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

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Abstract
The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of 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 limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q. We also verify that , which is consistent with a recent conjecture of Tsallis.