Cluster Monte Carlo study of the q-state potts model on hypercubic lattices

Abstract

The q-state Potts model (QPM) may be related to many interesting mathematical problems and physical systems. Previous studies of the QPM have been restricted mainly to small q values. Using the connection between the QPM and the q-state bond-correlated percolation model (QBCPM) and Swendsen and Wang's cluster Monte Carlo simulation method, in this paper we study the QPM on d-dimensional hypercubic lattices with d being 3, 4, 5 and 6, and q being 2, 3, 4, 8, 16, 32, 64, 128, 256, 512, 1024 and 2048. For q=2 where phase transitions are second order, a percolation Monte Carlo renormalization group method is used to determine more accurate critical points. We find that in the studied space dimensions, the critical points increase slowly with q. As q increases, the critical points for different d approach to each other such that when q=2048 the absolute values of the differences between such critical points are less than 1.1%. We also discuss some related theoretical problems, including implications of our results to the percolation theory of supercooled water.