环面上Potts模型与RSOS模型的关系

Four-point geometrical correlation functions in the two-dimensional $Q$-state Potts model: connections with the RSOS models

The "bootstrap determination" of the geometrical correlation functions in the two-dimensional Potts model proposed in a paper [arXiv:1607.07224] was later shown in [arXiv:1809.02191] to be incorrect, the actual spectrum of the model being considerably more complex than initially conjectured. We provide in this paper a geometrical interpretation of the four-point functions built in [arXiv:1607.07224], and explain why the results obtained by these authors, albeit incorrect, appeared so close to those of their numerical simulations of the Potts model. Our strategy is based on a cluster expansion …

环面上Potts模型的本征值振幅

Eigenvalue amplitudes of the Potts model on a torus

我们认为Q-状态Potts模型的随机集群制定，定义在有限的二维晶格的大小L × N与环形边界条件。由于簇的非局部性，配分函数Z（L，N）不能简单地写成转移矩阵TL的迹。利用组合方法建立了分解Z（L，N）= \sum 1，D k} b^{1，D k} K 1，D k}，其中特征标K 1，D k} = \sum i（λ i）^N是简单迹.在该分解中，T\_L的本征值λ\_i的振幅b^{l，D\_k}用数字l= 0，1，.标记L和循环群Cl的一个表示Dk.严格地得到了b^{l，Dk}的一个一般表达式，并利用数论结果证明了它与Read和Saleur在连续极限中得到的一个表达式是一致的.

Ergodicity of the Wang--Swendsen--Kotecký algorithm on several classes of lattices on the torus

We prove the ergodicity of the Wang--Swendsen--Kotecký (WSK) algorithm for the zero-temperature $q$-state Potts antiferromagnet on several classes of lattices on the torus. In particular, the WSK algorithm is ergodic for $q\ge 4$ on any quadrangulation of the torus of girth $\ge 4$. It is also ergodic for $q \ge 5$ (resp. $q \ge 3$) on any Eulerian triangulation of the torus such that one sublattice consists of degree-4 vertices while the other two sublattices induce a quadrangulation of girth $\ge 4$ (resp.~a bipartite quadrangulation) of the torus. These classes include many lattices of inte…

Conformally invariant boundary conditions in the antiferromagnetic Potts model and the $SL(2,\mathbb{R})/U(1)$ sigma model

We initiate a study of the boundary version of the square-lattice $Q$-state Potts antiferromagnet, with $Q \in [0,4]$ real, motivated by the fact that the continuum limit of the corresponding bulk model is a non-compact CFT, closely related with the $SL(2,\mathbb{R})_k/U(1)$ Euclidian black-hole coset model. While various types of conformal boundary conditions (discrete and continuous branes) have been formally identified for the the $SL(2,\mathbb{R})_k/U(1)$ coset CFT, we are only able in this work to identify conformal boundary conditions (CBC) leading to a discrete boundary spectrum. The co…

Potts model coupled to causal triangulations

In this work we study the annealed Potts model coupled to two dimensional causal triangulations with periodic boundary condition. Using duality on a torus, we provide a relation between the free energy of the Potts model coupled CTs and its dual. This duality relation follows from the FK representation for the Potts model. In order to determine a region where the critical curve for the model can be located we use the duality relation and the high-temperature expansion. This is done by outlining a region where the infinite-volume Gibbs measure exists and is unique and a region where the finite-…

正方形格点上宽边长条上的精确Potts模型配分函数

Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice

本文精确计算了具有周期性纵向边界条件的任意长度L_x，宽度L_y=3的正方形格子上的Potts模型的配分函数，其中q为一般q，温度为一般q。（i）（FBC_y，PBC_x）=循环，（ii）（FBC_y，TPBC_x）=莫比乌斯，（iii）（PBC_y，PBC_x）=环形，（iv）（PBC_y，TPBC_x）=克莱因瓶，其中FBC和（T）PBC指自由和（扭曲）周期性边界条件。文中还包括了L_y=2圆环和Klein瓶带的计算结果。在无限长极限下，讨论了热力学性质，并给出了任意大宽度带的低温行为的一些一般性结果。我们确定了q和温度的{ \mathbb C}^2空间中的子流形，其中对于这些带，自由能是奇异的。我们的计算也被用来计算一定数量的图论兴趣。