Emergence of a non trivial fluctuating phase in the XY model on regular networks

We study an XY-rotor model on regular one dimensional lattices by varying the number of neighbours. The parameter $2\ge \gamma \ge 1$ is defined. $\gamma =2$ corresponds to mean field and $\gamma =1$ to nearest neighbours coupling. We find that for $\gamma <1.5$ the system does not exhibit a phase transition, while for $\gamma >1.5$ the mean field second order transition is recovered. For the critical value $\gamma ={\gamma }_c=1.5$, the systems can be in a non trivial fluctuating phase for whichthe magnetisation shows important fluctuations in a given temperature range, implying an infinite susceptibility. For all values of $\gamma$ the magnetisation is computed analytically in the low temperatures range and the magnetised versus non-magnetised state which depends on the value of $\gamma$ is recovered, confirming the critical value ${\gamma }_c=1.5$.