Duality in the Ising Model on the Quasicrystals

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 (2 J / T c )sinh (2 J / 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.