Universality away from Critical Points: Collapse of Observables in a Thermostatistical Model

The $p$-state clock model in two dimensions is a discrete model exhibiting, for $p>4$, a quasi-liquid phase in a region $T_14$ and all $T>T_{\rm eu}$, all thermal averages become identical to those of the continuous, planar rotor model ($p=\infty$). This {\em collapse of thermodynamic observables} amounts to an emergent symmetry, not present in the Hamiltonian. For $p \ge 8$, the collapse starts in the quasi-liquid phase and makes the transition at $T_2$ indistinguishable from the Berezinskii-Kosterlitz-Thouless (BKT) transition of the planar rotor. For $p \le 6$, we find $T_{\rm eu} > T_2$, and the transition at $T_2$ is no longer BKT. The results include a detailed analysis of the critical properties at $T_1$ and $T_2 $. Broader implications are discussed.