Triple critical point and emerging temperature scales in $SU(N)$ ferromagnetism at large $N$

The non-Abelian ferromagnet recently introduced by the authors, consisting of atoms in the fundamental representation of $SU(N)$, is studied in the limit where $N$ becomes large and scales as the square root of the number of atoms $n$. This model exhibits additional phases, as well as two different temperature scales related by a factor $N\!/\!\ln N$. The paramagnetic phase splits into a "dense" and a "dilute" phase, separated by a third-order transition and leading to a triple critical point in the scale parameter $n/N^2$ and the temperature, while the ferromagnetic phase exhibits additional structure, and a new paramagnetic-ferromagnetic metastable phase appears at the larger temperature scale. These phases can coexist, becoming stable or metastable as temperature varies. A generalized model in which the number of $SU(N)$-equivalent states enters the partition function with a nontrivial weight, relevant, e.g., when there is gauge invariance in the system, is also studied and shown to manifest similar phases, with the dense-dilute phase transition becoming second-order in the fully gauge invariant case.