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Publication# 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 epsilon-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 Gamma-convergence analysis consists of two steps: we first fix N and let the lattice spacing epsilon -> 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 -> +infinity. The final result of this two-step limit process is an anisotropic total variation of S-1-valued vector fields of bounded variation.

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Related concepts (4)

Total variation

In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b]. Functions whose total variation is finite are called functions of bounded variation.

Z N model

The model (also known as the clock model) is a simplified statistical mechanical spin model. It is a generalization of the Ising model. Although it can be defined on an arbitrary graph, it is integrable only on one and two-dimensional lattices, in several special cases. The model is defined by assigning a spin value at each node on a graph, with the spins taking values , where . The spins therefore take values in the form of complex roots of unity.

Reciprocal lattice

In physics, the reciprocal lattice represents the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system (usually a Bravais lattice). The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, where refers to the wavevector. In quantum physics, reciprocal space is closely related to momentum space according to the proportionality , where is the momentum vector and is the reduced Planck constant.