Phase transitions in the Potts model with hidden states: potential application to a shy voter model*

A dark (or hidden) state in which a spin does not interact with any other spins contributes to the entropy of an interacting spin system. Here, we analytically demonstrate that the Potts model with hidden states exhibits a rich phase diagram comprising various types of phase transitions and critical points with different natures. Adopting the Ginzburg–Landau formalism, we reveal that when the number of visible states q is less than 2, and the number of hidden states is sufficiently large, the transition from the paramagnetic (disordered) to ferromagnetic (ordered) phase occurs as temperature decreases. The transition can be continuous, discontinuous, hybrid, or two consecutive transitions. There exist several characteristic points, explosive critical point, critical endpoint, and tricritical point, at which two different types of hybrid transitions, and supercritical behaviors occur, respectively. Thus, various of types of phase transitions that appear in nonequilibrium complex systems can be understood within an integrated scheme. Finally, we discuss the potential applications of the hidden Potts model to social opinion formation with shy voters.