Potts Model with Long-Range Interactions in One Dimension

Abstract

We study the nature of the phase transition of the $q$-state Potts model with long-range ferromagnetic interactions decaying as $1/{r}^{d+\ensuremath{\sigma}}$, in dimension $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$, 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 $\ensuremath{\sigma}$, from ${q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8$ for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ to ${q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.3$. Detailed results for various $\ensuremath{\sigma}$ will be shown and discussed. Mean-field calculation confirms the tendency of our MC results.