Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

We demonstrate that the invaded cluster algorithm, introduced by Machta et al (1995 Phys. Rev. Lett. 75 2792-5), is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits, but only modest computational effort. On two quasiperiodic graphs which were not investigated in this respect before, the 12-fold symmetric square-triangle tiling and the 10-fold symmetric Tuebingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from the Onsager value. But notable exceptions to the criterion are found which consist not only of the exactly solvable cases, in agreement with a recent exact result, but also of the self-dual ones and maybe more. (author)