The one-dimensional Potts model with long-range interactions: A renormalization group approach

The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as l/r(alpha) is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature T-c(b, q, alpha) and the correlation length critical exponent nu(b, q, alpha) as a function of alpha for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the b --> infinity limit. We also obtain extrapolated critical temperatures (b = infinity) for arbitrary values of alpha > 1 and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near alpha = 2. Furthermore, the use of another extrapolation procedure suitable only in the vicinity of alpha = 2 led us to conjecture that the exact critical temperature T-c(q, alpha = 2) is the same for any value of q. We also verify that T-c(q, alpha --> 1) proportional to (alpha - 1)(-1) For All q, which is consistent with a recent conjecture of Tsallis.