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

@article{Cicalese2021CoarseGA,
  title={Coarse graining and large-\$N\$ behavior of the \$d\$-dimensional \$N\$-clock model},
  author={Marco Cicalese and Gianluca Orlando and Matthias Ruf},
  journal={Interfaces and Free Boundaries},
  year={2021}
}
We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $\varepsilon$-lattice in which the spin field is constrained to take values in a discretization $\mathcal{S}_N$ of the unit circle~$\mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $\Gamma$-convergence analysis consists of two steps: we first fix $N$ and let the lattice spacing $\varepsilon \to 0$, obtaining an interface energy in the continuum defined on… 

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References

SHOWING 1-10 OF 34 REFERENCES
The $N$-clock model: Variational analysis for fast and slow divergence rates of $N$
We study a nearest neighbors ferromagnetic spin system on the square lattice in which the spin field is constrained to take values in a discretization of the unit circle consisting of $N$ equi-spaced
Emergence of concentration effects in the variational analysis of the $N$-clock model
We investigate the relationship between the $N$-clock model (also known as planar Potts model or $\mathbb{Z}_N$-model) and the $XY$ model (at zero temperature) through a $\Gamma$-convergence analysis
Continuum limit and stochastic homogenization of discrete ferromagnetic thin films
We study the discrete-to-continuum limit of ferromagnetic spin systems when the lattice spacing tends to zero. We assume that the atoms are part of a (maybe) non-periodic lattice close to a flat set
Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach
This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar
Variational Analysis of a Two-Dimensional Frustrated Spin System: Emergence and Rigidity of Chirality Transitions
TLDR
Carrying out the $\Gamma$-convergence analysis of proper scalings of the energy, it is proved the emergence and characterize the geometric rigidity of the chirality phase transitions.
Domain Formation in Magnetic Polymer Composites: An Approach Via Stochastic Homogenization
We study the magnetic energy of magnetic polymer composite materials as the average distance between magnetic particles vanishes. We model the position of these particles in the polymeric matrix as a
Destruction of long range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems
The low-temperature state of two-dimensional classical systems, which in the three-dimensional case have an ordered phase with a spontaneous violation of a continuous symmetry (magnetic substances,
Vortices in the Magnetic Ginzburg-Landau Model
With the discovery of type-II superconductivity by Abrikosov, the prediction of vortex lattices, and their experimental observation, quantized vortices have become a central object of study in
Continuous spin mean-field models : Limiting kernels and Gibbs properties of local transforms
We extend the notion of Gibbsianness for mean-field systems to the setup of general (possibly continuous) local state spaces. We investigate the Gibbs properties of systems arising from an initial
Variational problems with percolation: dilute spin systems at zero temperature
We study the asymptotic behaviour of dilute spin lattice energies by exhibiting a continuous interfacial limit energy computed using the notion of Γ-convergence and techniques mixing Geometric
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