Duality in the Ising Model on the Quasicrystals

Abstract

The ferromagnetic Ising model on the dual Penrose lattice is simulated by the Monte Carlo method. Using a phenomenological Monte Carlo renormalization group based on finite-size scaling, the critical temperature and the magnetic critical exponent are estimated as T_c⁄J=2.150±0.006 and y_H=1.88±0.02. It is shown that the duality relation, sinh (2J⁄T_c) sinh (2J⁄T_c^*)=1, between the critical temperature of the Ising model on a lattice T_c and that on its dual lattice T_c^* holds for a quasiperiodic lattice in two dimensions as well as for a regular lattice.