论文标题
$ d $ d $ d $ n $ n $ clock型号的粗晶状和大$ n $行为
Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model
论文作者
论文摘要
我们研究了$ n $ -Clock模型的渐近行为，这是最近的邻居铁磁旋转模型上的$ d $二维立方$ \ VAREPSILON $ -LATTICE，其中旋转字段被限制为以iPtization $ \ natialcal $ \ Mathcal {s} _n $ $ \ $ \ $ \ n $ n $ n $ n} $ n} $ n} $ h.稳定点。我们的$γ$ -Convergence分析包括两个步骤：我们首先修复$ n $，然后让晶格间距$ \ varepsilon \ to 0 $，在连续的continuum中获得的界面能量在分段恒定的旋转字段中定义为$ \ Mathcal {s} _n $;在第二阶段，我们让$ n \ to +\ infty $。这个两步限制过程的最终结果是各向异性的总变化为$ \ mathbb {s}^1 $ - 价态变化的矢量字段。
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 $Γ$-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 piecewise constant spin fields with values in $\mathcal{S}_N$; at a second stage, we let $N \to +\infty$. The final result of this two-step limit process is an anisotropic total variation of $\mathbb{S}^1$-valued vector fields of bounded variation.
