Potts Model with Long-Range Interactions in One Dimension

We study the nature of the phase transition of the $q$-state Potts model with long-range ferromagnetic interactions decaying as $1/r^{d+\sigma }$, in dimension $d=1\mathrm{}$, using a histogram Monte Carlo (MC) technique. The model can exhibit a first-order transition or a second-order phase transition with nonstandard critical exponents. The critical value of $q$ above which a first-order transition occurs decreases with decreasing $\sigma$, from $q_c=8\mathrm{}$ for $\sigma =1\mathrm{}$ to $q_c=2\mathrm{}$ for $\sigma =0.3\mathrm{}$. Detailed results for various $\sigma$ will be shown and discussed. Mean-field calculation confirms the tendency of our MC results.