Two-transition conjecture for the three-dimensional symmetric N-clock model

Let $N\geq 2$ and consider the three-dimensional symmetric $N$-clock model, with inverse temperature $\beta$. Write $\beta_N^G$ for the low-temperature threshold and $\beta_N^{cr}$ for the lower transition threshold.

Two-transition conjecture. There exists $\bar N$ such that, for every $N\geq\bar N$, the model undergoes two phase transitions: for

$$\beta>\beta_N^G,$$

it has $N$ pure magnetized phases with exponential decay of truncated correlations, where $\beta_N^G\to\infty$ as $N\to\infty$; for

$$\beta_N^{cr}<\beta<\beta_N^G,$$

it has at least $N$ pure phases with nonzero magnetization and only algebraic correlation decay; and for

$$\beta<\beta_N^{cr},$$

it has only one Gibbs state, with exponential decay of correlations.

The conjecture proposes a three-dimensional analogue of the phase structure suggested by the two-dimensional clock model: a low-temperature phase with exponential decay, an intermediate phase with algebraic decay, and a high-temperature uniqueness regime. The stronger continuum-of-pure-phases proposal mentioned in the source is a separate conjecture and is not included here. The supplied evidence says that the analogous behavior for two-dimensional clock models was subsequently proved, so this candidate is marked solved.