Fang Sheng, Ke Da, Zhong Wei, Deng Youjin
MinJiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China.
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.
Phys Rev E. 2022 Apr;105(4-1):044122. doi: 10.1103/PhysRevE.105.044122.
We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional Q-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents d_{B}=1.7320(3) and 1.794(2) for Q=2,3, respectively. However, for large Q, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the O(n) loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with Q=n^{2}. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops d_{B}=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6) for Q=1,2,3,2+sqrt[3],4, respectively, and d_{min}=1.0945(2),1.0675(3),1.0475(3),1.0322(4) for Q=2,3,2+sqrt[3],4, respectively. Our estimates significantly improve over the existing results for both d_{B} and d_{min}. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of n for the leading correction exponent.
我们给出了在Fortuin-Kasteleyn键表示下二维Q态Potts模型的主链和最短路径指数的蒙特卡罗研究。我们首先使用团簇算法在正方形晶格上模拟临界Potts模型，分别得到Q = 2、3时的主链指数(d_{B}=1.7320(3))和(1.794(2))。然而，对于较大的Q，该研究存在严重的临界慢化和有限尺寸修正缓慢收敛的问题。为克服这些困难，我们考虑在密堆积相的蜂窝晶格上的O(n)环模型，它被认为对应于(Q = n^{2})的临界Potts模型。通过一种高效的团簇算法，我们从环所包围的区域确定，对于Q = 1、2、3、(2+\sqrt{3})、4，主链指数(d_{B}=1.64339(5))、(1.73227(8))、(1.7938(3))、(1.8384(5))、(1.8753(6))，对于Q = 2、3、(2+\sqrt{3})、4，最短路径指数(d_{min}=1.0945(2))、(1.0675(3))、(1.0475(3))、(1.0322(4))。我们的估计在(d_{B})和(d_{min})方面都显著优于现有结果。最后，通过研究与主链相关量的有限尺寸修正，我们推测了一个作为n的函数的主导修正指数的精确公式。
