A mean-field renormalisation-group approach to Ising and q=3-state Potts models with long-range interactions in one dimension

We study the one-dimensional Potts model with long-range interactions decaying with distance r as r1+σ. An extended mean-field renormalisation-group procedure is applied, such that three finite-size linear lattices are compared, in order to evaluate critical temperatures and exponents for the q=2 (Ising model) and q=3 (such that the transition is a continuous one) cases. Good results are obtained, whenever comparison with exact results or with other procedures is possible. Moreover, we evaluate the surface field exponent for these models. We have been able to go to rather large lattices and then a suitable finite-size scaling procedure is employed to obtain the results in the thermodynamic limit.