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

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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. Contents 1. Introduction 1 2. Notation and preliminaries 12 3. Currents 15 4. Proofs in the regime " θ" "j log "j 29 5. Proofs in the regime θ" ∼ "j log "j 66 6. Proofs in the regime "j log "j θ" 1 68 7. Proofs in the regime θ" " 72 References 79 1. Introduction Classical ferromagnetic spin systems on lattices represent fundamental models to under- stand 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 Fr¨ohlich 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 " > 0, we consider the 2 2 square lattice "Z and we define Ω" := Ω \ "Z . The classical XY model is defined on spin 1 fields u:Ω" ! S by X − "2u("i) · u("j) ; (1.1) hi;ji where a · b denotes the scalar product between a; b 2 R2 and the sum is taken over ordered 2 2 pairs of nearest neighbor hi; ji, i.e., (i; j) 2 Z ×Z such that ji − jj = 1 and "i; "j 2 Ω" . 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 N" 2 or, equivalently, θ" := . The N N" 1 integer N" represents the number of points in the discretization S" of the codomain S , while θ" is the smallest non-zero angle between two neighboring spin values. More precisely, we set S" := fexp(ιkθ"): k = 0;:::;N" − 1g ; where ι is the imaginary unit. The admissible spin fields we consider here are only those 1 taking values in the discrete set S" , i.e., we consider the energy defined for every u:Ω" ! S by 8 X 2 <>− " u("i) · u("j) if u:Ω" !S" ; F"(u) := hi;ji :>+1 otherwise: For N" = N 2 N, with N independent of ", the spin system described by the energy F" 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) = − "2 u("i) · u("j) − 1 = "2ju("i) − u("j)j2: " 2 hi;ji hi;ji This suggests to study the excess energy E"(u) := F"(u) − min F" , under those scalings 1 κ" ! 0 for which E" has a nontrivial variational limit in the sense of Γ-convergence, κ" cf. [28, 18]. In order to understand such scalings, it is convenient to recast the energy as ( XY"(u) if u:Ω" !S" ; E"(u) := (1.2) +1 otherwise: FROM THE N -CLOCK MODEL TO THE XY MODEL 3 One can expect that the relevant scaling κ" is affected by N" , as it emerges in the 1 two limiting scenarios N" = 2 and S" = S (formally corresponding to N" = +1). In fact, for the Ising system, i.e., N" = 2, it has been shown in [3] that, choosing κ" = ", a 1 configuration u" with equibounded energies " E"(u") can be identified, as " ! 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 " E"(u") approximates an interfacial-type energy, whose anisotropy reflects the symmetries of the underlying lattice. In contrast, for 1 the XY system, i.e., S" = S , it has been shown in [4, Example 1] that no interfacial-type 2 energy emerges at any scaling κ" " . 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) 2 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 u" ! u in L (Ω; ) such that XY"(u") ! 0 for all κ" " in S κ" the following way: for "i = ("i1; "i2) 2 Ω" we define 8 >v1 if "i1 ≤ 0 ; <> "i1 u"("i) := exp ι ('1 − '2) 1 − + '2 if 0 < "i1 ≤ η" ; (1.4) η" > :v2 if "i1 > η" ; where η" ! 0 is chosen below. Then 2 1 " η" " XY"(u") ∼ 1 − cos ('1 − '2) ∼ : (1.5) κ" η" κ" η"κ" 2 "2 Thanks to the assumption κ" " , it is always possible to find η" ! 0 such that η" . κ" As a consequence, the angle between two neighboring spins u"("(i + e1)) and u"("i) has " κ" 1 modulus j'1 − '2j . This construction, always possible in the case S" = , may η" " S 1 κ" not be feasible when S" 6= S if the minimal angle θ" satisfies θ" & " . Hence one cannot exclude the possibility of finding a nontrivial scaling κ" such that interfacial-type energies 1 arise in the limit of E"(u"). κ" To better understand the behavior of E" in the constrained setting u" :Ω" !S" , we " modify the construction in (1.4) by choosing η" so that j'1 − '2j = θ" , i.e., η" = η" " j'1 − '2j , see Figure 1. As a result θ" 1 " εθ" E"(u") ∼ 1 − cos(θ") j'1 − '2j ∼ j'1 − '2j : κ" θ"κ" κ" This suggests that the nontrivial scaling κ" = εθ" leads to a finite energy proportional to j'1 − '2j.
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