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

Abstract

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