bound (4.128) reads Z 1 − + 1 1 lim sup 2 Eε(uε; Ω) ≤ C η + 2πN + dS (un , un )|νun |1dH . ε→0 ε | log ε| SN λn λn Jun ∩(Ω\ h=1 Q0 (xh)) (5.1) Note that the left hand side agrees with the unconstrained scaled XY -model. In particular, by Proposition 2.4 we deduce that there exists a measure µn ∈ Mb(Ω) of the form µn = PKn f k=1 dk,nδxk,n with dk,n ∈ Z \{0} such that, up to a subsequence, µuε * µn . Thus the (already proven) lower bound of Theorem 1.5 and (5.1) yield Z (c) |∇˚un|2,1 dx + |D ˚un|2,1(Ω) + J (µn,˚un; Ω) + 2π|µn|(Ω) Ω Z − + 1 1 ≤ C η + 2πN + dS (un , un )|νun |1dH . SN λn λn Jun ∩(Ω\ h=1 Q0 (xh))

Since the last term in the above estimate is controlled via (4.125), we deduce that |µn|(Ω) is equibounded in n. In particular, up to a subsequence, there exists a measure µ0 ∈ Mb(Ω) PK f that has the structure µ0 = k=1 dkδxk for some dk ∈ Z \{0} such that µn * µ0 (and weakly* in the sense of measures). We next want to use a lower semicontinuity property of the left hand side. However, due to the mixed term J (µ, u; Ω), this is not straightforward, so we slightly estimate the left hand side from below with a negligible error when λn → 0. Indeed, from a similar reasoning as for Remark 4.9 applied to the deﬁnition of J (µ, u; Ω) in (4.23) we infer that Z − + 1 1 J (µn,˚un; Ω) ≥ dS (˚un ,˚un )|ν˚un |1dH .

J˚un Inserting this lower bound in the previous estimate we obtain that Z Z (c) − + 1 1 |∇˚un|2,1 dx + |D ˚un|2,1(Ω) + dS (˚un ,˚un )|ν˚un |1dH + 2π|µn|(Ω)

Ω J˚un Z − + 1 1 ≤ C η + 2πN + dS (un , un )|νun |1dH . SN λn λn Jun ∩(Ω\ h=1 Q0 (xh)) 1 2 The left hand side is lower semicontinuous with respect to the L (Ω; R )-convergence of ˚un and the weak*-convergence of µn (we have not found the ﬁrst point in the literature, so we prove it Proposition 6.1 below). The limit of the right hand side is given by (4.125). Due to (4.127) and the fact that u ∈ W 1,1(Ω; S1) (recall that we are in the setting of Proposition 4.22) we conclude that Z Z |∇u|2,1 dx + 2π|µ0| ≤ C η + |∇u|2,1 dx + 2πN , Ω Ω which implies that |µ0| ≤ N (recall that η < η0 can be chosen arbitrary small, while the constant C is bounded uniformly). We ﬁnish the proof by showing that all measures µn have mass 1 in a uniform neighborhood of each of the points xh given by the target measure PN µ = h=1 deg(u)(xh)δxh . Indeed, the inclusion (4.124) implies that uε = vε on each Bη/16(xh), with vε given by (4.66). Due to (4.80) we have for ε small enough 1 θ E (v ; B (x )) ≤ Cη ε + 2π| log ε| ≤ C| log ε| . ε2 ε ε η h ε

This allows to apply [7, Proposotion 5.2], so that the flat convergence of µvε Bη/16(xh) is equivalent to the flat convergence of the (normalized) Jacobians of the piecewise affine interpolation of vε on Bη/16(xh). Denote this piecewise affine interpolation and the one 68 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF associated to the function u on Bη/16(xh) by vbε and ub(ε), respectively. As shown in Step 4 of the proof of Proposition 4.22 we have

1 kv − u(ε)k 2 k∇v k 2 + k∇u(ε)k 2 ≤ Cθ | log ε| 2 . bε b L (Bρ/2(xh)) bε L (Bρ/2(xh)) b L (Bρ/2(xh)) ε

Also in the scaling regime θε = ε| log ε| the right hand side vanishes when ε → 0. Hence [7, ±1 f x−xh Lemma 3.1] yields that Jvε − Ju(ε) * 0. Since u = on Bη/16(xh), Step 1 of the b b |x−xh| proof of [4, Theorem 5.1 (ii)] implies that 1 Ju(ε) *f deg(u)(x )δ . Choosing an arbitrary π b h xh 0,1 ϕ ∈ Cc (Bη/16(xh)), the above arguments imply 1 1 hµn Bη/16(xh), ϕi = limhµu , ϕi = limh Jvε, ϕi = limh Ju(ε), ϕi = deg(u)(xh)ϕ(xh). ε→0 ε ε→0 π b ε→0 π b

Letting n → +∞ in the above equality we infer that µ0 Bη/16(xh) = deg(u)(xh)δxh . Now consider the decomposition of µ0 into the mutually singular measures N N ! X [ µ0 = deg(u)(xh)δxh + µ0 Ω \ Bη/16(xh) . h=1 h=1 From mutual singularity we deduce that N N ! X [ N ≥ |µ | = | deg(u)(x )δ | + µ Ω \ B (x ) 0 h xh 0 η/16 h h=1 h=1 N !

[ ≥ N + µ0 Ω \ Bη/16(xh) . h=1 SN PN Hence µ0 Ω \ h=1 Bη/16(xh) = 0 and therefore µ0 = h=1 deg(u)(xh)δxh = µ as claimed. Finally, the estimate (5.1) yields that Z 1 − + 1 1 Γ- lim sup Eε(˚un, µn) ≤ 2πN + C η + dS (un , un ) dH . ε→0 εθε SN λn λn Jun ∩(Ω\ h=1 Q0 (xh)) Combining the lower semicontinuity of the Γ- lim sup 11 with respect to L1 -convergence in the u-variable and ﬂat convergence in the µ-variable with (4.125), (4.127), and the fact that f µn * µ, we infer that 1 Z Γ- lim sup Eε(u, µ) ≤ 2π|µ|(Ω) + C η + |∇u|2,1 dx , ε→0 εθε Ω where we used that |µ|(Ω) = N . The arbitrariness of η allows to conclude the proof by repeating the density arguments of Proposition 4.24 which are independent of the size of θε .

6. Proofs in the regime ε| log ε| θε 1

In the present scaling regime the discrete vorticity measures µuε for sequences with bounded energy are not necessarily compact. Hence we cannot use the parametric integral as a comparison, but we will work directly with the spin variable uε . First we need to show that our candidate for the Γ-limit is lower semicontinuous. To this end, we would like to apply the relaxation result [6] for functionals deﬁned on S1 -valued BV maps. Notice however that the representation of the relaxation in [6] is explicit only for isotropic functionals. For this reason it cannot be directly applied to prove the lower semicontinuity of our candidate for the Γ-limit, which features the anisotropic norm | · |2,1 instead. Hence we use a slicing argument since in the one dimensional setting the functionals are isotropic.

11 1 2 As in footnote 10 at p. 65 one can do as follows: for u ∈ L (Ω; R ) and µ ∈ Mb(Ω) set Eeε(u, µ) < +∞ 1 if and only if u ∈ PCε(Sε) and µ = µu , where Eε(u, µ) = Eε(u). The results in Theorem 1.5 correspond e εθε 1 to computing the Γ-limit of Eeε with respect to the L -convergence of u and the ﬂat convergence of µ. FROM THE N -CLOCK MODEL TO THE XY MODEL 69

Proposition 6.1. For every open set A ⊂ Ω deﬁne the functional E(·; A): L1(A; R2) → [0, +∞] by Z Z (c) + − 1 1 |∇w| dx + |D w| (A) + d 1 u , u |ν | dH , if u ∈ BV (A; ) ,  2,1 2,1 S u 1 S E(u; A) := A J ∩A  u +∞ , otherwise.

Then E(·; A) is lower semicontinuous with respect to strong convergence in L1(A; R2). Proof. For an open set I ⊂ R let E1d : L1(I; R2) → [0, +∞] be deﬁned by Z 0 (c) X + − 1 1  |w | dt + |D w|(I) + dS w (t), w (t) , if w ∈ BV (I; S ) , E1d(w; I) := I t∈Jw∩I +∞ , otherwise.

By [6, Theorem 3.1] (see also [6, Remark 4.3]), the functional E1d is the relaxation of Z 0 1,1 1 |w | dt , w ∈ W (I; S ) I with respect to the strong topology of L1(I; R2). In particular, it is lower semicontinuous. 1 2 Next ﬁx an open set A ⊂ Ω and vn, v ∈ L (A; R ) such that vn → v strongly in L1(A; R2). We want to prove that

E(v; A) ≤ lim inf E(vn; A) . (6.1) n→+∞ Without loss of generality, we assume that the right-hand side in (6.1) is ﬁnite and that the 1 ∗ lim inf is actually a limit. Since |Dvn|(A) ≤ E(vn; A) we obtain v ∈ BV (A; S ) and vn * v weakly* in BV (A; R2). Note further that Z Z (c) + − 1 1 E(vn; A) = |∇vn e1| dx + |D vn e1|(A) + dS (vn , vn )|νvn · e1| dH

A Jvn ∩A Z Z (6.2) (c) + − 1 1 + |∇vn e2| dx + |D vn e2|(A) + dS (vn , vn )|νvn · e2| dH .

A Jvn ∩A

Let us ﬁx a direction ξ ∈ S1 , which plays the role of one of the two coordinate direc- tions e1 , e2 . In the following we use the notation and the properties of slicing recalled in Subsection 2.1. We start by observing that the coarea formula (cf. [12, formula (272)] with + − 1 g = dS (vn , vn ), E = Jvn ∩ A and f the projection onto the orthogonal complement of ξ ) implies Z Z + − 1 X ξ + ξ − 1 1 dS (vn , vn )|νvn · ξ| dH = dS (vn)z (t), (vn)z (t) dz . ξ Jv ∩A Πξ t∈J ξ ∩Az n (vn)z Hence, by the equality above and Fatou’s Lemma, we deduce that Z Z (c) + − 1 1 lim inf |∇vn ξ| dx + |D vn ξ|(A) + d (vn , vn )|νvn · ξ| dH n→+∞ S

A Jvn ∩A Z Z ξ 0 (c) ξ ξ X ξ + ξ − = lim inf (vn)z dt + |D (vn)z|(Az) + d 1 (vn)z (t), (vn)z (t) dz n→+∞ S ξ Πξ ξ t∈J ξ ∩Az Az (vn)z Z Z ξ 0 (c) ξ ξ X ξ + ξ − ≥ lim inf (vn)z dt + |D (vn)z|(Az) + d 1 (vn)z (t), (vn)z (t) dz . n→+∞ S ξ Πξ ξ t∈J ξ ∩Az Az (vn)z (6.3) 70 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

1 2 Since vn → v strongly in L (A; R ), Fubini’s Theorem implies that there exists a subsequence (possibly depending on ξ and which we do not relabel) such that ξ ξ 1 ξ 2 1 ξ (vn)z → vz strongly in L (Az; R ) , for H -a.e. z ∈ Π . ξ ξ 1 1 ξ Moreover vz ∈ BV (Az; S ) for H -a.e. z ∈ Π . Thus from the one-dimensional lower semicontinuity result we infer that Z ξ 0 (c) ξ ξ X ξ + ξ − lim inf (vn)z dt + |D (vn)z|(Az) + d 1 (vn)z (t), (vn)z (t) n→+∞ S ξ ξ t∈J ξ ∩Az Az (vn)z 1d ξ ξ 1d ξ ξ = lim inf E (vn)z; Az ≥ E (vz ; Az) n→+∞ Z ξ0 (c) ξ ξ X ξ + ξ − = v dt + |D v |(A ) + d 1 (v ) (t), (v ) (t) z z z S z z ξ ξ t∈J ξ ∩Az Az vz for H1 -a.e. z ∈ Πξ . Integrating the inequality above with respect to z ∈ Πξ , again by the coarea formula, and by (6.3) we obtain that Z Z (c) + − 1 1 lim inf |∇vn ξ| dx + |D vn ξ|(A) + d (vn , vn )|νvn · ξ| dH n→+∞ S

A Jvn ∩A Z Z (c) + − 1 1 ≥ |∇v ξ| dx + |D v ξ|(A) + dS (v , v )|νv · ξ| dH .

A Jv ∩A

We conclude the proof of (6.1) by evaluating the last inequality for ξ = e1, e2 , by (6.2), and employing the superadditivity of the lim inf . Proof of Theorem 1.1. The compactness result in i) follows directly from Proposition 4.1 since it is only required that θε 1. 1 To prove ii), it suﬃces to consider u ∈ BV (Ω; S ) and a sequence uε ∈ PCε(Sε) such 1 2 that uε → u strongly in L (Ω; R ) and 1 lim inf Eε(uε) ≤ C. ε→0 εθε Let us ﬁx an open set A ⊂⊂ Ω and σ > 0. By Lemma 4.2 , for ε small enough we have that 1 1 X 1 Eε(uε) ≥ (1 − σ) ε dS uε(εi), uε(εj) εθε 2 hi,ji εi,εj∈Ω Z + − 1 1 ≥ (1 − σ) dS (uε , uε )|νuε |1 dH = (1 − σ)E(uε; A) .

Juε ∩A Hence Proposition 6.1 yields that 1 lim inf Eε(uε) ≥ (1 − σ)E(u; A) . ε→0 εθε We conclude the proof of the lower bound letting σ → 0 and A % Ω. In order to show the Γ-limsup inequality in iii), we start by extending u ∈ BV (Ω; S1) slightly outside Ω. Using a local reﬂection along the Lipschitz boundary ∂Ω (see the construction in (3.8)), we extend u (with a slight abuse of notation) to a function u ∈ BV (Ω;e S1), Ωe ⊃⊃ Ω, such that |Du|(∂Ω) = 0. Next we approximate the function u ∈ BV (Ω;e S1) with W 1,1 functions without increasing the energy. To do so, observe that by [6, Theorem 3.1], the L1(Ω;e R2)-relaxation of the functional Z 1,1 1 v ∈ W (Ω;e S ) 7→ |∇v|2,1 dx

Ωe FROM THE N -CLOCK MODEL TO THE XY MODEL 71

(extended to +∞ elsewhere in L1(Ω;e R2)) is given by Z Z 1 (c) + − 1 v ∈ BV (Ω;e S ) 7→ |∇v|2,1 dx + |D v|2,1(Ω)e + K(v , v , νv) dH

Ωe Jv ∩Ωe

(extended to +∞ elsewhere in L1(Ω;e R2)), with the surface energy K : S1×S1×S1 → R deﬁned by n Z o K(a, b, ν) := inf |∇ψ|2,1 dx : ψ ∈ P(a, b, ν) ,

Qν where Qν is the unit cube centered at the origin with two sides parallel to ν and P(a, b, ν) is 1,1 1 1 1 the collection of all ψ ∈ W (Qν ; S ) with ψ(x) = a if x·ν = − 2 , ψ(x) = b if x·ν = 2 , and ψ is periodic with period 1 in the direction orthogonal to ν . In particular, P(a, b, ν) contains the collection of functions with a one-dimensional proﬁle in the direction ν , i.e., functions 1,1 1 1,1 1 1 1 1 ψ ∈ W (Qν ; S ) such that there exists a curve γ ∈ W ((− 2 , 2 ); S ) with γ(− 2 ) = a, 1 0 γ( 2 ) = b satisfying ψ(x) = γ(x · ν). For such functions we have ∇ψ(x) = γ (x · ν) ⊗ ν and 0 0 therefore, since |γ (x · ν) ⊗ ν|2,1 = |γ (x · ν)| |ν|1 ,

1 Z Z Z2 0 0 K(a, b, ν) ≤ |∇ψ|2,1 dx = |ν|1 |γ (x · ν)| dx = |ν|1 |γ (t)| dt .

1 Qν Qν − 2 1,1 1 1 1 1 1 Taking the inﬁmum over all such curves γ ∈ W ((− 2 , 2 ); S ) with γ(− 2 ) = a, γ( 2 ) = b, we conclude that

1 K(a, b, ν) ≤ dS (a, b)|ν|1 . Let us ﬁx σ > 0. By the deﬁnition of relaxation and thanks to the property |Du|(∂Ω) = 0 1,1 1 1 2 there exists a sequence uk ∈ W (Ω;e S ) such that uk → u in L (Ω;e R ) and Z Z Z (c) − + 1 1 |∇uk|2,1 dx ≤ |∇u|2,1 dx + |D u|2,1(Ω) + dS (u , u )|νu|1 dH + σ . (6.4)

Ω Ω Ju∩Ω

1,1 1 for k large enough. We approximate uk ∈ W (Ω;e S ) applying a known density result for Sobolev functions with values in manifolds. By [15, Theorem 2] there exists a sequence j,k ∞ j,k 1 1,1 1 of finite sets V ⊂ Ωe and a sequence uj,k ∈ C (Ωe \ V ; S ) ∩ W (Ω;e S ) such that 1,1 1 uj,k → uk strongly in W (Ω;e S ) as j → +∞. In particular, Z Z |∇uj,k|2,1 dx ≤ |∇uk|2,1 dx + σ (6.5) Ω Ω for j large enough (depending on k ). Then we apply Lemma 4.18 to uj,k to split the singularities (note that the construction therein can still be performed if the function uj,k happens to have degree zero in some j,k singularities, see Remark 4.19). For 0 < τ 1 we find finite sets Vτ ⊂ Ωe and functions τ ∞ j,k 1 1,1 1 τ 1 2 τ uj,k ∈ C (Ωe \ Vτ ; S ) ∩ W (Ω;e S ) such that uj,k → uj,k in L (Ω;e R ) as τ → 0, uj,k j,k has degree ±1 around each point of Vτ , and Z Z τ |∇uj,k|2,1 dx ≤ |∇uj,k|2,1 dx + σ (6.6) Ω Ω for τ small enough. 2 −n We consider the lattice λnZ , λn = 2 . Applying Lemma 4.20 we can assume that j,k 2 Vτ ⊂ λnZ ∩ Ω.e However, we need to rule out points on ∂Ω. To this end, note that the 2 proof of Lemma 4.20 is constructive and starts by considering points of the lattice λnZ that j,k j,k approximate points in Vτ . Since any point x ∈ Vτ ∩ ∂Ω can be approximated by points 2 j,k in λnZ ∩ Ω, we can assume that, after the application of Lemma 4.20, Vτ ∩ ∂Ω = Ø. j,k 0 j,k 0 Since Vτ is a finite set, we can find Ω ⊃⊃ Ω such that Vτ ∩ (Ω \ Ω) = Ø. 72 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

τ ρ,τ We now modify uj,k close to the singularities. Lemma 4.21 gives a sequence uej,k and ±1 η = η (ρ) such that for every x ∈ V j,k we have uρ,τ (x) = x−x for x ∈ B (x), 0 0 τ ej,k |x−x| η0 uρ,τ (x) = uτ (x) for x ∈ B (x) \ B√ (x), and ej,k j,k ρ η0 Z Z ρ,τ τ |∇uej,k |2,1 dx ≤ |∇uj,k|2,1 dx + σ , (6.7) Ω Ω ρ,τ τ 1 2 for ρ small enough (depending on τ ). Moreover uej,k → uj,k strongly in L (Ω; R ) as ρ → 0. ρ,τ ∞ 0 j,k 1 We are ﬁnally in a position to apply Proposition 4.22 to uej,k ∈ C (Ω \ Vτ ; S ) ∩ W 1,1(Ω0; 1), keeping in mind Remark 4.23. Since | log ε| ε → 0, the estimates (6.4)–(6.7) S θε yield Z Z 1 ρ,τ (c) − + 1 Γ- lim sup E (u ) ≤ |∇u| dx + |D u| (Ω) + d 1 (u , u )|ν | dH + C σ . ε ej,k 2,1 2,1 S u 1 ε→0 εθε Ω Ju∩Ω We conclude thanks to the L1 -lower semicontinuity of the Γ- lim sup, letting ρ → 0, τ → 0, j → +∞, k → +∞, and eventually σ → 0.

7. Proofs in the regime θε ε We now come to the scaling regime which yields a discretization of S1 that is ﬁne enough to commit asymptotically no error compared to the XY -model up to the ﬁrst order devel- opment. Throughout this section we shall always assume that

θε ε . (7.1) 7.1. Renormalized and core energy. We recall that the renormalized energy correspond- PM ing to the conﬁguration of vortices µ = h=1 dhδxh is deﬁned by X X W(µ) = −2π dhdk log |xh − xk| − 2π dhR0(xh) , h6=k h PM where R0 is harmonic in Ω and R0(x) = − h=1 dh log |x − xh| for x ∈ ∂Ω. The renormalized energy can also be recast as (µ) = lim m(η, µ) − 2π|µ|(Ω)| log η| , (7.2) W η→0 e where n Z d o 2 x−xh h m(η, µ) := min |∇w| dx: w(x) = αh for x ∈ ∂Bη(xh) , |αh| = 1 . (7.3) e |x−xh| η Ωµ

To define the core energy, we introduce the discrete minimization problem in a ball Br n 1 o γ(ε, r) := min E (v; B ): v : ε 2 ∩ B → 1, v(x) = x for x ∈ ∂ B , (7.4) ε2 ε r Z r S |x| ε r where ∂εBr is the discrete boundary of Br , defined for a general open set by 2 ∂εA = {εi ∈ εZ ∩ A: dist(εi, ∂A) ≤ ε} . 2 Note that ∂εBr ⊂ εZ ∩ Br \ Br−ε . Then the core energy of a vortex is the number γ given in the following lemma. The result is analogous to [8, Theorem 4.1] with some differences: here we consider rε → 0 depending on ε and we use a different notion of discrete boundary of a set. The modifications in the proof are minor, but we give the details for the convenience of the reader.

Lemma 7.1. Let rε be a family of radii such that ε rε ≤ C . Then there exists

h ε i lim γ(ε, rε) − 2π log =: γ ∈ , (7.5) ε→0 rε R where γ is independent of the sequence rε . FROM THE N -CLOCK MODEL TO THE XY MODEL 73

Proof. We introduce the function

n 2 1 x o I(t) = min E1(v; B 1 ): v : ∩ B 1 → , v(x) = for x ∈ ∂1B 1 . t Z t S |x| t Let us show that t2 I(t1) ≤ I(t2) + 2π log + %t for 0 < t1 ≤ t2 , (7.6) t1 2 where %t2 is a generic sequence (which may change from line to line) satisfying %t2 → 0 as 2 1 x t2 → 0. To this end, let v2 : Z ∩ B 1 → S be such that v(x) = |x| for x ∈ ∂1B 1 and t2 t2 E1(v2; B 1 ) = I(t2). We extend v2 to B 1 setting t2 t1 ( 2 v2(i) , if i ∈ Z ∩ B 1 , t2 v1(i) := i 2 |i| , if i ∈ Z ∩ B 1 \ B 1 . t1 t2

To reduce notation, we deﬁne A(t1, t2) := B 1 \ B 1 . Next, note that if i ∈ B 1 and −2 t2 t1 t2 1 1 j∈ / B 1 with |i − j| = 1, then |i| ≥ t − 1 > t − 2. Hence t2 2 2

1 X i j 2 I(t1) ≤ E1(v1; B 1 ) ≤ E1(v2; B 1 ) + E1(v1; A(t1, t2)) = I(t2) + |i| − |j| . t1 t2 2 hi,ji i,j∈A(t1,t2)

To estimate the sum in A(t1, t2), we use the notation of the trimmed quadrants (cf. (4.71)) already used in Step 1 in the proof of Proposition 4.22. Summing over all pairs of signs s ∈ {(+, +), (−, +), (+, −), (−, −)}, a similar argument as for (4.72) and (4.74) leads to

d 1 e t1 2 1 X i j 2 1 X X i j 2 X k − ≤ − + C 2 |i| |j| 2 |i| |j| (k − 1)4 hi,ji s hi,ji k=b 1 −4c s t2 i,j∈A(t1,t2) i,j∈Q3∩A(t1,t2) Z 1 t2 ≤ dx + %t ≤ 2π log + %t , |x|2 2 t1 2

B 1 \B 1 −4 t1 t2

1 1 t2 where we also used that by the mean value theorem | log t − log( t − 4)| ≤ 4| 1−4t |. 2 2 2 This proves (7.6). As a consequence, the limit limt→0 I(t) − 2π| log t| =: γ exists. Since ε γ(ε, rε) = I , it only remains to show that γ 6= −∞. To this end, we show that the rε boundary conditions in the definition of γ(ε, 1) force concentration of the Jacobians, so that 2 1 we can use localized lower bounds. Let vε : εZ ∩ B1 → S be an admissible minimizer for 2 εi the problem defining γ(ε, 1) and extend it to εZ \ B1 via vε(εi) = |εi| . Then, using the boundary conditions imposed on vε and (4.83), we deduce that 2 1 1 1 X 2 εi εj Eε(vε; B3) ≤ Eε(vε; B1) + ε − ε2 ε2 2ε2 |εi| |εj| hi,ji εi,εj∈B3\B1/2 C X ≤ γ(ε, 1) + ε4 ≤ γ(ε, 1) + C. (7.7) ε2 hi,ji εi,εj∈B3\B1/2 Since we already proved that γ(ε, 1) − 2π| log ε| remains bounded when ε → 0, Proposi- f tion 2.4 implies that (up to a subsequence) µvε B3 * µ for some µ = d1δx1 with d1 ∈ Z and x1 ∈ B3 . We claim that µ 6= 0. Indeed, let us denote by vbε the piecewise affine interpolation of vε and let η : [0, 3] → R be the piecewise affine function such that η = 1 on [0, 1], 0,1 η = 0 on [2, 3] and η is affine on [1, 2]. Then define the Lipschitz function ϕ ∈ Cc (B3) via ϕ(x) = η(|x|). Using the flat convergence of µvε , which transfers to the scaled Jacobian −1 π Jvbε due to [7, Proposition 5.2], we infer that Z Z 1 1 (vε)1∂2(vε)2 − (vε)2∂2(vε)1 hµ, ϕi = lim Jvε ϕ dx = − lim · ∇ϕ dx , ε→0 b ε→0 −(v ) ∂ (v ) + (v ) ∂ (v ) π B3 2π B2\B1 ε 1 1 ε 2 ε 2 1 ε 1 74 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF where in the last equality we integrated by parts due to the fact that in dimension two the Jacobian can be written as a divergence (in two ways). Note that on B2 \ B1 the function vε agrees with the discrete version of x/|x|. Hence one can pass to the limit in 1 ε as vbε converges to x/|x| on B2 \ B1 weakly in H (B2 \ B1). Moreover, it holds that ∇ϕ(x) = −x/|x| a.e. in B2 \ B1 . An explicit computation shows that 1 Z hµ, ϕi = |x|−1 dx = 1 . (7.8) 2π B2\B1 1 Consequently µ B3 = δx1 . Let σ < 2 dist(x1, ∂B3). Then [8, Theorem 3.1(ii)] yields 1 1 σ lim inf Eε(vε; B3) − 2π| log ε| ≥ lim inf Eε(vε; Bσ(x1)) − 2π log + 2π log σ ε→0 ε2 ε→0 ε2 ε ≥ −C + 2π log σ for some constant C . Combining this lower bound with (7.7) yields that −C + 2π log σ ≤ lim (γ(ε, 1) − 2π| log ε|) + C. ε→0 This shows that γ > −∞ and concludes the proof.

Below we will use a shifted version of γ(ε, rε). More precisely, given x0 ∈ Ω, set n 1 o 2 1 x−x0 γx0 (ε, r) := min Eε(v; Br(x0)) : v : εZ ∩Br(x0) → S , v(x) = for x ∈ ∂εBr(x0) . ε2 |x−x0|

As one would expect the asymptotic behavior does not depend on x0 as shown in the next lemma.

Lemma 7.2. Let γ ∈ R be given by Lemma 7.1 and let ε rε ≤ C . Then it holds that

ε lim γx0 (ε, rε) − 2π log = γ . ε→0 rε 2 Proof. Consider a point xε ∈ εZ such that |x0 − xε| ≤ 2ε. Then, given a minimizer 2 1 v : εZ ∩ Brε → S for the problem defining γ(ε, rε) (extended via the boundary conditions on ε 2 \ B ) we define v(εi) = v(εi − x ). This function is admissible in the definition of Z rε e ε γx0 (ε, rε + 4ε). Hence 2 ε ε 1 X 2 εi εj γx0 (ε, rε + 4ε) − 2π log ≤ γ(ε, rε) − 2π log + ε − . rε+4ε rε 2ε2 |εi| |εj| hi,ji εi,εj∈Brε+6ε εi,εj∈ /Brε−ε The last sum can be bounded applying (4.83) which leads to

2 2 1 X 2 εi εj C X 2 C rεε + ε ε − ≤ ε ≤ |Brε+8ε \ Brε−3ε| ≤ C , 2ε2 |εi| |εj| r2 r2 r2 ε 2 ε ε hi,ji εiεZ εi,εj∈Brε+6ε εi∈Brε+6ε εi,εj∈ /Brε−ε εi/∈Brε−ε which vanishes due to the assumption that ε rε . Thus we proved that

ε lim sup γx0 (ε, rε) − 2π log r ≤ γ . ε→0 ε The reverse inequality for the lim inf can be proven by a similar argument. 7.2. Compactness and Γ-convergence. We recall the compactness result and the Γ- liminf inequality obtained in [8, Theorem 4.2]. We emphasize that these results also hold in 2 1 our setting, regarding uε ∈ PCε(Sε) as a spin ﬁeld uε : εZ → S , that means, neglecting the Sε constraint. Theorem 7.3 (Theorem 4.2 in [8]). The following results hold: FROM THE N -CLOCK MODEL TO THE XY MODEL 75

2 1 1 i) (Compactness) Let M ∈ N and let uε : εZ ∩ Ω → S be such that ε2 Eε(uε) − 2πM| log ε| ≤ C . Then there exists a subsequence (which we do not relabel) such that f PN µuε * µ for some µ = h=1 dhδxh with |µ|(Ω) ≤ M . Moreover, if |µ|(Ω) = M , then |dh| = 1. 2 1 f ii)( Γ-liminf inequality) Let uε : εZ ∩ Ω → S be such that µuε * µ with µ = PN h=1 dhδxh , |dh| = 1. Then h 1 i lim inf Eε(uε) − 2πM| log ε| ≥ W(µ) + Mγ . ε→0 ε2 For the construction of the recovery sequence our arguments slightly diﬀer from the proof of [8, Theorem 4.2]. For the reader’s convenience we give here the detailed proof, which together with Theorem 7.3 establishes Theorem 1.4. PM Proposition 7.4 (Γ-limsup inequality). Let µ = h=1 dhδxh with |dh| = 1. Then there 2 f exists a sequence uε : εZ ∩ Ω → Sε with µuε * µ such that h 1 i lim sup 2 Eε(uε) − 2πM| log ε| ≤ W(µ) + Mγ . (7.9) ε→0 ε

Proof. To avoid confusion among inﬁnitesimal sequences, in this proof we denote by %ε a sequence, which may change from line to line, such that %ε → 0 when ε → 0. Step 1 (Construction of the recovery sequence) 0 Let us ﬁx 0 < η < η < 1 with η small enough such that the balls Bη(xh) are pairwise disjoint and their union is contained in Ω. We denote by wη a solution to the minimum η η η problem (7.3) in Ωµ . For every h = 1,...,M there exists αh ∈ C with |αh| = 1 such that d 0 d η η x−xh h η η /2 η η x−xh h w (x) = α for x ∈ ∂Bη(xh). Extend w to Ωµ by w (x) := α h |x−xh| h |x−xh| η for x ∈ Bη(xh) \ Bη0/2(xh). To reduce notation, we set Aη0 (xh) := Bη(xh) \ Bη0 (xh). The η 1,2 η0 1 extension w then belongs to W (Ωµ ; S ) and its Dirichlet energy is given by M Z Z X Z 1 |∇wη|2 dx = |∇wη|2 dx + dx = m(η, µ) + 2πM log η . (7.10) 2 e η0 |x − xh| h=1 η0 Ωη Aη (x ) Ωµ µ η0 h

− 1 2 1 Set r = | log ε| 2 ε and let u : ε ∩ B (x ) → be a function that agrees with ε eε Z rε h S d η x−xh h α on ∂εBr (xh) and such that, cf. Lemmata 7.1 and 7.2, h |x−xh| ε 1 E (u ; B (x )) = γ (ε, r ) = 2π log rε + γ + % ≤ 2π| log ε| + γ + % . (7.11) ε2 ε eε rε h xh ε ε ε ε 2 We now extend ueε to εZ ∩ Ω distinguishing two cases: we set d η εi−xh h uε(εi) := α if εi ∈ Bη0 (xh) \ Br (xh) ; (7.12) e h |εi−xh| ε 2 η0 η on εZ ∩ Ωµ the definition is more involved since w has only Sobolev regularity up the (Lipschitz)-boundary and we are not aware of any density results preserving the traces on 1 η η part of the boundary and the S -constraint. First we need to extend w to Ωeµ for some open set Ωe ⊃ Ω with Lipschitz boundary. This can be achieved via a local reflection as in (3.8), so η 1,2 η0 1 2 that we may assume from now on that w ∈ W (Ωeµ ; S ). We further extend it to R with 0 1 η /2 compact support (neglecting the S constraint outside Ωeµ ). Now let us define the discrete η x 2 approximation of this extended w . Consider the shifted lattice Zε = x + εZ with x ∈ Bε η 1,2 2 2 and denote by wbε,x ∈ W (R ; R ) the piecewise affine interpolation of (the quasicontinuous η x representative of) w on a standard triangulation associated to Zε . As shown in the proof of [50, Theorem 1] there exists x ∈ B such that wη → wη strongly in W 1,2( 2; 2) ε ε bε,xε R R (the proof is given in the scalar-case, but the argument also works component-wise; see also 2 η0 1 [50, Section 3.1]). Thus it is natural to define ueε : εZ ∩ Ωµ → S by u (εi) = wη (εi + x ) . (7.13) eε ε,xε ε 76 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

0 η η /2 1 Observe that since w is deﬁned on Ωeµ with values in S , for ε small enough ueε is indeed S1 -valued. Moreover, the strong convergence of the aﬃne interpolations ensures that

2 Z Z 1 X 2 ueε(εi) − ueε(εj) X η 2 η ε ≤ |∇w | dx ≤ |∇w |dx + %ε . (7.14) 2 ε bε,xε hi,ji T triangle η0 0 εT +xε η0 η Ωµ εi,εj∈Ωµ εT ∩Ωµ 6=Ø

2 Finally, we deﬁne the global sequence uε := Pε(ueε): εZ ∩ Ω → Sε . Note that the piecewise 0 constant maps uε and ueε actually depend on η and η . For the following computations however we drop the dependence on these parameters to simplify notation. We start estimating the error in energy due to the projection Pε . By (4.68) we have that

2 1 1 X 2 1 θε X Eε(uε) = uε(εi)−uε(εj) ≤ Eε(uε)+C|Ω| +2 θε uε(εi)−uε(εj) . (7.15) ε2 2 ε2 e ε2 e e hi,ji hi,ji

1 We shall prove that ε2 Eε(ueε) carries the whole energy, that means, h 1 i lim sup 2 Eε(uε) − 2πM| log ε| ≤ W(µ) + Mγ , (7.16) ε→0 ε e whereas the remainder satisﬁes

X lim θε uε(εi) − uε(εj) = 0 . (7.17) ε→0 e e hi,ji

Inequalities (7.15), (7.16), and (7.17) then yield (7.9) thanks to the assumption (7.1). In order to prove both (7.16) and (7.17), we estimate separately the contribution of the η0 0 energy and that of the remainder in the regions Brε (xh),Ωµ , and (Bη +ε(xh)\Brε−ε(xh)). We remark that this decomposition of εZ2 ∩ Ω takes into account all the nearest-neighbors interactions.

Step 2 (Estimates close to the singularities)

Let us start with the estimates inside Brε (xh). Notice that (7.11) already gives explicitly the value of 1 E (u ; B ), so we only have to estimate the remainder in B (x ). ε2 ε eε rε rε h Combining the Cauchy-Schwarz inequality with (7.11), (7.1), and taking into account that − 1 rε = | log ε| 2 , we obtain that

1 1 2 2 X X 2 X 2 θε ueε(εi) − ueε(εj) ≤ θε ueε(εi) − ueε(εj) hi,ji hi,ji hi,ji εi,εj∈Brε (xh) εi,εj∈Brε (xh) εi,εj∈Brε (xh) 1 (7.18) θ 1 2 ≤ Cr ε E (u ; B (x )) ε ε ε2 ε eε rε h θ 1 ≤ Cr ε 2π| log ε| + γ + % 2 → 0 as ε → 0 . ε ε ε

Step 3 (Estimates in the perforated domain) η0 We go on with the estimates inside Ωµ . In this set the function ueε is given by (7.13). In particular, by (7.10) and (7.14),

1 0 Z E (u ;Ωη ) ≤ |∇wη|2 dx + % = m(η, µ) + 2πM log η + % . (7.19) ε2 ε eε µ ε e η0 ε η0 Ωµ FROM THE N -CLOCK MODEL TO THE XY MODEL 77

Concerning the remainder, the Cauchy-Schwarz inequality, (7.19), and (7.1) imply that

1 1 2 2 X X 2 X 2 θε ueε(εi) − ueε(εj) ≤ θε ueε(εi) − ueε(εj) hi,ji hi,ji hi,ji η0 η0 η0 εi,εj∈Ωµ εi,εj∈Ωµ εi,εj∈Ωµ 1 1 θε 1 η0 2 ≤ C|Ω| 2 E (u ;Ω ) ε ε2 ε eε µ 1 1 θε η 2 ≤ C|Ω| 2 m(η, µ) + 2πM log + % → 0 as ε → 0 . ε e η0 ε (7.20)

Step 4 (Estimates in the annulus) Finally, we resort to an argument analogous to the one presented in Step 1 in the proof of 1 0 Proposition 4.22 to estimate the energy ε2 Eε in the set (Bη +ε(xh) \ Brε−ε(xh)), where d η x−xh h uε(x) = α (or slightly shifted in Bη0+ε(xh) \ Bη0 (xh) due to (7.13), for which e h |x−xh| d η η x−xh h we recall that w = α in Bη(xh) \ B 0 (xh)). To simplify notation, for h |x−xh| η /2 R R > r > 0 we denote in this step Ar (x) := BR(x) \ Br(x). We ﬁrst estimate the contribution involving the shifted function. For any εi, εj with η0+ε |i − j| = 1 and εi ∈ Aη0 (xh) the condition |xε| ≤ ε and (4.83) imply that 4ε |u (εi) − u (εj)| ≤ . eε eε η0 − ε Summing this estimate we deduce that

1 X (η0 + 3ε)2 − (η0 − 2ε)2 εη0 + ε2 ε2|u (εi) − u (εj)|2 ≤ C ≤ C → 0 as ε → 0 . ε2 eε eε (η0 − ε)2 (η0 − ε)2 hi,ji 0 εi∈Aη +ε(x ) η0 h Hence we can write

0 1 η0+ε 1 η Eε(uε; A (xh)) ≤ Eε(uε; A (xh)) + %ε . (7.21) ε2 e rε−ε ε2 e rε−ε

η0 d 2 η x−xh h Note that on the set ε ∩ A (xh) the function uε coincides with x 7→ α , Z rε−ε e h |x−xh| so that the invariance of the discrete energy under orthogonal transformations implies that

0 2 1 η 1 X εi − xh εj − xh E (u ; A (x )) = ε2 − . 2 ε eε rε−ε h 2 ε ε |εi − xh| |εj − xh| hi,ji 0 εi,εj∈Aη (x ) rε−ε h Using a shifted version of the trimmed quadrants deﬁned in (4.71) and summing over all possible pairs of signs s ∈ {(+, +), (−, +), (+, −), (−, −)}, we can split the energy as

∞ 0 0 1 η X 1 s η X 1 Eε(uε; A (xh)) ≤ Eε(uε;(εQ + xh) ∩ A (xh)) + C , ε2 e rε−ε ε2 e 3 rε−ε k2 (7.22) s rε k= ε −2 where we used the bound (4.83) to estimate the contributions not fully contained in one of P+∞ −2 rε the trimmed quadrants by the last sum. Since the sum k=1 k is ﬁnite and ε → +∞, the second term in the right-hand side is inﬁnitesimal as ε → 0. On each trimmed quadrant we use a shifted version of (4.70) and a monotonicity argument as in (4.74) to deduce that

1 η0 X 1 Z 1 E (u ;(εQs + x ) ∩ A (x )) ≤ ε2 ≤ dx , 2 ε eε 3 h rε−ε h 2 2 ε |εi − xh| |x| η0 η0 εi∈A (xh) s rε−ε εQ2∩Ar −3ε 2 s ε εi∈εZ ∩(Q3+xh) 78 MARCO CICALESE, GIANLUCA ORLANDO, AND MATTHIAS RUF

η0 with the annulus A = B 0 \ B centered at 0. Summing over all four quadrants, rε−3ε η rε−3ε since ε rε , we get that 0 Z X 1 s η 1 η0 η0 Eε(uε;(εQ3 + xh) ∩ Ar −ε(xh)) ≤ dx = 2π log = 2π log + %ε . ε2 e ε |x|2 rε−3ε rε s 0 Aη rε−3ε In combination with (7.21) and (7.22) we conclude that

1 η0+ε η0 Eε(uε; Ar −ε(xh)) ≤ 2π log + %ε . (7.23) ε2 e ε rε 0 We now estimate the remainder term in Aη +ε (x ), for which applying the Cauchy-Schwarz rε−ε h inequality as in (7.18) is too rough. However, note that for any i ∈ Z2 and x ∈ εi + [0, ε)2 with |εi − xh| ε, we have for ε small enough that √ 1 |εi − x | ≥ |x − x | − 2ε ≥ |x − x | . h h 2 h Hence, using (4.83) and a change of variables we obtain that Z X X ε θε 1 θε ueε(εi) − ueε(εj) ≤ C θε ≤ C dx . (7.24) |εi − xh| ε |x| η0+ε hi,ji εi∈ε 2∩A (x ) η0+3ε 0 Z rε−ε h A εi,εj∈Aη +ε (x ) rε−3ε rε−ε h Since the last integral is proportional to η0 , inserting assumption (7.1) shows that the right- hand side of (7.24) is inﬁnitesimal as ε → 0. Step 5 (Proof of (7.16) and (7.17) and conclusion) To prove (7.16), we employ (7.11), (7.19), and (7.23) to split the energy as follows

1 1 0 E (u ) − 2πM| log ε| ≤ E (u ;Ωη ) − 2πM log η + 2πM log η ε2 ε eε ε2 ε eε µ η0 M X h 1 i + E (u ; B (x )) − 2π log rε ε2 ε eε rε h ε h=1 M X h 1 η0+ε η0 i + Eε(uε; Ar −ε(xh)) − 2π log ε2 e ε rε h=1 ≤ me (η, µ) + 2πM log η + Mγ + %ε . 0 Now we stress again the dependence of ueε on η and η , denoting the sequence by ueε,η (set for instance η0 = η/2). Letting ε → 0, for η < 1 we deduce that h 1 i lim sup 2 Eε(uε,η) − 2πM| log ε| ≤ m(η, µ) − 2πM| log η| + Mγ . ε→0 ε e e Moreover, (7.17) follows from (7.18), (7.20), and (7.24) splitting the remainder in the same way. Hence, for each η < 1 we found a sequence uε,η ∈ PCε(Sε) such that h 1 i lim sup 2 Eε(uε,η) − 2πM| log ε| ≤ m(η, µ) − 2πM| log η| + Mγ . (7.25) ε→0 ε e Before we conclude via a diagonal argument, we have to identify the flat limit of the vorticity measure µuε,η . From the above energy estimate and Proposition 2.4 we deduce that, passing f M η P η to a subsequence, µuε,η * µη for some µη = k=1 dkδx with |µη|(Ω) ≤ M (we allow for η k dk ∈ Z to sum up to M ). Moreover, due the logarithmic energy bound we can apply [7, Proposition 5.2] and deduce that π−1Ju − µ *f 0, where u denotes the piecewise bε,η uε,η bε,η affine interpolation associated to uε,η . Let now vbε,η be the function defined via piecewise 2 affine interpolation of the values ueε,η(εi), i ∈ Z . As a consequence of [7, Lemma 3.1], we have that f J(ubε,η) − J(vbε,η) * 0 . (7.26) Indeed, because of (7.25) and (7.16) this can be justified by the same reasoning as in (4.133). Hence it suffices to study the limit of the Jacobians of vbε,η . We show that the limit carries FROM THE N -CLOCK MODEL TO THE XY MODEL 79

mass in each ball Bρ(xh) for all ρ > 0. To this end, we test the ﬂat convergence against the Lipschitz function h 2 ϕρ (x) := min{max{ ρ (ρ − |x − xh|), 0}, 1} . Using the distributional divergence form of the Jacobian and the fact that vbε,η agrees with d the piecewise aﬃne interpolation of the map x 7→ αη x−xh h on the support of the h |x−xh| h gradient of ϕρ provided ρ < η/4 and rε ρ/2, we infer that h −1 h hµη, ϕρ i = limhπ Jvε,η, ϕρ i ε→0 b Z 1 (vbε,η)1∂2(vbε,η)2 − (vbε,η)2∂2(vbε,η)1 h = − lim ∇ϕρ dx = dh , ε→0 2π −(vε,η)1∂1(vε,η)2 + (vε,η)2∂1(vε,η)1 ρ b b b b Aρ/2(xh) where the limit can be calculated similar to (7.8). From this equality and the arbitrariness of

ρ > 0, we deduce that {x1, . . . , xN } ⊂ supp(µη) and µ {xh} = dhδxh . Since |µη|(Ω) ≤ M , it follows that µη = µ independent of η and the subsequence of ε. Since the flat convergence is given by a metric, we can thus use a diagonal argument with η = ηε to find a sequence f uε := uε,ηε satisfying µuε * µ and, due to (7.25), also the claimed inequality (7.9). Acknowledgments. The work of M. Cicalese was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”. G. Orlando has been supported by the Alexander von Humboldt Foundation. M. Ruf acknowledges financial support from the European Research Council under the European Community’s Seventh Framework Program (FP7/2014-2019 Grant Agreement QUANTHOM 335410).

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(Marco Cicalese) Technische Universitat¨ Munchen,¨ Munich, Germany Email address: [email protected]

(Gianluca Orlando) Technische Universitat¨ Munchen,¨ Munich, Germany Email address: [email protected]

(Matthias Ruf) Universite´ libre de Bruxelles, Brussels, Belgium Email address: [email protected]