We reconsider the (q+r)-state 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. The number of invisible states r introduced into the model plays the role of a parameter that regulates the order of the phase transition. For the particular case q = 2 it had already been shown that on the square lattice the change occurs at large r (for d = 2) and that 3 < r_c < 4 within the mean field analysis [1]. Within the mean field approach, we give a more precise estimate r_c(q=2) = 3.65(5). Moreover, we find the novel mechanism of changing the order of the phase transition in the region 1 = q < 2. It is characterized by two marginal dimensions, r_{c_1} and r_{c_2} [2]. These dimensions indicate how the discontinuity in characteristics of the first order phase transitions emerges. The above region of q is relevant for description of bond percolation (q=1), some intermediate values of q also are known to have physical realisations.

[1] R. Tamura, S. Tanaka, and N. Kawashima. Prog. Theor. Phys. 124, 381 (2010); arXiv:1111.6509.

[2] M. Krasnytska, P. Sarkanych, B. Berche, Yu. Holovatch, and R. Kenna. J. Phys. A: Math. Theor. 49, 255001 (2016).