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{\bf Marginal dimensions of the Potts model with invisible states}

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{\bf M. Krasnytska$^{1,4}$}, P. Sarkanych$^{1,2,4}$, B. Berche$^{3,4}$, Yu. Holovatch$^{1,4}$, R. Kenna$^{2,4}$ \\

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$^1$\textit{Institute for Condensed Matter Physics, National Acad.
Sci. of Ukraine, UA--79011 Lviv, Ukraine}

$^2$\textit{Applied Mathematics Research Centre, Coventry
University, Coventry CV1 5FB, United Kingdom}

$^3$\textit{ Institut Jean Lamour, CNRS/UMR 7198, Groupe de Physique
Statistique, Universit\'e de Lorraine, BP 70239, F-54506 Vand\oe
uvre -les-Nancy Cedex, France }

$^4$\textit{${\mathbb L}^4$ Collaboration \& Doctoral College for
the Statistical Physics of Complex Systems,
Leipzig-Lorraine-Lviv-Coventry, D-04009 Leipzig, Germany}



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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 $r_c(q=2) \simeq 3.65(5)$.
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.

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\par \noindent [1]\quad R. Tamura, S. Tanaka, and N. Kawashima. Prog. Theor. Phys. {\bf 124},
381 (2010); arXiv:1111.6509.

\par \noindent [2]\quad M. Krasnytska, P. Sarkanych, B. Berche, Yu. Holovatch, and R. Kenna. J. Phys. A: Math. Theor. {\bf 49}, 255001
(2016).


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