Marginal dimensions of the Potts model with invisible states

We reconsider the mean-field Potts model with q interacting and r non-interacting (invisible) states. The model was recently introduced to explain discrepancies between theoretical predictions and experimental observations of phase transitions in some systems where the Z q -symmetry is spontaneously broken. We analyse the marginal dimensions of the model, i.e., the value of r at which the order of the phase transition changes. In the q = 2 case, we determine that value to be ${r}_{{\rm{c}}}=3.65(5);$ there is a second-order phase transition there when $r\lt {r}_{{\rm{c}}}$ and a first-order one at $r\gt {r}_{{\rm{c}}}$. We also analyse the region $1\leqslant q\lt 2$ and show that the change from second to first order there is manifest through a new mechanism involving two marginal values of r. The q = 1 limit gives bond percolation. Above the lower value r c1, the order parameters exhibit discontinuities at temperature $\tilde{t}$ below a critical value t c. The larger value r c2 marks the point at which the phase transition at t c changes from second to first order. Thus, for ${r}_{{\rm{c}}1}\lt r\lt {r}_{{\rm{c}}2}$, the transition at t c remains second order while at $\tilde{t}$ the system undergoes a first order phase transition. As r increases further, $\tilde{t}$ increases, bringing the discontinuity closer to t c. Finally, when r exceeds r c2 $\tilde{t}$ coincides with t c and the phase transition becomes first order. This new mechanism indicates how the discontinuity characteristic of first order phase transitions emerges.