Phase transitions in one-dimensional Riesz gases with long-range interaction

We provide numerical evidence for the existence of phase transitions with respect to the temperature in the one-dimensional Riesz gases with non-singular pair interaction, that is particles on the line interacting via the potential $-|r|^{-s}$, where $s \in (-1, 0)$. Our numerics hint for the existence of two distinct phase transitions whose critical temperatures depend on $s$, namely a first transition which separates between a fluid and a quasisolid phase reminiscent of the Berezinski-Kosterlitz-Thouless (BKT) transition, and a second transition below which freezing occurs and the system is in a solid phase. We determine the phase diagram with respect to $s$ and the temperature $T$, which we find to be consistent with the known (or expected) results on the 1D Coulomb gas ($s = -1$), known to be a solid at all temperature, and the Dyson log--gas ($s = 0$) which exhibits a BKT transition at $T = 1/2$ and which is believed to be a fluid at all positive temperature.