High-Precision Monte Carlo Simulation of the Ising Models on the Penrose Lattice and the Dual Penrose Lattice

Abstract

We study the Ising models on the Penrose lattice and the dual Penrose lattice by means of the high-precision Monte Carlo simulation. Simulating systems up to the total system size N = 20633239, we estimate the critical temperatures on those lattices with high accuracy. For high-speed calculation, we use the generalized method of the single-GPU-based computation for the Swendsen–Wang multi-cluster algorithm of Monte Carlo simulation. As a result, we estimate the critical temperature on the Penrose lattice as Tc/J = 2.39781 ± 0.00005 and that of the dual Penrose lattice as \(T_{\text{c}}^{*}/J = 2.14987 \pm 0.00005\). Moreover, we definitely confirm the duality relation between the critical temperatures on the dual pair of quasilattices with a high degree of accuracy, \(\sinh (2J/T_{\text{c}})\sinh (2J/T_{\text{c}}^{*}) = 1.00000 \pm 0.00004\).