Effect of long-range interactions on the critical behavior of the continuous Ising model

Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\sigma }$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $-1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\sigma .$ Results for various values of $\sigma$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.