关于6顶点模型无序相和反铁电相之间的转变
On the transition between the disordered and antiferroelectric phases of the 6-vertex model
AI总结：
本文研究对称六顶点模型在$a+b=c$与$a+b<c$区域间的转变，证明两区域高度函数分别为离域化与局域化，借助Baxter-Kelland-Wu耦合等方法给出证明，并得到其与自对偶线处Ashkin-Teller模型的耦合及相关自旋序结果。
摘要：
参数满足$a,b,c>0$的对称六顶点模型（six-vertex model）预计在$a+b<c$（反铁电相）、$|a-b|<c\leq a+b$（无序相）和$|a-b|>c$（铁电相）区域表现出不同行为。本文研究了$a+b=c$与$a+b<c$两个区域之间转变的呈现方式。\n 当$a+b<c$时，我们证明相关的高度函数是局域化的，其极值周期Gibbs态可以用整数参数化：在第$n$个态中，高度$n$和$n+1$发生渗流，而其补集的连通分量直径具有指数衰减的尾部。当$a+b=c$时，高度函数是离域化的。\n 证明依赖于六顶点模型与随机团簇模型（random-cluster model）之间的Baxter-Kelland-Wu耦合，以及后者的最新研究结果。我们通过修改团簇权重，引入了自由边界条件与有线边界条件之间的插值方法。利用三角格点围道（$\mathbb{T}$-回路），我们描述了另一种高度函数耦合方式，该方法特别给出了$a=b=c$时离域化的新证明。\n 最后，我们强调了六顶点模型的一种自旋表示，并得到了它与$\mathbb{Z}^2$上自对偶线$\sinh 2J = e^{-2U}$处的Ashkin-Teller模型（Ashkin-Teller model）的耦合。当$J<U$时，我们证明两个Ising构型各自的关联呈指数衰减，而它们的乘积具有铁磁序。
英文摘要：
The symmetric six-vertex model with parameters $a,b,c>0$ is expected to exhibit different behavior in the regimes $a+b<c$ (antiferroelectric), $|a-b|<c\leq a+b$ (disordered) and $|a-b|>c$ (ferroelectric). In this work, we study the way in which the transition between the regimes $a+b=c$ and $a+b<c$ manifests. When $a+b<c$, we show that the associated height function is localized and its extremal periodic Gibbs states can be parametrized by the integers in such a way that, in the $n$-th state, the heights $n$ and $n+1$ percolate while the connected components of their complement have diameters with exponentially decaying tails. When $a+b=c$, the height function is delocalized. The proofs rely on the Baxter-Kelland-Wu coupling between the six-vertex and the random-cluster models and on recent results for the latter. An interpolation between free and wired boundary conditions is introduced by modifying cluster weights. Using triangular lattice contours ($\mathbb{T}$-circuits), we describe another coupling for height functions that in particular leads to a novel proof of the delocalization at $a=b=c$. Finally, we highlight a spin representation of the six-vertex model and obtain a coupling of it to the Ashkin-Teller model on $\mathbb{Z}^2$ at its self-dual line $\sinh 2J = e^{-2U}$. When $J<U$, we show that each of the two Ising configurations exhibits exponential decay of correlations while their product is ferromagnetically ordered.
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