  In this paper, we investigate the energy landscape of the two-component spin
systems, known as the Curie-Weiss-Potts model, which is a generalization of the
Curie-Weiss model consisting of $q\ge3$ spins. In the energy landscape of a
multi-component model, the most important element is the relative strength
between the inter-component interaction strength and the component-wise
interaction strength. If the inter-component interaction is stronger than the
component-wise interaction, we can expect all the components to be synchronized
in the course of metastable transition. However, if the inter-component
interaction is relatively weaker, then the components will be desynchronized in
the course of metastable transition. For the two-component Curie-Weiss model,
the phase transition from synchronization to desynchronization has been
precisely characterized in studies owing to its mean-field nature. The purpose
of this paper is to extend this result to the Curie-Weiss-Potts model with
three spins. We observe that the nature of the phase transition for the
three-spin case is entirely different from the two-spin case of the Curie-Weiss
model, and the proof as well as the resulting phase diagram is fundamentally
different and exceedingly complicated.
