论文标题
三角形晶格上的关键位点渗透：从集成性到共形分区功能
Critical site percolation on the triangular lattice: From integrability to conformal partition functions
论文作者
论文摘要
三角形晶格上的关键位点渗透由阳式溶解的稀释稀释$ a_2^{（2）} $循环模型具有交叉参数，专门为$λ= \ frac \ pi3 $，对应于订约的宽带fugacity fugacity $β= -2 \cos4λ= -2 \cos4λ= 1 $。我们研究了该模型的通勤转移矩阵和相关的bethe ansatz方程所满足的功能关系。单行传输矩阵分别具有条纹和周期性边界条件，是普通稀释的temperley temperley-lieb代数的元素。这些代数的标准模块由$ d $的缺陷次数标记，在后一种情况下，也由twist $ e^{iγ} $标记。非线性积分方程技术用于分析以中央电荷$ C = 0 $的缩放限度和保形权重$δ，\barδ$的分析求解bethe ansatz功能方程。对于地面，我们发现$δ=δ_{1，d+1} $对于条边界条件，$（δ，\barδ）=（δ_{γ/π，d/2}，δ__{γ/π，-d/π，-d/2}）$在周期性边界条件下$δ_{r，s} = \ frac1 {24}（（（3R-2S）^2-1）$。我们给出了每个标准模块中传输矩阵迹线的缩放限制的明确猜想。对于$ d \ le8 $，这些猜想得到了伯特·安萨兹方程的对数形式的数值解决方案，用于领先的$ 20 $或更多的保形素质。通过这些猜想，我们应用马尔可夫痕迹以获得圆柱和圆环上的共形分区函数。这些正是与我们先前在密度$ a_1^{（1）} $ loop模型上与$λ= \ frac \ pi3 $描述的正方形晶格上的关键债券渗透结果相吻合。所有这些共同数据的同意提供了令人信服的证据，支持这两个随机模型作为对数CFT之间的强烈普遍性。
Critical site percolation on the triangular lattice is described by the Yang-Baxter solvable dilute $A_2^{(2)}$ loop model with crossing parameter specialized to $λ=\frac\pi3$, corresponding to the contractible loop fugacity $β=-2\cos4λ=1$. We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley-Lieb algebras. The standard modules for these algebras are labeled by the number of defects $d$ and, in the latter case, also by the twist $e^{iγ}$. Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge $c=0$ and conformal weights $Δ,\barΔ$. For the groundstates, we find $Δ=Δ_{1,d+1}$ for strip boundary conditions and $(Δ,\barΔ)=(Δ_{γ/π,d/2},Δ_{γ/π,-d/2})$ for periodic boundary conditions, where $Δ_{r,s}=\frac1{24}((3r-2s)^2-1)$. We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For $d\le8$, these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading $20$ or more conformal eigenenergies. With these conjectures, we apply the Markov traces to obtain the conformal partition functions on the cylinder and torus. These precisely coincide with our previous results for critical bond percolation on the square lattice described by the dense $A_1^{(1)}$ loop model with $λ=\frac\pi3$. The concurrence of all this conformal data provides compelling evidence supporting a strong form of universality between these two stochastic models as logarithmic CFTs.
