Rich phase structure and multiple critical temperatures in a coupled Ising–Potts model

We study a coupled Ising–Potts model on Cayley trees of order k≥2, where each site carries a two-component spin (s,σ) with s=±1 and σ∈{1,…,q}. This coupling unifies and extends the classical Ising and Potts models and is relevant for systems with competing order parameters, such as multi-component alloys, spin glasses, and network-type models. We establish an explosive proliferation of translation-invariant phases (TIPs). For the case k = 2 and q = 2, 3, 4, we determine the exact critical temperatures and number of TIPs. In particular, for q = 4, we show that the model exhibits 15 distinct critical temperatures – five times more than in the pure Potts case – at which the number of phases changes sharply. At low temperatures, the phase multiplicity exceeds 2q+1, reaching more than 17 times the corresponding Potts bound. Remarkably, some phases exist at intermediate temperatures, a new phenomenon absent in both parent models. These intermediate-temperature phases occur for q≥3, but not for q = 2. These results reveal a fundamentally richer phase structure of translation-invariant Gibbs measures, demonstrating how coupling distinct spin components qualitatively reshapes the critical behavior of lattice systems.