利用Wang-Swendsen-Koteckmann Monte Carlo算法和一种新的有限尺度外推方法，研究了正方晶格上反铁磁$q$态Potts模型（$q=3$和$q=4$）.对于$q=3$，我们获得了良好的控制相关长度$ξ\sim 5000$ ;数据与$ξ(β) = A e^{2β} β(1 + a_1 e^{-β} + \ldots)$一致，如$β\to\infty$。对于$q=4$，即使在零温度下，模型也是无序的。
英文摘要：
We study the antiferromagnetic $q$-state Potts model on the square lattice for $q=3$ and $q=4$, using the Wang-Swendsen-Kotecký Monte Carlo algorithm and a new finite-size-scaling extrapolation method. For $q=3$ we obtain good control up to correlation length $ξ\sim 5000$; the data are consistent with $ξ(β) = A e^{2β} β(1 + a_1 e^{-β} + \ldots)$ as $β\to\infty$. For $q=4$ the model is disordered even at zero temperature.
We give a general condition for a discrete spin system with nearest-neighbor interactions on the $\mathbb{Z}^d$ lattice to exhibit long-range order. The condition is applicable to systems with residual entropy in which the long-range order is entropically driven. As a main example we consider the antiferromagnetic $q$-state Potts model and rigorously prove the existence of a broken sub-lattice symmetry phase at low temperature and high dimension -- a new result for $q\ge 4$. As further examples, we prove the existence of an ordered phase in a clock model with hard constraints and extend the kn…
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
本文导出了具有自由边界和圆柱边界条件的正方格子Potts模型的传递矩阵的一些新的结构结果。特别是，我们获得显式的封闭形式的表达式占主导地位的（在大|q|）在传输矩阵中的对角条目，任意宽度m，作为一个特殊的一维聚合物模型的解决方案。我们还得到了体积和表面的大q展开式（分别为。角）自由能的零温度反铁磁体（=色多项式）通过阶q^{-47}（分别。q^{-46}）。最后，我们计算了宽度为9 <= m <= 12的带在自由边界条件下的色根，并粗略地定位了极限曲线。
It is widely believed that the phase transition for the four-state ferromagnetic Potts model on the square lattice is of the pseudo-first order. Specifically, it is expected that first-order phase transition behavior is found on small lattices and that the true nature of second-order phase transition only emerges with large system sizes. It is also intuitively expected that for other geometries, the types of the associated phase transitions should be identical to that of the square lattice. However, after simulating more than 16 million spins for the four-state Pott model, we observe that a fe…
A universal (supervised) neural network (NN), which is only trained once on a one-dimensional lattice of 200 sites, is employed to study the phase transition of the two-dimensional (2D) 5-state ferromagnetic Potts model on the square lattice. In particular, the NN is obtained by using merely two artificially made configurations as the training set. Due to the elegant features of the considered NN, results associated with systems consisting of over 4000000 spins can be obtained with ease, and convincing evidence showing the investigated phase transition is weakly first order is reached. The out…
