One-dimensional Ising models with long-range interactions

We consider Ising models with long-range ferromagnetic pair interactions decaying as 1/r(theta) for 1.0 < theta less than or equal to 1.5. We first find approximate values for the critical temperature. We use a cluster mean-field approach combined with finite-size scaling and Vanden Broeck and Schwartz transformations. For theta = 1.10 we find T-c = 21.000 97 which can be compared with recent results of Luijten and Blote who found T-c = 21.000 99 +/- 0.000 26, and which is two orders of magnitude more accurate than any previous results. Since we use a mean-field cluster approximation as part of our approach, the accuracy for larger values of theta decreases significantly. In addition to T-c we obtain approximate values for the critical exponents beta, gamma and delta using the coherent anomaly method. For theta = 1.10 we obtain beta = 0.4995, gamma = 1.0008, and delta = 2.9947-all extremely close to the predictions of renormalization group calculations which say that these exponents should take on their classical values for this value of theta.