One-dimensional Ising model with long-range interactions: A renormalization-group treatment

The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\alpha }}$ 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 α. 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→∞. We obtain a b=∞ extrapolation of the critical temperature for arbitrary values of α>1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value ${\mathit{T}}_{\mathit{c}}$/J=${\mathrm{\pi }}^2$/12 for the 1/${\mathit{r}}^2$ model.