First-order phase transition in $1d$ Potts model with long-range interactions

The first-order phase transition in the one-dimensional $q$-state Potts model with long-range interactions decaying with distance as $1/r^{1+\sigma}$ has been studied by Monte Carlo numerical simulations for $0 < \sigma \le 1$ and integer values of $q > 2$. On the basis of finite-size scaling analysis of interface free energy $\Delta F_L$, specific heat and Binder's fourth order cumulant, we obtain the first-order transition which occurs for $\sigma$ below a threshold value $\sigma_c(q)$.