Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models

A critical review is given of status and perspectives of Monte Carlo simulations that address bulk and interfacial phase transitions of ferromagnetic Ising models. First, some basic methodological aspects of these simulations are briefly summarized (single-spin flip vs. cluster algorithms, finite-size scaling concepts), and then the application of these techniques to the nearest-neighbor Ising model in d=3 and 5 dimensions is described, and a detailed comparison to theoretical predictions is made. In addition, the case of Ising models with a large but finite range of interaction and the crossover scaling from mean-field behavior to the Ising universality class are treated. If one considers instead a long-range interaction described by a power-law decay, new classes of critical behavior depending on the exponent of this power law become accessible, and a stringent test of the ε-expansion becomes possible. As a final type of crossover from mean-field type behavior to two-dimensional Ising behavior, the interface localization–delocalization transition of Ising films confined between “competing” walls is considered. This problem is still hampered by questions regarding the appropriate coarse-grained model for the fluctuating interface near a wall, which is the starting point for both this problem and the theory of critical wetting.