The histogram Monte Carlo renormalization group method proposed by Hu is used to determine the critical point of the q-state Potts model on the Kagome lattice. Our results are compared with the predictions of conjectures by Wu and Tsallis.
The determination of the critical point of the q -state Potts model on the Kagomé lattices has been an outstanding unsolved problem. Here we study this problem by regarding the Potts model as generating correlated bond percolations. By applying a combination of Monte Carlo renormalization group and finite-size scaling analyses to the percolation problem, numerical estimates with an accuracy of are obtained for the Kagomé critical point for q = 1,2,3,4. Our results for q = 1 are consistent with a recent highly accurate numerical estimate by Ziff and Suding, and results for q = 2 agree with the known exact result within the numerical accuracy. Compared with results obtained in a recent series analysis by Jensen et al , our numbers differ from theirs slightly for q = 3, and agree with theirs with a slightly better accuracy for q = 4. Our numbers also confirm that a conjecture due to Wu is extremely accurate.
The critical behavior of the three-state Potts model in two dimensions is investigated with the use of the Monte Carlo renormalization group (MCRG), and the results are compared with conjectured values as well as with other methods. Recent advances in the MCRG enable greatly improved convergence, as well as greater insight into the structure of the renormalization transformation. The introduction of a four-point interaction is used to control the chemical potential of effective vacancies in a purely Potts Hamiltonian.
Based on the subgraph expansions for phase transition models, we present a histogram Monte Carlo method to calculate the free energy and other physical quantities for such models as continuous functions of the system parameter. The method does not have any critical slowing down. We then use physical quantities thus obtained in a percolation renormalization group method to calculate critical point, critical exponents, and thermodynamic order parameter. Our method gives quite accurate results and may be applied to many phase transition models.
We present a histogram Monte Carlo method to calculate the existence probability Ep, the percolation probability P, and the mean-cluster size S for percolation problems as continuous functions of the bond or site probability p. We then use Ep and P in a percolation renormalization-group method to calculate the critical point and exponents. Our method gives quite accurate results for percolation problems.