具有竞争相互作用的自旋模型中的棋盘、条纹和角能
Checkerboards, stripes and corner energies in spin models with competing interactions
摘要：
我们研究了存在竞争相互作用、长程反铁磁和强度为J的最近邻铁磁的二维伊辛自旋系统的零温相图。我们首先引入了“角能”的概念，它表明，当反铁磁相互作用衰减快于距离的四次方时，当J接近J_c时，相对于棋盘状态，条纹状态是有利的，向铁磁状态的转变，即，当均匀磁化的磁畴的长度尺度变大时。接下来，我们对反铁磁相互作用的条纹态和棋盘态的能量进行了详细的解析计算，其中反铁磁相互作用具有指数衰减和幂律衰减r^{-p}，p>2，这取决于曼哈顿距离而不是欧几里得距离。我们证明了当基态结构的尺度很大时，条纹相总是比棋盘相更有利。如果p>3，则对于J \lesssim J_c发生这种情况，并且如果2<p<=3，则对于J足够大。我们的许多考虑，包括严格的边界结转到尺寸大于2和更一般的短程铁磁相互作用。
英文摘要：
We study the zero temperature phase diagram of Ising spin systems in two dimensions in the presence of competing interactions, long range antiferromagnetic and nearest neighbor ferromagnetic of strength J. We first introduce the notion of a "corner energy" which shows, when the antiferromagnetic interaction decays faster than the fourth power of the distance, that a striped state is favored with respect to a checkerboard state when J is close to J_c, the transition to the ferromagnetic state, i.e., when the length scales of the uniformly magnetized domains become large. Next, we perform detailed analytic computations on the energies of the striped and checkerboard states in the cases of antiferromagnetic interactions with exponential decay and with power law decay r^{-p}, p>2, that depend on the Manhattan distance instead of the Euclidean distance. We prove that the striped phase is always favored compared to the checkerboard phase when the scale of the ground state structure is very large. This happens for J\lesssim J_c if p>3, and for J sufficiently large if 2<p<=3. Many of our considerations involving rigorous bounds carry over to dimensions greater than two and to more general short-range ferromagnetic interactions.
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