Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model

Abstract

We study the asymptotic behavior of the N -clock model, a nearest neighbors ferromagnetic spin model on the d-dimensional cubic ε-lattice in which the spin field is constrained to take values in a discretization S N of the unit circle S 1 consisting of N equispaced points. Our Γ-convergence analysis consists of two steps: we first fix N and let the lattice spacing ε → 0, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in S N ; at a second stage, we let N → +∞. The final result of this two-step limit process is an anisotropic total variation of S 1 -valued vector fields of bounded variation.

The antiferromagnetic XY model on the triangular lattice: topological singularities

“…More precisely, it Γ-converges to an anisotropic perimeter of the phase boundary. At this energy scaling, the asymptotic behavior of the AFXY model shares similarities with systems having finitely many phases, such as Ising systems [17,2,16] or Potts systems [20].…”

The $N$-clock model: Variational analysis for fast and slow divergence rates of $N$

“…In this table we summarize our results. By "interfaces" we mean that the energy concentrates on 1 -dimensional domain walls that separate the different phases [26], while " BV " denotes a BV -type total variation, Theorem 1.1. The expression " BV +concentration" indicates the presence in a BV -type energy of a surface term of the form J (µ, u; Ω) which accounts for concentration effects on 1-dimensional surfaces, as in [25] and Theorem 1.2.…”

“…The results in this paper and in [25,26] concern a related problem regarding the behavior of low-energy states of the two systems in the discrete-to-continuum variational analysis as the lattice spacing vanishes and N diverges simultaneously. With the help of fine concepts in geometric measure theory and in the theory of cartesian currents, these results show to which extent the coarse-grain limit of the N -clock model resembles the one of the XY model obtained in [4,7].…”

“…→ 0 as ε → 0 . (7 26). Step 4 (Estimates in the annulus) Finally, we need to estimate the energy1 ε 2 E ε in the set (B η +ε (x h ) \ B rε−ε (x h )) , where u ε (x) = α η h x−x h |x−x h | d h (or slightly shifted in B η +ε (x h ) \ B η (x h ) due to(7.18), for which…”