  A hidden state in which a spin does not interact with any other spin
contributes to the entropy of an interacting spin system. Using the
Ginzburg-Landau formalism in the mean-field limit, we explore the $q$-state
Potts model with extra $r$ hidden states. We analytically demonstrate that when
$1 < q \le 2$, the model exhibits a rich phase diagram comprising a variety of
phase transitions such as continuous, discontinuous, two types of hybrids, and
two consecutive second- and first-order transitions; moreover, several
characteristics such as critical, critical endpoint, and tricritical point are
identified. The critical line and critical end lines merge in a singular form
at the tricritical point. Those complex critical behaviors are not wholly
detected in previous research because the research is implemented only
numerically. We microscopically investigate the origin of the discontinuous
transition; it is induced by the competition between the interaction and
entropy of the system in the Ising limit, whereas by the bi-stability of the
hidden spin states in the percolation limit. Finally, we discuss the potential
applications of the hidden Potts model to social opinion formation with shy
voters and the percolation in interdependent networks.
