论文标题
parafermonic模型中自旋簇的旋转界面和交叉概率
Spin interfaces and crossing probabilities of spin clusters in parafermionic models
论文作者
论文摘要
我们考虑二维$ z_n $旋转晶格模型中的分形曲线。这些是n个状态的自旋模型，其经历了由Zn parafermionic场理论描述的连续铁磁 - 磁相变。这里的主要动机是研究Schramm-Loewner演变（SLE）与结构田间理论之间的对应关系。通过使用Monte-Carlo模拟，我们计算了n = 3和n = 4个自旋模型的不同自旋界面的分形维，该模型分别对应于3个状态POTTS模型以及Fateev-Zamolodchikov Point的Ashkin-Teller模型。这些数字度量改进并完成了先前作品中提出的数字度量与SLE/ECFT预测一致。然后，我们考虑矩形结构域中自旋簇的交叉概率。使用多个SLE方法，我们为Zn Parafarmion理论提供交叉概率公式。输入交叉概率公式的副共振块是通过求解knhiznik-Zamolodchikov等级3的3。在3个州Potts模型案例中，副含量与Virasoro相吻合的3个州，我们与S.M.M.Flores等人相当。对于n> = 4，交叉概率满足三阶微分方程而不是二阶方程，我们的公式是新的。将理论预测与在n = 4处采取的蒙特卡洛措施进行了比较，并找到了公平的一致性。
We consider fractal curves in two-dimensional $Z_N$ spin lattice models. These are N states spin models that undergo a continuous ferromagnetic-paramagnetic phase transition described by the ZN parafermionic field theory. The main motivation here is to investigate the correspondence between Schramm-Loewner evolutions (SLE) and conformal field theories with extended conformal algebras (ECFT). By using Monte-Carlo simulation, we compute the fractal dimension of different spin interfaces for the N=3 and N=4 spin models that correspond respectively to the 3 states Potts model and to the Ashkin-Teller model at the Fateev-Zamolodchikov point. These numerical measures, that improve and complete the ones presented in the previous works, are shown to be consistent with SLE/ECFT predictions. We consider then the crossing probability of spin clusters in a rectangular domain. Using a multiple SLE approach, we provide crossing probability formulas for ZN parafarmionic theories. The parafermionic conformal blocks that enter the crossing probability formula are computed by solving a Knhiznik-Zamolodchikov system of rank 3. In the 3 states Potts model case, where the parafermionic blocks coincide with the Virasoro ones, we rederive the crossing formula found by S.M.Flores et al., that is in good agreement with our measures. For N>=4 where the crossing probability satisfies a third order differential equation instead of a second order one, our formulas are new. The theoretical predictions are compared to Monte-Carlo measures taken at N=4 and a fair agreement is found.
