Potts Model with Invisible States: A Review

The Potts model with invisible states was introduced to explain discrepancies between theoretical predictions and experimental observations of phase transitions in some systems where $Z_q$ symmetry is spontaneously broken. It differs from the ordinary $q$-state Potts model in that each spin, besides the usual $q$ visible states, can be also in any of $r$ so-called invisible states. Spins in an invisible state do not interact with their neighbours but they do contribute to the entropy of the system. As a consequence, an increase in $r$ may cause a phase transition to change from second to first order. Potts models with invisible states describe a number of systems of interest in physics and beyond and have been treated by various tools of statistical and mathematical physics. In this paper we aim to give a review of this fundamental topic.