ONE-DIMENSIONAL ISING-MODEL WITH LONG-RANGE INTERACTIONS - A RENORMALIZATION-GROUP TREATMENT

The one-dimensional Ising model with ferromagnetic interactions which decay as 1/r(alpha) is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of alpha. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b --> infinity. We obtain a b = infinity extrapolation of the critical temperature for arbitrary values of alpha > 1, which we conjecture approximates with high precision the exact one. In particular, we obtain the value T-c/J = pi(2)/12 for the 1/r(2) model.