Translational invariance in critical phenomena Ising model on a quasi-lattice

The Ising model on a two-dimensional quasi-crystal (the Penrose tiling) is studied. Using the correlation inequality and the duality transformation bounds for the critical temperature are obtained as 1.82<T c <7. Monte Carlo simulations on finite lattices give an estimate of T c =2.41+or-0.02. Finite size scaling analysis of the Monte Carlo data shows that the system belongs to the same universality class as the Ising model on the two-dimensional Bravais lattices. However, the finite size scaling forms do not reproduce the asymptotic limits in the range studied whereas, in the same range, the periodic lattices are known to behave as expected.