# Entropy-Induced Phase Transitions in a Hidden Potts Model

Cook Hyun Kim  Center for Complex Systems, KI of Grid Modernization, Korea
Institute of Energy Technology, Naju, Jeonnam 58217, Korea  D.-S. Lee  School
of Computational Sciences, and Center for AI and Natural Sciences, Korea
Institute for Advanced Study, Seoul, 02455, Korea  B. Kahng
<EMAIL_ADDRESS>Center for Complex Systems, KI of Grid Modernization,
Korea Institute of Energy Technology, Naju, Jeonnam 58217, Korea

###### Abstract

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 $1<q\leq
2$, 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.

###### pacs:

89.75.Hc, 64.60.ah, 05.10.-a

##  I Introduction

Phase transitions and critical phenomena in spin models in thermally
equilibrium states [1, 2, 3] can help understand collective behavior in non-
equilibrium complex systems through mathematical correspondence. For instance,
whereas the $q$-state Potts (denoted as Q-Potts) model widely used in
statistical physics mainly to explore phase transitions in various magnetic
materials can be mapped in the $q\to 1$ limit to the percolation transition
model, where a giant cluster emerges as links are added in complex networks
[4, 5, 6, 7, 8]. The formalism of the Bose$–$Einstein condensation in thermal
quantum systems can be applied to describe the evolving structure in non-
equilibrium complex networks [9]. The phase transition of the Ising model
emerging at critical temperatures [10, 11, 12, 13, 14, 15, 16] provides
potential insights into understanding the formation of consensus in the voter
model [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32],
despite the inherent differences between thermal and stochastic noises. Each
spin represents a voter, and each spin’s up or down state represents the left
or right wing of the voter’s opinion state.

Conversely, phase transitions in non-equilibrium systems [17, 33, 34] inspire
motivation to study corresponding spin models. Recently, various types of
phase transitions, such as continuous and discontinuous transitions, but also
hybrid and consecutive transitions have been discovered in non-equilibrium
complex systems, particularly in multilayer networks. However, no universal
theory has been established in the non-equilibrium systems, so underlying
mechanisms are not yet fully understood. Under this circumstance, analytical
methods established in the corresponding equilibrium systems can give a clue
for a unified theory or framework of the non-equilibrium systems, as phase
transitions and critical phenomena in many non-equilibrium systems with the
Ising spin, such as the majority-vote model, belong to the same universality
class as the Ising model [35]. Therefore, for example, understanding opinion
separation or formation on two different issues in social networks can be
inspired by the study of the Ashkin–Teller (denoted as AT) model on scale-free
networks [38, 36, 37, 39]. Its Hamiltonian comprises two types of Ising spins
with intra- and inter-type interactions. The AT model’s phase diagram,
particularly within the mean-field limit, exhibits various types of phase
transitions and critical points. A hybrid (or mixed order) phase transition
occurs at a critical endpoint at which a second-order transition line intrudes
into a first-order transition line. Through the complex phase diagram, the AT
spin model can illustrate the causes leading to hybrid phase transitions.
Moreover, the AT model in the mean-field limit can give an insight into the
hybrid percolation transition in interdependent networks [40] where the origin
cannot be explained without such an integrated scheme. Therefore, studying
spin models in equilibrium systems motivated by real-world phenomena in non-
equilibrium systems and vice versa is interesting and meaningful;
nevertheless, this approach may not fully capture transient behaviors before a
system reaches a steady state.

With this background, this paper explores phase transitions in the Potts model
with hidden states. This modified Potts model provides a rich phase diagram
with various types of phase transitions and critical points, extending the
model’s ability to capture complex phenomena in systems beyond traditional
thermodynamics. This study is motivated by the presence of so-called shy
voters who do not readily express their opinions or candidate preferences
until they cast their votes or forever and rarely interact with other voters
[41, 42]. Consequently, opinion polls may struggle to predict the election
outcome accurately. The Potts model, incorporating hidden states as defined
below, can provide insights into the potential complexity of election results
influenced by the presence of these shy voters.

This Potts model with hidden states was initially proposed to explain
experimental results in physically disordered systems [43, 44, 45, 46, 47],
studied recently [48]. The hidden Potts model comprises $N$ spins, each spin
taking one of $q$ visible or $r$ hidden states,
$s_{i}=1,\cdots,q,q+1,\cdots,q+r$. Hence, it is called the $(q,r)$-state Potts
model [44, 46, 48] and denoted as the QR-Potts model. The Hamiltonian is

$\mathcal{H}=-J\sum_{\langle
i,j\rangle}\sum_{\alpha=1}^{q}\delta_{s_{i},\alpha}\delta_{s_{j},\alpha}-H\sum_{i}\delta_{s_{i},1},$
(1)

where $\langle i,j\rangle$ is the nearest neighbors, $\delta_{a,b}$ is the
Kronecker delta function, $J>0$ is a coupling constant, and $H$ is an external
field in the direction of the first state $s_{i}=1$. The spin $s_{i}$
interacts with the neighboring spins and contributes to the energy only in one
of the visible states $s_{i}=1,2\cdots,q$. The hidden states contribute only
to the entropy. This Hamiltonian can be rewritten as

$\mathcal{H}=-J\sum_{\langle
i,j\rangle}\sum_{\alpha=1}^{q}\delta_{s_{i},\alpha}\delta_{s_{j},\alpha}-H\sum_{i}\delta_{s_{i},1}-T\ln
r\sum_{i}\delta_{s_{i},q+1}.$

This Hamiltonian with $q=2$ was proposed to describe the two-step transition
exhibited by a bi-nuclear spin-crossover complex [49] and the solid-fluid
transitions of lipidic chains [50].

The mean-field (MF) solution has been obtained at $q=2$ to demonstrate a
discontinuous phase transition for $r>r_{c}=4e/3$ [43, 44, 45, 46]. This
result was surprising because the Ising model is known to exhibit only a
continuous transition even in the MF limit. The authors of Ref. [48]
numerically studied the phase transitions for $1\leq q<2$ to discover two
characteristic values, $r_{\ell}$ and $r_{h}$, such that for
$r_{\ell}<r<r_{h}$, as temperature $T$ decreases, the order parameter first
undergoes a continuous transition and then a discontinuous transition while
only a discontinuous transition occurs at $r=r_{h}$. Despite discovering such
interesting transition behaviors, a rigorous theoretical study has been absent
to elucidate the nature and origin of various phase transitions. Note that $q$
is regarded as a non-integer number, even though $q$ represents the number of
states. This generalization allows us to consider the crossover behavior from
the Ising limit to the percolation limit. Moreover, the Potts model with non-
integer $q$ was considered in association with the cluster-weighted
percolations [51, 52].

Here, we employ the Ginzburg–Landau (GL) formalism to the QR-Potts model to
reveal a much richer phase diagram, as illustrated in Fig 1, than the one
known before [44, 45, 46, 48]. The two lines comprising critical points (CP)
and the critical endpoints (CE), which merge at a tri-critical point (TP),
divide the parameter space into the regimes exhibiting continuous,
discontinuous, consecutive, and hybrid transitions, respectively, in the
parameter space. The GL formalism allows us to trace the stable and metastable
states, revealing physical mechanisms underlying the consensus formation and
the influence of shy voters. These analytic results provide a deeper
understanding of the critical phenomena than numerical results. Interestingly,
we find that the phase diagram of the hidden Potts model is similar to that of
the AT model on scale-free networks [38, 39], in which two types of Ising
spins located on each node are subject to intra- and inter-type interactions.
This similarity suggests that the results obtained here can be universal to
various systems.

The paper is organized as follows: In Sec. II, we introduce the GL free energy
of the QR-Potts model and present the method to obtain two order parameters
$m$ and $m_{r}$, representing the fraction of spins in a given visible state
and any hidden states, respectively. In Sec. III, we employ the analytic and
numerical methods based on Sec. II to obtain the phase diagram illustrating
different phase transitions of the order parameter $m$. The other order
parameter for hidden states is studied in Sec. IV. The implication of our
results for social opinion dynamics is discussed in Sec. V. The results are
summarized in Sec. VI.

Figure 1:  Phase schematic of QR-Potts model. (a) Three-dimensional phase
diagram in parameter space ($x=r/q$, $q$, $K=T^{-1}$). I represent a region
with $x\in[0,x_{\ell}]$, where the second-order transition occurs at $T_{c}$
from para to ferro phases for fixed $x$ and $q$. II represents a region with
$x\in[x_{\ell},x_{h}]$, where second-order transition $T_{c}$ and first-order
transition at $T_{f}$ occur subsequently as $T$ decreases. III represents
region with $x\in[x_{h},\infty]$, where first-order transition occurs at
$T_{f}$. (b) Phase diagram in space ($x$,$q$). They display phase transition
type in parameter space $[q,r]$. Solid and dotted lines and dashed lines
represent continuous and discontinuous transitions. CP and CE represent
critical points and endpoints, which can be illustrated more clearly in (c).
TP is located at $q=2$ and $x_{c}=2e/3$ at which CP and CE lines are merged.
CP line reaches $x_{1\ell}=e^{2}$ as $q\to 1$. CE line reaches $x_{1h}\simeq
9.60$ as $q\to 1$. (c) Phase diagram in space $[x,K=T^{-1}]$ for $1\leq q<2$.
Various types of phase transitions occur depending on $x=r/q$. (d) When $q=2$
and $x\leq x_{c}$, continuous transition occurs, whereas when $x>x_{c}$,
discontinuous transition occurs. (e) When $q>2$, first-order transition occurs
regardless of $x$ value.

##  II Model and Formalism

Approximating the two-spin interaction in 1 in terms of interaction with MFs,
valid for spatial dimension larger than upper critical dimension, one can
obtain the GL free energy, which allows further analysis, numerical and
analytical, for the QR-Potts model. We introduce the average probability of a
spin to be in the first visible state, the other visible states, and hidden
states as

$\displaystyle\langle\delta_{s_{i},\alpha}\rangle=\left\\{\begin{array}[]{lll}\mu&~{}~{}\textrm{for}{}&~{}~{}\alpha=1,\cr\nu_{1}&~{}~{}\textrm{for}{}&~{}~{}\alpha=2,\cdots,q,\cr\nu_{2}&~{}~{}\textrm{for}{}&~{}~{}\alpha=q+1,\cdots,q+r,\end{array}\right.$
(5)

respectively, which will play the role of the MFs. Notice that
$\mu+(q-1)\nu_{1}$ and $r\nu_{2}$ are the probability that a spin is in
visible and hidden states. They satisfy $\mu+(q-1)\nu_{1}+r\nu_{2}=1$,
allowing us to represent $\nu_{2}$ in terms of $\mu$ and $\nu_{1}$.

The partition function,
$\mathcal{Z}\equiv\sum_{\\{s_{i}\\}}e^{-\beta\mathcal{H}}$, where $\beta\equiv
1/k_{B}T$ with the Boltzmann constant $k_{B}$, can be decomposed as
$\mathcal{Z}\simeq\sum_{\mu,\nu_{1}}\mathcal{Z}_{\mu,\nu_{1}}$ with the
constrained partition function
$\mathcal{Z}_{\mu,\nu_{1}}\equiv\sum_{\\{s_{i}\\}}^{\prime}e^{-\beta\mathcal{H}}$,
where the prime indicates that the summation runs under the constraint.
Neglecting the quadratic fluctuations such as
$(\delta_{s_{i},\alpha}-\langle\delta_{s_{i},\alpha}\rangle)(\delta_{s_{j},\alpha}-\langle\delta_{s_{j},\alpha}\rangle)$,
one obtains the effective MF Hamiltonian

$\displaystyle-\beta\mathcal{H}^{\textrm{mf}}_{\mu,\nu_{1}}=-K\sum_{\langle
i,j\rangle}\left(\mu^{2}+\sum_{\alpha=2}^{q}\nu_{1}^{2}\right)$ (6)
$\displaystyle\hskip
20.0pt+K\sum_{i}\sum_{j\in\text{n.n.}(i)}\left((h+\mu)\delta_{s_{i},1}+\sum_{\alpha=2}^{q}\nu_{1}\delta_{s_{i},\alpha}\right),$
(7)

where $K\equiv\beta J=1/T$ (we set $J/K_{B}\equiv 1$), $\text{n.n.}(i)$ is the
set of the nearest neighbors of site $i$, and $h\equiv H/(Jz)$ is the re-
scaled external field. We assume that every site has the same number $z$ of
the nearest neighbors. The free energy $f_{\mu,\nu_{1}}^{\rm
mf}\equiv-N^{-1}\ln\mathcal{Z}_{\mu,\nu_{1}}^{\textrm{mf}}$ is obtained from
the partition function
$\mathcal{Z}_{\mu,\nu_{1}}^{\textrm{mf}}=\sum_{\\{s_{i}\\}}e^{-\beta\mathcal{H}^{\textrm{mf}}_{\mu,\nu_{1}}}$
in the MF limit.

Our main interest covers two issues: (i) whether spins are more likely to
align in the direction of the first state, and (ii) how likely a spin is in
hidden states as a function of inverse temperature $K$. Therefore we consider

$\displaystyle m\equiv\mu-\nu_{1}\text{ and }m_{r}\equiv r\nu_{2}$ (8)

instead of $\mu$ and $\nu_{1}$. $m$ is close to $1$ if a spin is most likely
to be in the first state and $0$ if it is equally likely in all the visible
states or most likely in hidden states. $m_{r}$ distinguishes the latter two
cases, representing the probability of being in hidden states. Rewriting
$f_{\mu,\nu_{1}}^{\rm mf}$ as a function of $m$ and $m_{r}$, we obtain the GL
free energy

$\displaystyle
f(m,m_{r},h)\simeq\dfrac{1}{2\tau}\left\\{(1-m_{r})^{2}+(q-1)m^{2}\right\\}-\ln\mathcal{X},$
(9)

where,

$\displaystyle\mathcal{X}=e^{(h+1-m_{r}+(q-1)m)/\tau}+(q-1)e^{(1-m_{r}-m)/\tau}+r.$
(10)

Here, we introduce a re-scaled temperature $\tau\equiv{q/Kz}={qT/z}$ and call
it temperature hereafter. We set $h\to 0^{+}$ unless stated otherwise and
denote the GL free energy simply by $f(m,m_{r})$.

In the thermodynamic limit $N\to\infty$, the GL free energy of the whole
system in the MF limit can be approximated by
$\bar{f}\equiv-N^{-1}\ln\mathcal{Z}\simeq\textrm{min}_{m,m_{r}}f(m,m_{r})$ and
the global minimum location gives the order parameters
$(\bar{m},\bar{m}_{r})=\arg\textrm{min}_{m,m_{r}}f(m,m_{r})$. They satisfy
${\partial f}/{\partial m}\big{|}_{\bar{m},\bar{m}_{r}}={\partial f}/{\partial
m_{r}}\big{|}_{\bar{m},\bar{m}_{r}}=0$, yielding

$\displaystyle\bar{m}={{e^{q\bar{m}/\tau}-1}\over{e^{q\bar{m}/\tau}+q-1+re^{-(1-\bar{m}-\bar{m}_{r})/\tau}}},$
(11)

and

$\displaystyle\bar{m}_{r}={re^{-(1-\bar{m}-\bar{m}_{r})/\tau}\over
e^{q\bar{m}/\tau}+q-1+re^{-(1-\bar{m}-\bar{m}_{r})/\tau}}.$ (12)

Practically, we first obtain $m_{r}(m)$ from 12 and utilize it in 9 to define
a single-variable function $f(m)\equiv f(m,m_{r}(m))$. Then, the order
parameters are given by

$\bar{m}=\arg\textrm{min}_{m}f(m)\ {\rm and}\ \bar{m}_{r}=m_{r}(\bar{m}).$
(13)

Also, the expansion of $f(m)$ near $m=0$ can offer an analytic understanding
of the order parameters. We first expand 12 with respect to $m$ to obtain
$m_{r}(m)=m_{r}(0)+\sum_{n=2}^{\infty}B_{n}\left(\dfrac{qm}{\tau}\right)^{n}$
with $B_{n}$’s constants and inserting it into 9, one finds

$\displaystyle f(m)=\sum_{n=0}^{\infty}C_{n}\left(qm\over\tau\right)^{n}.$
(14)

The coefficients $B_{n}$’s and $C_{n}$’s are in Supporting information (SI.
7-6. Hereafter, we drop the bar notation in $\bar{m}$ and $\bar{m}_{r}$ for
simplicity.

Figure 2:  Plot of order parameters $m$ (left axis) and $m_{r}$ (right axis)
vs temperature $T=z\tau/q$ where $z=4$ for all the cases: (a)–(e) for
$1<q=1.5<2$, and (a) $x<x_{\ell}$, (b) $x=x_{\ell}$, (c) $x_{\ell}<x<x_{h}$,
(d) $x=x_{h}$, and (e) $x>x_{h}$. (f)–(h) at $q=2$ and for (f) $x<x_{c}$, (g)
$x=x_{c}$, and (h) $x>x_{c}$. (i) For $q=2.5>2$ and $x>0$. Continuous
transitions occur in (a)–(c), (f), and (g). Discontinuous transitions occur in
(c)–(e), (h), and (i). A hybrid transition occurs in (d). Note that in (b),
another type of continuous transition occurs at $T_{c}^{\prime}$ in which
$\partial^{2}f(m,T)/\partial m^{2}\to 0$. This implies $\partial m/\partial
T\to\infty$.

Figure 3:  Plots of free energy densities $f(m)$ vs $m$ for several cases of
$1<q=1.5<2$, where $f(m)$ is reset to zero when $m=0$ for illustration. (a) A
continuous transition occurs in Regime I with $x=3.0<x_{\ell}$. At $T_{c}$,
$\partial^{2}f(m)/\partial m^{2}=0$ at $m=0$. As $T$ decreases below $T_{c}$,
the minimum of $f(m)$ decreases continuously. Thus, a continuous transition
occurs. (b) At $x=x_{\ell}\approx 4.0$, global minimum of $f(m)$ occurs near
$m=0$ at $T_{c}$ for second-order transition. As $T$ decreases and reaches
$T_{c}^{\prime}$, position $m$ of global minimum increases rapidly but
continuously. Order parameter increases rapidly, as presented in Fig. 2(b).
(c) When $x_{\ell}=4.1<x<x_{h}$, global minimum of $f(m)$ occurs at $m=0$ at
$T_{c}$ for second-order transition. As $T$ decreases below $T_{c}$, minimum
position $m_{1}$ slowly increases and becomes finite value $m_{1}$. Order
parameter increases continuously. As $T$ decreases further, another minimum
occurs at $m_{2}$. At $T_{f}$, $f(m_{1})=f(m_{2})$ and discontinuous
transition occurs at $m_{2}$. Order parameter jumps from $m_{1}$ to $m_{2}$.
(d) At $x=x_{h}\approx 4.685$, $T_{c}=T_{f}$. The first and second derivatives
of $f(m)$ at $m=m_{1}=0$ are zero. So, susceptibility diverges at $T_{c}$. At
the same temperature, another minimum occurs at $m_{2}$ as
$f(m_{1})=f(m_{2})$. Thus, the discontinuous transition occurs. Hybrid
transition occurs. This point is referred to as CE. (e) For $x=5.20>x_{h}$,
discontinuous transition occurs at $T_{f}$.

##  III Phase Transition for visible states

In this section, we utilize both the numerical solutions [ 13] and the
analytic expansion [ 14] with the coefficients given in Supplementary
Information (SI) 7-6 to trace the global minimum of $f(m)$ varying with
$\tau,q,$ and $x$ to obtain the phase diagram [Fig. 1]. The change of the sign
of $C_{3}$ at $q=2$ [SI. 7-6] is responsible for different types of phase
transitions between $q<2$ and $q>2$ in the Q-Potts model ($r=0$) [2] and also
in the QR-Potts model. Therefore, we consider the cases of $q>2$, $1<q<2$, and
$q=2$.

###  III.1 $q>2$: Discontinuous transition

When $q$ is close to and larger than $2$, the fourth-order expansion $f(m)$ in
14 with $C_{3}\propto-(q-2)$ and $C_{2},C_{4}>0$ can approximate well $f(m)$
and reveal the change of the global minimum at temperature $\tau_{f}$ [Fig.
2(a)], which is given for small $x$ by $\tau_{f}\simeq\tau_{f}^{(0)}-{x\over
e}$ with $\tau_{f}^{(0)}\simeq 1+{(q-2)^{2}\over 6(q-1)}$ and the order
parameter

${m}\simeq\begin{cases}0&\ {\rm for}\ \tau>\tau_{f},\\\
\tau_{f}\dfrac{q-2}{q-1}\left(1+{3x\over 2e}\right)&\ {\rm for}\
\tau<\tau_{f}.\end{cases}$ (15)

The transition temperature $\tau_{f}$ decreases as $x$ increases because the
energetic effect, favoring the ordered state, can dominate the entropic
effect, favoring hidden states only at low temperatures as $x$ increases.

###  III.2 $1<q<2$: Continuous, successive, hybrid, and discontinuous
transitions

When $1<q<2$, $C_{3}>0$ but $C_{4}$ varies with $x$, which underlies different
types of phase transitions occurring in three regimes, denoted as I for
$x<x_{\ell}(q)$, II for $x_{\ell}(q)<x<x_{h}(q)$, and III for $x>x_{h}(q)$
with the CP $x_{\ell}(q)$ and at the CE $x_{h}(q)$ [Fig. 1(c)].

$-$ Continuous transition: In Regimes I and II ($x<x_{h}$), the global minimum
of $f(m)$ at $m_{1}$ is shifted continuously from zero to an infinitesimally
positive value as temperature decreases passing the critical temperature
$\tau_{c}$ [Figs. 2(b) and (d), and 3(a) and (c)]. It is related to the change
of the sign of $C_{2}$ at $\tau_{c}$, leading to $\tau_{c}={1\over 1+xe^{-1}}$
for $x<e^{2}$. Near $\tau_{c}$, $m$ is zero or quite small, and one can employ
the third-order expansion in 14 with $C_{2}\propto\tau-\tau_{c}$ to find that

${m}\simeq\begin{cases}0&\ {\rm for}\quad\tau>\tau_{c},\\\
\dfrac{2}{2-q}\dfrac{\tau_{c}-\tau}{\tau_{c}}&\ {\rm
for}\quad\tau<\tau_{c}.\end{cases}$ (16)

$-$ Discontinuous transition: In Regime II and III ($x>x_{\ell}$), $C_{3}$ is
positive but $C_{4}$ is negative such that the free energy $f(m)$ increases
with $m$ owing to $C_{3}>0$ and then decreases owing to $C_{4}<0$ and
increases again because of positive higher-order terms, resulting in two local
minima at $m_{1}$ and $m_{2}$ and the change of the global minimum between
them at a temperature $\tau_{f}$ [Fig. 3(c)-(e)]. The discontinuous
transitions in these regimes are presented in Fig. 2(d) (II) and Fig. 2(f)
(III).

Because a continuous transition also occurs in Regime II, one finds double
transitions in Regime II [Fig. 1(c)]: As the temperature decreases, the order
parameter $m$ changes from zero to an infinitesimally positive value at
$\tau_{c}$ and then discontinuously jumps to a finite positive value ($m_{2}$)
at $\tau_{f}$, which is less than $\tau_{c}$.

In Regime II, $\tau_{c}$ is higher than $\tau_{f}$. As $x$ increases, however,
$\tau_{c}$ decreases faster than $\tau_{f}$, and therefore, they meet at
$x_{h}$, called the CE. For $x>x_{h}$ (Regime III), the continuous increment
of the local minimum $m_{1}$ at the temperature $\tau_{c}$ less than
$\tau_{f}$ does not affect the global minimum maintained at $m_{2}$.
Therefore, one can see just a discontinuous transition at $\tau_{f}$.

$-$ Explosive hybrid transition at CP ($x_{\ell}$): At $x=x_{\ell}$ (CP), no
discontinuous jump is seen [Fig. 2(c)], but as the temperature is further
lowered than $\tau_{c}$, the order parameter $m$ increases continuously but
explosively [Fig. 2(c)] near the characteristic temperature
$\tau_{c}^{\prime}$, i.e, $dm/d\tau|_{\tau_{c}^{\prime}}\to\infty$. As
temperature is near $\tau_{c}^{\prime}$, $m$ increases continuously as

$\displaystyle m-m_{c}^{\prime}\sim(\tau_{c}^{\prime}-\tau)^{\beta^{\prime}}$
(17)

with $\beta^{\prime}=1/2$. The detailed derivation is presented in SI. 8-6.
Hence, a hybrid transition occurs. At $m_{c}^{\prime}$, $\partial f/\partial
m=\partial^{2}f/\partial^{2}m=0$ as shown in Fig. 3(b). Therefore, critical
behavior appears at $(\tau_{c}^{\prime},m_{c}^{\prime})$.

$-$ Hybrid transition at CE ($x_{h}$): At $x=x_{h}$ (CE), the change of the
global minimum from $m_{1}=0$ to $m_{2}>0$ and the gradual shift of the former
local minimum from $m_{1}=0$ to finite $m_{1}$ coincide at $\tau_{f}=\tau_{c}$
[Figs. 2(e) and 3(d)]. Interestingly, the susceptibility $\chi\equiv{\partial
m\over\partial h}$ is finite in the side $\tau\to\tau_{c}^{-}$ but diverges in
the other side $\tau\to\tau_{c}^{+}$ in Fig. S1 in SI. 9-6. It implies that
the transition is hybrid or mixed-order, exhibiting the properties of
continuous and discontinuous transitions simultaneously [53, 54, 55, 56]. We
reveal that $\partial f/\partial m=0$ at $m=m_{1}=0$ and $m=m_{2}>0$ and
$\partial^{2}f/\partial^{2}m=0$ only at $m=m_{1}=0$ as shown in Fig. 3(d).
Therefore, critical behavior appears at $(\tau_{c}^{+},0)$.

###  III.3 $q=2$: Continuous and discontinuous transitions

When $q=2$, the CP and CE lines are merged at $x_{\rm TP}$. See Fig. 1(b) and
SI. 10-6. The free energy is expanded as $f(m)\simeq
C_{2}(qm/\tau)^{2}+C_{4}(qm/\tau)^{4}+C_{6}(qm/\tau)^{6}$. When $x<x_{\rm
TP}$, $C_{4}>0$, and a single minimum at $m_{1}=0$ is shifted continuously to
finite $m_{1}$ at temperature near $\tau_{c}$ [Fig. 2(g)]. Because
$C_{2}\sim\tau-\tau_{c}$, one finds $m\sim(\tau_{c}-\tau)^{1/2}$ for
$\tau<\tau_{c}$, with the critical exponent $\beta=1/2$ different from
$\beta=1$ in 16 for $1<q<2$. When $x>x_{\rm TP}$, $C_{2}>0$ but $C_{4}<0$, and
thus $f(m)$ can have multiple minima over a wide range of temperatures. A
discontinuous transition occurs at $\tau_{f}$ [Fig. 2(i)], which is
approximately $\tau_{c}+{C_{4}^{2}\over 4C_{6}}$ when the finite jump $\Delta
m\simeq\sqrt{|C_{4}|\over C_{6}}$ at $\tau_{f}$ is small. Therefore, one finds
that $\tau_{c}=\tau_{f}$ and $\Delta m=0$ if $C_{4}=0$, which holds at
$x=x_{\rm TP}={2e/3}$ and $\tau=\tau_{\rm TP}={3/5}$ [SI. 10-6], implying that
it is a TP. At $x_{\rm TP}$, the free energy is given by $f(m)\simeq
C_{2}(qm/\tau)^{2}+C_{6}(qm/\tau)^{6}$, and therefore $m$ indicates a
continuous transition at $\tau_{c}$ [Fig. 2(h)] as

${m}\sim\begin{cases}0&\ {\rm for}\quad\tau>\tau_{\rm TP},\\\ \left(\tau_{\rm
TP}-\tau\right)^{1/4}&\ {\rm for}\quad\tau<\tau_{\rm TP}.\end{cases}$ (18)

Near TP, the CP and CE lines exist, exhibiting a super-criticality:
$x_{\ell}-x_{\rm TP}\sim(2-q)^{2/3}$ along the CP line and $x_{h}-x_{\rm
TP}\sim(2-q)^{2/3}$ along the CE line. The sixth-order expansion of $f(m)$ is
adopted to derive these results as presented in SI. 10-6. Therefore, we find
that the gap $\Delta x\equiv x_{h}-x_{\ell}$ shrinks as $\Delta
x\sim(2-q)^{2/3}$. Note that this gap is not the discontinuity of the order
parameter as revealed in the conventional critical behavior, such as the
liquid–gas transition, but the discontinuity between the two transition lines
$x_{\ell}$ and $x_{h}$. Moreover, this novel supercritical behavior appears in
the AT model with the same critical exponent value. Interestingly, the CP and
CE lines in the AT model with $q$ states (Q-AT) on the random graph also
exhibit the supercritical behavior with the exponent $2/3$, suggesting a
super-universality of the QR-potts and Q-AT model in the MF limit.

The phase diagram of the QR-Potts model spanned by $(q,x=r/q,1/T)$ is similar
to that of the AT model on scale-free (SF) networks [38, 39]. The phase
diagrams in Fig. 1(b) and (c) appear similar to Fig. 2(a) in Ref.[38],
commonly displaying the CE and CP lines. The parameter space of the AT model
on SF networks is spanned by $(\lambda,x_{\rm AT},1/T)$. $\lambda$ is the
degree exponent of the SF network and $x_{\rm AT}$ is the intra- and inter-
type interaction ratio. From the perspective of the phase diagram,
$(\lambda,x_{\rm AT})$ corresponds to ($q,x=r/q$) in the QR-Potts model. The
correspondence between ($\lambda,x_{\rm AT}$) and ($q,x$) seems non-trivial.
However, it is clear that quantities ($\lambda,x_{\rm AT}$) affect both the
interaction energy and entropy to change the free energy landscape as ($q,x$)
in the QR-Potts model.

##  IV Phase Transition for the hidden states in $q\to 1^{+}$

Figure 4:  (a)–(c) Plots of $m_{r}(0)\equiv y$ vs $T$ for various values of
$x$ but with fixed $q=1$ $\in(1,2)$:(a) when $x<e^{2}$, (b) when $x=e^{2}$,
and when $x>e^{2}$. (d)–(f) Plots of ad hoc potential vs $y$ for various
values of $x$: (d) when $x<e^{2}$, (e) $x=e^{2}$, and (f) $x>e^{2}$. The green
(dark) circle point in (b) is located at $\tau_{c}^{\prime}$, at which
$d^{2}f(m_{r})/dm_{r}^{2}|_{m_{r}=m_{r}^{*}}=0$
($\tau_{c}^{\prime}=m_{r}^{*}(1-m_{r}^{*})$). Hence, the potential is flat at
$m_{r}^{*}(\tau_{c}^{\prime})$. This point corresponds to the TP of a
discontinuous percolation in the limit $q\to 1$.

The Kasteleyn-Fortuin mapping [4, 5] relates the Q-Potts model to the bond
percolation in the limit $q\to 1$. It is still valid for the QR-Potts model.
In the QR-Potts model with $q\to 1^{+}$, a discontinuous transition of $m$
occurs for $x>x_{1\ell}=e^{2}$ [Fig. 1(c)]. With such large $x$, there can be
multiple solutions $m_{r}$ of 12, resulting in a discontinuous jump of
$m_{r}$, which can drive the discontinuous transition of $m$ for $q\to 1^{+}$.

Consider the GL free energy in 9 with $q\to 1$,

$\displaystyle f(m,m_{r})\to f(m_{r})={m_{r}^{2}-1\over
2\tau}-\ln(q+re^{-{1-m_{r}\over\tau}}).$ (19)

Its global minimum location $\bar{m}_{r}=\arg\textrm{min}_{m_{r}}f(m_{r})$
satisfies $\partial f(m_{r})/\partial m_{r}|_{m_{r}=\bar{m}_{r}}\equiv
f_{m_{r}}(m_{r})|_{m_{r}=\bar{m}_{r}}=0$ or equivalently

$\bar{m}_{r}={xe^{-(1-\bar{m}_{r})/\tau}\over
1+xe^{-{(1-\bar{m}_{r})/\tau}}}.$ (20)

If $x<x_{1\ell}=e^{2}$, $f(m_{r})$ has a single minimum at $\bar{m}_{r}$ for
all $\tau$ [Fig. 4(a)]. If $x>x_{1\ell}$, two local minima $\bar{m}_{r-}$ and
$\bar{m}_{r+}$ exist for $\tau_{-}<\tau<\tau_{+}$ [Fig. 4(c)]. Notice that at
$x_{1\ell}=e^{2}$, $f_{m_{r}}=f_{m_{r}m_{r}}=f_{m_{r}m_{r}m_{r}}=0$ at
$m_{r}={1\over 2}$ and $\tau={1\over 4}$ [Fig. 4(b)]. The range of temperature
having both $\bar{m}_{r-}$ and $\bar{m}_{r+}$ can be obtained by using the
condition $f_{m_{r}}=f_{m_{r}m_{r}}=0$ at $m_{r}=\bar{m}_{r\pm}$ and
$\tau=\tau_{\mp}$ yielding $\tau_{\mp}=\bar{m}_{r\pm}(1-\bar{m}_{r\pm})$ and
20 for given $x$. When $x$ is large, $\bar{m}_{r-}\sim 1/\ln(x\ln x)$ and
$\bar{m}_{r+}\sim 1-e/x$ and therefore is $m_{r}$ behaves as

$m_{r}\simeq\begin{cases}{x}e^{-{1}\over{\tau}}&\ {\rm for}\
\tau\ll\tau_{f}^{\prime},\\\ {1\over\ln\left({x}\ln x\right)}&\ {\rm for}\
\tau\to\tau_{f}^{\prime-},\\\ 1-{e\over x}&\ {\rm for}\
\tau>\tau_{f}^{\prime},\end{cases}$ (21)

where $\tau_{f}^{\prime}$ is the temperature located between $\tau_{-}$ and
$\tau_{+}$ [SI. 11-6].

When $q\to 1^{+}$, 12 is independent of $m$, indicating that the solution of
20 can be utilized in place of $m_{r}$ in 11 to determine $m$ as

$\displaystyle m=1-e^{-m/\tau}-mxe^{-(1-m_{r})/\tau}.$ (22)

If $r=0$, then 22 reduces to $m=1-e^{-m/\tau}$, known for the bond percolation
in random graphs [8]. From 22, one can obtain the features of bond percolation
transition for the QR-Potts model. Consequently, in case $m_{r}$ is
discontinuous, $m(r\neq 0)$ exhibits a discontinuous transition.

##  V Discussions: Potential applications to social opinion formation

The QR-Potts model renders intriguing insights into the interplay between
conformity and individualism in social dynamics and opinion formation. The QR-
Potts model explores how individuals in a social network choose between
following the majority opinion and keeping their ideas hidden. The model
reveals that this competition can lead to a sudden emergence of consensus or
the prevalence of hidden opinions under certain conditions. Suppose every
agent has two options: following the majority opinion or selecting a random
opinion. The former option contributes to the free energy as much as
$qe^{\beta J}$, whereas the latter option incurs a free energy cost of $q+r$.
Thus, there exists a characteristic temperature $T_{\circ}\approx
1/\ln(1+r/q)$ such that when the strength of social fluctuation $T$ is lower
(higher) than $T_{\circ}$, each agent should take the ordered (hidden)
opinion. If the number of hidden states $r$ is much larger than unity, then
the characteristic temperature $T_{\circ}$ is much lower than $1$. This
implies that even at relatively low temperatures, agents are prone to hide
their opinions rather than follow to the majority.

When the system contains many hidden states ($x>e^{2}$), the order parameter
$m_{r}(m)$ exhibits bi-stable states [SI. 12-6], including a hidden super-
cooling state between them. As we know, in the liquid–gas transition, the
system remains in a meta-stable state for an extended period. This implies
that an unstable, highly disordered social system, akin to a hidden super-
cooling state, can endure significant social suppression. This resilience is
particularly striking given high suppression factors, such as authoritarian
regimes (in the limit $q\to 1^{+}$) or oppressive political conditions.

Comparing swing agents with zealots [57, 58, 59, 60, 61, 62] in the context of
opinion dynamics provides valuable insights into how two different types of
agents affect the dynamics of opinion propagation. swing agents and zealots
can be mirror images of each other, with one group actively influencing others
and the other group remaining passive and unresponsive. Developing an opinion
dynamics model that accommodates both swing agents and zealots can offer a
better understanding of how diverse agent behaviors shape the evolution of
opinions within a society.

Finally, we note that shy voters change their states because of thermal
fluctuations as noise-induced updating in the noisy voter model [61, 62, 63,
64, 65], where an opinion changes spontaneously and independently of the
neighbors’ state. However, the shy voter in our model affects the entropy
change of the system, followed by the change of the free energy landscape.
Therefore, the state updating in the shy voters cannot be regarded as a
finite-size effect. In the continuum limit, the shy voters do not behave
randomly; however, they make a new local minimum of the system in the
thermodynamic limit. Recently, a variant of the voter model has been shown to
display a change in entropy production [35], leading to a change of entropy
and free energy landscape.

##  VI Summary

We studied the phase transition of the QR-Potts model comprising $q$ visible
states and $r$ hidden states. The Hamiltonian includes the interactions
between two spins in visible states but no interaction with a spin in hidden
states. Applying the Ginzburg–Landau formalism in the MF limit, we obtained a
rich phase diagram including a variety of phase transitions, such as
continuous, discontinuous, and hybrid (or mixed-order) transitions, and
diverse kinds of transition points such as CP, CE, and TP in the parameter
space comprising temperature $T$, the number of visible states $q$, and the
ratio $x=r/q$. This phase diagram is close to that of the AT model, implying
that the two types of spin states in the QR-Potts model play the role of two
layers of different spin species in the AT model.

The phase transition of the QR-Potts model highly depends on $q$ and $x$. We
analytically uncovered diverse features of the phase transitions. When $q>2$,
a discontinuous transition occurs regardless of the value of $x$. When
$1<q<2$, the transition is continuous in Regime I ($x<x_{\ell}$) while it is
discontinuous in Regime III ($x>x_{h}$). Between the two regimes,
($x_{\ell}<x<x_{h}$) called Regime II, a continuous and then a discontinuous
transition occurs consecutively as $T$ decreases. Along the curve,
$x_{\ell}(q)$ for $1\leq q<2$, the order parameter increases continuously with
decreasing $T$ lower than $T_{c}$, but its derivative with respect to
temperature diverges at $T_{c}^{\prime}$. As temperature is near
$T_{c}^{\prime}$, the order parameter behaves
$m-m_{c}\prime\sim(T_{c}^{\prime}-T)^{\beta^{\prime}}$. So, the singular
behavior of the hybrid transition occurs near $m_{c}>0$. On the other hand,
along the CE curve at $x_{h}(q)$ for $1\leq q<2$, another type of hybrid
transition occurs, in which the order parameter is discontinuous, but the
susceptibility $\partial m/\partial h$ diverges at $m=0$. Note that the hybrid
transition at the CP line can be found in other non-equilibrium models such as
$k$-core percolation [66] and the hybrid percolations in cluster merging
process [67, 68]. On the other hand, the hybrid transition at the CE line can
be found in the epidemic model [69]. Thus, our results integrate such
scattered previous results.

The two characteristic transition lines are merged at $x_{\rm TP}$ and
$q_{c}=2$. We revealed that the two transition lines exhibit a singular
behavior $x_{h}-x_{\ell}(q)\sim(q_{c}-q)^{2/3}$. In percolation limit ($q\to
1^{+}$), the discontinuous transition for $x>x_{1\ell}=e^{2}$ is induced by
the bi-stability of the hidden spin states, whereas in Ising limit ($q\to 2$),
the discontinuous transition for $x_{\ell}<x<e^{2}$ is induced by the
competition between the interaction and entropy of the system.

This paper highlights how introducing hidden states to the Potts model can
lead to a more versatile and powerful tool for modeling a wide range of
complex phenomena in theoretical and practical contexts, including those
related to non-equilibrium systems and political elections. This research
could significantly improve our understanding of real-world systems and make
more accurate predictions or decisions in various fields.

###### Acknowledgements.

We would like to thank Prof. János Kertész for introducing this problem and
Prof. Maxi San Miguel for discussing voter models. This work was supported by
the National Research Foundation of Korea by Grant No. NRF-2014R1A3A2-069005
and RS-2023-00279802 and the KENTECH Research Grant No. KRG-2021-01-007 (BK),
and a KIAS Individual Grant(No. CG079902) from Korea Institute for Advanced
Study (DSL).

Summplementary Information for
Entropy-Induced Phase Transitions in a Hidden Potts Model

Supplemental Information for
Entropy-Induced Phase Transitions in a Hidden Potts Model

##  7-6 Coefficients of $B_{n}$ and $C_{n}$ in the free energy expansion

We first expand $m_{r}(m)$, the solution to (12), for small $m$ as
$m_{r}(m)=m_{r}(0)+\sum_{n=2}^{\infty}B_{n}\left(\dfrac{qm}{\tau}\right)^{n}$
with the coefficients given by

$\displaystyle B_{2}$ $\displaystyle=-{(q-1)m_{r}(0)(1-m_{r}(0))\over
2q^{2}\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}},$ (S1)
$\displaystyle B_{3}$ $\displaystyle=-{(q-1)(q-2)m_{r}(0)(1-m_{r}(0))\over
6q^{3}\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}},$ (S2)
$\displaystyle B_{4}$ $\displaystyle=-{(q-1)m_{r}(0)(1-m_{r}(0))\over
24q^{4}\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}}\left[q^{2}-6q+6-{3(q-1)(1-2m_{r}(0))\over\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}^{2}}\right],$
(S3) $\displaystyle B_{5}$
$\displaystyle=-{(q-1)(q-2)m_{r}(0)(1-m_{r}(0))\over
120q^{5}\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}}\left[q^{2}-12q+12-{10(q-1)(1-2m_{r}(0))\over\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}^{2}}\right],$
(S4) $\displaystyle B_{6}$ $\displaystyle=-{(q-1)m_{r}(0)(1-m_{r}(0))\over
720q^{6}\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}}\left[q^{4}-30q^{3}+150q^{2}-240q+120+{45(1-q)^{2}(1-4\tau)\over\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}^{4}}\right.$
(S5) $\displaystyle\hskip
75.0pt\left.-{30(1-q)^{2}(1-9\tau)\over\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}^{3}}-{5(q-1)\\{18(q-1)\tau+(1-2m_{r}(0))(5q^{2}-26q+26)\\}\over\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}^{2}}\right].$
(S6)

Here $m_{r}(0)$ can be obtained by solving (12) with $m=0$. Inserting this
expansion into (11), one can obtain (14) with the coefficients given by

$\displaystyle C_{0}$ $\displaystyle={1\over
2\tau}(1-m_{r}(0))^{2}-\ln\left({r\over m_{r}(0)}\right),$ (S7) $\displaystyle
C_{1}$ $\displaystyle=0,$ (S8) $\displaystyle C_{2}$ $\displaystyle={q-1\over
2q^{2}}\left\\{{\tau}-1+m_{r}(0)\right\\},$ (S9) $\displaystyle C_{3}$
$\displaystyle=-{(q-1)(q-2)(1-m_{r}(0))\over 6q^{3}},$ (S10) $\displaystyle
C_{4}$ $\displaystyle=-{(q-1)(1-m_{r}(0))\over
24q^{4}}\left[q^{2}-6q+6+{3(q-1)m_{r}(0)\over
1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)}\right],$ (S11) $\displaystyle
C_{5}$ $\displaystyle=-{(q-1)(q-2)(1-m_{r}(0))\over
120q^{5}}\left[q^{2}-12q+12+{10(q-1)m_{r}(0)\over
1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)}\right],$ (S12) $\displaystyle
C_{6}$ $\displaystyle=-{(q-1)(1-m_{r}(0))\over
720q^{6}}\left[q^{4}-30q^{3}+150q^{2}-240q+120-15{\tau}{(1-q)^{2}(2-m_{r}(0)/\tau)\over\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}^{3}}\right.$
(S13) $\displaystyle\left.\hskip
150.0pt+30{\tau}{(1-q)^{2}\over\\{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)\\}^{2}}+5{(q-1)(5q^{2}-26q+26)m_{r}(0)\over
1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)}\right].$ (S14)

##  8-6 Derivation of $\beta^{\prime}$ at CP

The order parameter $m^{*}(\tau)$ is the value of $m$ where the free energy
$f(m,\tau)$ is minimized for given temperature $\tau$, as given in (13),
satisfying ${\partial f\over\partial m}|_{m^{*}}=0$. Its derivative
${dm^{*}\over d\tau}$ with respect to temperature $\tau$ can be represented in
terms of the second-order derivative of $f$ as

$\displaystyle{dm^{*}\over d\tau}=-{{\partial^{2}f\over\partial
m\partial\tau}\over{\partial^{2}f\over\partial m^{2}}}\bigg{|}_{m^{*}(\tau)},$
(S15)

because

$\displaystyle{d\over d\tau}\left({\partial f\over\partial
m}\bigg{|}_{m^{*}(\tau)}\right)={\partial^{2}f\over\partial
m^{2}}\bigg{|}_{m^{*}}{dm^{*}\over d\tau}+{\partial^{2}f\over\partial
m\partial\tau}\bigg{|}_{m^{*}}=0.$ (S16)

At CP ($x=x_{\ell}$ and $\tau=\tau_{c}^{\prime}$), the derivative
${dm^{*}\over d\tau}$ diverges; i.e. $dm^{*}/d\tau\to\infty$, and so both the
first- and second-order derivative of the free energy $f(m,\tau)$ to $m$ are
zero there, i.e., ${\partial f\over\partial m}\bigg{|}_{\textrm{CP}}={\partial
f^{2}\over\partial m^{2}}\bigg{|}_{\textrm{CP}}=0$. Let us denote the order
parameter at CP by $m_{c}^{\prime}$. Near CP, the free energy $f$ will be
expanded around $m_{c}^{\prime}$ and $\tau_{c}^{\prime}$ as

$\displaystyle f(m$ $\displaystyle=m_{c}^{\prime}+\delta
m,\tau=\tau_{c}^{\prime}+\delta\tau)=f(m_{c}^{\prime},\tau_{c}^{\prime})+\dfrac{\partial
f}{\partial\tau}\bigg{|}_{\textrm{CP}}\delta\tau+\dfrac{\partial^{2}f}{\partial
m\partial\tau}\bigg{|}_{\textrm{CP}}\delta\tau\delta
m+\dfrac{\partial^{3}f}{\partial
m^{3}}\bigg{|}_{\textrm{CP}}\dfrac{1}{6}\delta m^{3}+\textrm{ H.O.},$ (S17)

with $\delta\tau=\tau-\tau_{c}^{\prime}$ and $\delta m=m-m_{c}^{\prime}$.
Differentiating with respect to $\delta m$, we obtain

$\displaystyle\dfrac{\partial f}{\partial
m}\simeq\dfrac{\partial^{2}f}{\partial
m\partial\tau}\bigg{|}_{\textrm{CP}}\delta\tau+\dfrac{\partial^{3}f}{\partial
m^{3}}\bigg{|}_{\textrm{CP}}\dfrac{1}{2}\delta m^{2}+\textrm{ H.O.}$ (S18)

which can be zero at $m^{*}$ if $\delta\tau\sim-\delta m^{2}$. Therefore, one
can see that $m^{*}-m_{c}^{\prime}\sim(\tau_{c}^{\prime}-\tau)^{1/2}$, which
indicates $\beta^{\prime}=1/2$ at CP in (17).

Figure S1:  Plots of the susceptibility $\chi_{m}$ vs $T(=z\tau/q)$ for $z=4$
and $1<q=1.5<2$. (a) When $x=3.0<x_{\ell}$, $\chi_{m}$ diverges; (b) when
$x=x_{\ell}\approx 4.1$, $\chi_{m}$ as evidence of the second-order
transition, this is a figure showing how the susceptibility $\chi_{m}$ changes
with temperature $T$ according to various $r$ values. (a) For $x=3.0$,
$\chi_{m}$ diverges to $\infty$ at $T_{c\pm}$. (b) For $r=4.1$, $\chi_{m}$
diverges to $\infty$ at $T_{c\pm}$ and also shows discontinuous jump at
$T_{f}$ in which $m$ so does. (c) For $x=4.685$, $\chi_{m}$ shows
discontinuous jump at $T\rightarrow T_{c-}$ ($m$ abruptly jumps to $0$) and
divergence at $T\rightarrow T_{c+}$. This is why we call this type of PT the
HPT. (d) For $x=5.2$, $\chi_{m}$ shows discontinuous jump at $T_{f}(>T_{c})$.

##  9-6 Susceptibility

For a non-zero external field, the order parameter $m^{*}(h)$ can be
determined by searching for the value of $m$ minimizing the free energy
$f(m,h)\equiv f(m,\tilde{m}_{r}(m),h)$ with $\tilde{m}_{r}(m)$ the solution to
(12) for given $m$ and $h$. With $f(m,h)$, one can characterize the response
of the order parameter (magnetization) to an external field (magnetic field)
by the susceptibility defined as

$\displaystyle\chi_{m}={\partial m^{*}(h)\over\partial h}\bigg{|}_{h=0}.$
(S19)

From (12) one can obtain
$\tilde{m}_{r}(m,h)=\tilde{m}_{r}(0,h)+\sum_{n=1}^{\infty}\tilde{B}_{n}(h)\left(\dfrac{qm}{\tau}\right)^{n}$
where $\tilde{m}_{r}(0,h)$ is given as

$\displaystyle\tilde{m}_{r}(0,h)$ $\displaystyle=m_{r}(0)+{h\over
q}\dfrac{m_{r}(0)(1-m_{r}(0))}{1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)}+O(h^{2})$
(S20)

up to the linear order of $h$ and $\tilde{B}_{n}(h)=B_{n}+O(h)$. Inserting
$\tilde{m}_{r}(m,h)$ into (9), we obtain

$\displaystyle
f(m,H)=f(m,m_{r}(m,h),h)=\sum_{n=0}^{\infty}\tilde{C}_{n}(h)\left(\dfrac{qm}{\tau}\right)^{n}$
(S21)

with the coefficients given by

$\displaystyle\tilde{C}_{0}$ $\displaystyle=C_{0}$ (S22)
$\displaystyle\tilde{C}_{1}$ $\displaystyle=-{h\over
q^{2}}(q-1)(1-m_{r}(0))+O(h^{2})$ (S23) $\displaystyle\tilde{C}_{2}$
$\displaystyle=C_{2}-{h\over 2q^{3}}(q-1)(q-2)(1-m_{r}(0))+O(h^{2})$ (S24)
$\displaystyle\tilde{C}_{3}$ $\displaystyle=C_{3}-{h\over
6q^{4}}(q-1)(1-m_{r}(0))\left[q^{2}-6q+6+{3(q-1)m_{r}(0)\over
1-{1\over\tau}m_{r}(0)\left(1-m_{r}(0)\right)}\right]+O(h^{2})$ (S25)

Since $f(m,h)$ is minimized at $m^{*}(h)$, it holds that $\partial f/\partial
m=0$ at $m^{*}(h)$. Considering that ${d\over dh}\left({\partial
f\over\partial m}|_{m^{*}(h)}\right)=f_{mm}{dm^{*}\over dh}+f_{mh}=0$ and
using the expansion of the free energy in the above, we can represent the
susceptibility near $\tau_{c}$ for $1<q<2$ as follows:

$\displaystyle\chi_{m}$
$\displaystyle=-\dfrac{f_{mh}}{f_{mm}}\bigg{|}_{m=m^{*}(0),h=0}\simeq{\tau}{(q-1)(1-m_{r}(0))\over
q^{3}\left\\{2C_{2}+6C_{3}(m/\tau)\right\\}},$ (S26)

where

$\displaystyle f_{mh}\equiv\frac{\partial^{2}f(m,h)}{\partial m\partial
h}\quad{\textrm{and}}\quad f_{mm}\equiv\frac{\partial^{2}f(m,h)}{\partial
m^{2}}.$ (S27)

Using (16), we obtain that for small $x$,

$\displaystyle\chi_{m}$
$\displaystyle\simeq\begin{cases}\dfrac{\left(1-m_{r}(0)\right)\tau}{q^{2}\\{1-(1-m_{r}(0))/\tau\\}}\sim\dfrac{1}{\tau-\tau_{c}}&\
{\rm for}\ \tau>\tau_{c}\\\ \\\
\dfrac{-\left(1-m_{r}(0)\right)\tau}{q^{2}\\{1-(1-m_{r}(0))/\tau\\}}\sim\dfrac{1}{\tau_{c}-\tau}&\
{\rm for}\ \tau<\tau_{c},\end{cases}$ (S28)

for the continuous transition. At the CE point ($x=x_{h}$ and
$\tau=\tau_{c}$), $m$ shows a discontinuous jump at $\tau_{c}^{-}$, and
therefore, the susceptibility behaves as

$\displaystyle\chi_{m}$
$\displaystyle\simeq\begin{cases}\dfrac{\left(1-m_{r}(0)\right)\tau}{q^{2}\\{1-(1-m_{r}(0))/\tau\\}}\sim\dfrac{1}{\tau-\tau_{c}}&\
{\rm for}\ \tau>\tau_{c}\\\ \\\ {\rm constant}&\ {\rm for}\
\tau<\tau_{c}\end{cases}$ (S29)

as shown in Fig. 4(c). Accordingly, the critical exponent $\gamma_{m}^{+}=1$.
Then, the scaling relation $\alpha+2\beta_{m}+\gamma_{m}^{+}=2$ holds (see
Table 1).

Range of $q$ | $\alpha_{m}$ | $\alpha_{m_{r}}$ | $\beta_{m}$ | $\beta_{m_{r}}$ | $\gamma_{m\pm}$ | $\gamma_{{m_{r}}\pm}$
---|---|---|---|---|---|---
$1\leq q<2$ | $-1$ | $-1$ | $1$ | $2$ | $1$ | $0$
$q=2,r<r_{c}$ | $0$ | $0$ | $\frac{1}{2}$ | $1$ | $1$ | $\frac{1}{2}$
$q=2,r=r_{c}$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{1}{4}$ | $\frac{1}{2}$ | $1$ | $\frac{3}{4}$
Table 1:  Critical exponents for various types of CPT: Here, $\alpha$ is the
exponent of the specific heat, $\beta_{m}$ ($\beta_{R}$) is the exponent of
the magnetization $m$ ($R$) at zero external magnetic fields, and $\gamma_{m}$
($\gamma_{R}$) is the exponent of the susceptibility for $m$
($R$)-magnetization near the transition temperature.

##  10-6 Phase transitions and supercritical behavior near TP

In this section, we study the phase transitions at and near the TP considered
in Sec. III.3 shown in Fig. 1(b). Let us first consider the case of $q=2$.
Then $C_{3}=C_{5}=0$ from (S14) and one can represent the free energy as

$\displaystyle f(m)\simeq
C_{2}\left(\dfrac{qm}{\tau}\right)^{2}+C_{4}\left(\dfrac{qm}{\tau}\right)^{4}+C_{6}\left(\dfrac{qm}{\tau}\right)^{6}.$
(S30)

The two coefficients $C_{2}$ and $C_{4}$ become zero when $x=x_{\rm TP}$ and
$\tau=\tau_{\rm TP}$ with

$\displaystyle x_{\rm TP}={2e\over 3},\,\tau_{\rm TP}={3\over 5}\ {\rm and}\
m_{r}(0,\tau_{\rm TP}).$ (S31)

At $x_{\rm TP}$, using (12), both $\epsilon_{\tau}\equiv\tau-\tau_{\rm TP}$
and $\epsilon_{r}\equiv m_{r}(0,\tau)-m_{r}(0,\tau_{\rm TP})$ have the same
order of magnitude, and thus within the first order of $\epsilon_{\tau}$, we
obtain the following:

$\displaystyle C_{2}\propto\epsilon_{\tau},\ \,\ C_{4}\propto\epsilon_{\tau},\
\,\ \textrm{ and }\ \,\ C_{6}\propto\mathcal{O}(1).$ (S32)

When $\tau\to\tau_{\rm TP}$, because the order of two coefficients $C_{2}$ and
$C_{4}$ are the same, the fourth-order term can be ignored compared with
$C_{2}$. When $C_{2}>0$, the global minimum locates at $m=0$, while $C_{2}<0$,
and $C_{6}>0$ generate the global minimum at finite $m$ as

$\displaystyle m\sim\begin{cases}0&\ {\rm for}\quad\tau>\tau_{\rm TP},\\\
(-\epsilon_{\tau})^{1/4}&\ {\rm for}\quad\tau<\tau_{\rm TP}.\end{cases}$ (S33)

This result is equivalent to (18).

As $q$ and $x$ deviate slightly from $q_{c}=2$ and $x_{\rm TP}$ along with the
CP and CE lines, $C_{3}$ and $C_{4}$ terms increase gradually and contribute
to the local minimum of $f(m)$, together with $C_{2}$ and $C_{6}$ terms. This
implies that the four terms would have the same order of magnitudes. Moreover,
using $C_{6}\sim\mathcal{O}(1)$, we presume that $C_{2}=\mathcal{O}(m^{4})$,
$C_{3}=\mathcal{O}(m^{3})$, and $C_{4}=\mathcal{O}(m^{2})$. From (S14),
$C_{4}\sim\epsilon_{\tau}$ and $C_{3}\sim(2-q)=\epsilon_{q}$, and thus
$\epsilon_{\tau}\sim\mathcal{O}(m^{2})$ and
$\epsilon_{q}\sim\mathcal{O}(m^{3})$. On the other hand,
$C_{2}\approx\epsilon_{\tau}+\epsilon_{r}$, and hence,
$\epsilon_{r}=-\epsilon_{\tau}+\mathcal{O}(m^{4})$. Using (12),
$\mathcal{O}(\epsilon_{x})=\mathcal{O}(\epsilon_{\tau})=\mathcal{O}(m^{2})$.
Therefore, we obtain that
$\mathcal{O}(\epsilon_{x})=\mathcal{O}(\epsilon_{q}^{2/3})$ on the CP and CE
lines.

$\displaystyle x_{\ell}-x_{\rm TP}\sim(q_{c}-q)^{2/3}\ {\rm and}\ x_{h}-x_{\rm
TP}\sim(q_{c}-q)^{2/3}.$ (S34)

##  11-6 Phase transitions of the order parameter in hidden states
in the $q\to 1^{+}$ limit

To obtain $m_{r}$ in the $q\to 1^{+}$ limit, we consider the constrained free
energy in (9) with $m$ set to zero,

$\displaystyle f(0,m_{r})\equiv f(m_{r})={m_{r}^{2}-1\over
2\tau}-\ln(q+re^{-{1-m_{r}\over\tau}}),$ (S35)

and the global minimum location

$\displaystyle\bar{m}_{r}=\arg\textrm{min}_{m_{r}}f(m_{r}).$ (S36)

It should satisfies $f_{m_{r}}(m_{r})|_{m_{r}=\bar{m}_{r}}=0$, which is
equivalent to (12) with $m=0$ or

$\displaystyle\bar{m}_{r}={xe^{-(1-\bar{m}_{r})/\tau}\over
1+xe^{-{(1-\bar{m}_{r})/\tau}}}.$ (S37)

Also, the global minimum condition requires

$\displaystyle
f_{m_{r}m_{r}}(m_{r})\bigg{|}_{m_{r}=\bar{m}_{r}}={1\over\tau}\left[1-{\bar{m}_{r}(1-\bar{m}_{r})\over\tau}\right]>0.$
(S38)

The solution $\bar{m}_{r}$ of (20) implies the following: ($x=0$,
$\bar{m}_{r}=0$) is the solution of the original Q-Potts model representing
the disordered (or paramagnetic) phase. For the QR-Potts model.

(i) When $0<x\ll 1$, $xe^{(1-m_{r})/\tau}$ is small in the entire range of
$m_{r}$ and $\tau$. Therefore, $m_{r}$ is also always small compared with
$\mathcal{O}(1)$. The free energy exhibits a single global minimum, and only
one stable solution $\bar{m}_{r}$ appears near zero as $\bar{m}_{r}\approx
xe^{-(1-\bar{m}_{r})/\tau}\approx xe^{-1/\tau}$. As shown in Fig. S2(a),
$m_{r}$ increases gradually with $\tau$ increasing.

(ii) As $x$ is increased further and arrives at $e^{2}$, there exists
$\bar{m}_{r}$ such that

$\displaystyle
f_{m_{r}}(m_{r})|_{m_{r}=\bar{m}_{r}}=f_{m_{r}m_{r}}(m_{r})|_{m_{r}=\bar{m}_{r}}=0.$
(S39)

This is a transition point corresponding to CP. Similar to $q\to 2^{-}$ case,
this critical point can be obtained from the condition
$f_{m_{r}m_{r}}(m_{r})|_{\tau=\tau_{c}^{\prime}}=0$. Therefore,
$\tau_{c}^{\prime}$ is obtained as follows:

$\displaystyle\tau_{c}^{\prime}=\bar{m}_{r}(x,\tau_{c}^{\prime})(1-\bar{m}_{r}(x,\tau_{c}^{\prime}))=\dfrac{1}{4}.$
(S40)

At $\tau_{c}^{\prime}$, $m_{r}$ increases most rapidly as shown in Fig. S2(b),
because $dm_{r}/d\tau\propto(f_{m_{r}m_{r}}(m_{r}))^{-1}\to\infty$ diverges at
$\tau_{c}^{\prime}$. As a result, $x_{1\ell}$ ($x_{\ell}$ for $q=1$) is
$e^{2}$ as shown in Fig. 1(b).

(iii) When $x>e^{2}$, there exist two solutions $\bar{m}_{r-}$ and
$\bar{m}_{r+}$ which satisfy the conditions $f_{m_{r}}(m_{r})=0$ and
$f_{m_{r}m_{r}}(m_{r})>0$. When $\tau$ is relative low (high),
$f(\bar{m}_{r-})$ is smaller (larger) than $f(\bar{m}_{r+})$. So the global
minimum occurs at $\bar{m}_{r-}$ and $\bar{m}_{r+}$ for lower and higher
$\tau$, respectively. As the position of the global minimum changes from
$\bar{m}_{r-}$ to $\bar{m}_{r+}$ at $\tau_{f}$, the order parameter
$\bar{m}_{r}$ shows a discontinuous jump, and thus the first-order transition
occurs.

We consider several extreme cases for which $m_{r}$ is analytically obtained.
In case (i), when $x\equiv r/q\ll 1$, using Eq. (20), we obtain

$\displaystyle m_{r}\simeq xe^{-1/\tau}$ (S41)

for all $\tau$. In case (iii), when $x>e^{2}$, Eq. (20) has two solutions
$\bar{m}_{r-}$ and $\bar{m}_{r+}$ when $\tau$ is between two characteristic
$\tau_{-}$ and $\tau_{+}$. (see Fig. S2(c)). At $\tau_{-}$, $\bar{m}_{r+}$
appears as a local (not global yet) minimum of $f(m_{r})$. Thus,

$\displaystyle
f_{m_{r}}(m_{r})|_{\bar{m}_{r+}}=f_{m_{r}m_{r}}(m_{r})|_{\bar{m}_{r+}}=0$
(S42) $\displaystyle\tau_{-}=\bar{m}_{r+}(1-\bar{m}_{r+}),\text{ with
}\bar{m}_{r+}=\dfrac{xe^{-{1/\bar{m}_{r+}}}}{1+xe^{-{1/\bar{m}_{r+}}}}.$ (S43)

At $\tau_{+}$, the smaller solution $\bar{m}_{r-}$ disappears, and thus the
larger solution $\bar{m}_{r-}$ becomes a unique local minimum of $f(m_{r})$.
Thus,

$\displaystyle
f_{m_{r}}(m_{r})|_{\bar{m}_{r-}}=f_{m_{r}m_{r}}(m_{r})|_{\bar{m}_{r-}}=0$
(S44)

Using this relation, we can also find that

$\displaystyle\tau_{+}=\bar{m}_{r-}(1-\bar{m}_{r-}),\ {\rm with}\
\bar{m}_{r-}=\dfrac{xe^{-{1/\bar{m}_{r-}}}}{1+xe^{-{1/\bar{m}_{r-}}}}.$ (S45)

From Eqs. (S43) and (S45), we find that $\bar{m}_{r-}$ and $\bar{m}_{r+}$ are
obtained as follows:

$\displaystyle\bar{m}_{r-}\simeq{1\over\ln\left({x}\ln x\right)}\ {\rm
and}\quad\bar{m}_{r+}\simeq 1-\dfrac{e}{x}$ (S46)

and we can obtain (21).

Figure S2:  (a) Plots of the order parameters $m$ (left scale) and $m_{r}(0)$
(right scale) vs $T=z\tau/q$ for $1<q<2$ and $x>e^{2}$. The order parameter
$m$ ($m_{r}(0)=0$) decreases (increases) continuously from $m=1$ ($m_{r}(0)$)
as $T$ is increased. At $T_{f}$, they exhibit a discontinuous drop to zero
(jump) and then stay at zero (continuously increases). (b)–(g) Plot of the
order parameter $m_{r}(m)$ that the system is in hidden states vs $m$ for
$q=1.5$ and $x=8>e^{2}$. (b) For $\tau=0.62<\tau_{c}^{\ell}$, only one
$m_{r}(m)$ (lower) curve appears in system. (c) For
$\tau_{c}^{\ell}<\tau=0.64<\tau_{c}^{h}$, another $m_{r}(m)$ curve emerges
above the lower $m_{r}(m)$ curve in $m\in(0,m_{\text{up}})$. (d) For
$\tau_{c}^{h}<\tau=0.65<\tau_{f}^{\prime\prime}$, the lower $m_{r}(m)$ curve
separates from $m=0$ line ($y$-axis), and so exists only in
$m\in(m_{\text{lo}},1)$. (e) For $\tau=0.66=\tau_{f}^{\prime\prime}$, two
disconnected $m_{r}(m)$ curves begin to merge into one curve. (f) For
$\tau_{f}^{\prime\prime}<\tau=0.75<\tau_{f}$, only one $m_{r}(m)$ curve
appears in the system. However, the value of $m_{r}(m)$ is much larger than
that of case (b). (g) For $\tau=0.9>\tau_{f}$, a position of $m$ and $m_{r}$
on $m_{r}(m)$ curve jump from an ordered state to a disordered state.

##  12-6 Bi-stability of the order parameter in hidden states with large
$x>e^{2}$

We investigate the behavior of $m_{r}(m)$ as a function of $\tau$, but $x$ is
fixed in the regime $x>e^{2}$. As one compares Fig. S2(b) with (c), a new
$m_{r}(m)$ curve appears at $m=0$ as $\tau\to\tau_{c}^{\ell}$. This emergence
implies that a discontinuous transition of $m_{r}(m)$ occurs at $m=0$ as
$\tau$ increases. As $\tau$ increases beyond $\tau_{c}^{\ell}$, the upper
curve of $m_{r}(m)$ is somewhat extended to $m_{\text{up}}$ as shown in Fig.
S2(c). As $\tau\to\tau_{c}^{h}$, the lower curve is detached from a vertical
line of $m=0$, and so, the lower curve spans from $m_{\text{lo}}$ to $1$. Note
that $m_{\text{lo}}$ is less than $m_{\text{up}}$ for
$\tau_{c}^{h}<\tau<\tau_{f}^{\prime\prime}$ as shown in Fig. S2(d). At
$\tau_{f}^{\prime\prime}$, $m_{\text{up}}$ and $m_{\text{lo}}$ meet as shown
in Fig. S2(e). Finally, when $\tau$ is higher than $\tau_{f}^{\prime\prime}$,
two curves are merged into one curve as shown in Fig. S2(f), and thus, the
local minimum with a high degree of $m_{r}$ state disappears. From this
behavior of $m_{r}$ curves, we find that the meta-stable state sustains in a
range of $\tau_{c}^{\ell}<\tau<\tau_{f}^{\prime\prime}$ only for $x>e^{2}$.

## References

  * [1] R. B. Potts, Some Generalized Order-Disorder Transformations, Math. Proc. Camb Philos. Soc. 48, 106 (1952).
  * [2] F. Y. Wu, The Potts model. Rev. Mod. Phys. 54, 235 (1982).
  * [3] D.-S. Lee, K. I. Goh, B. Kahng, D. Kim, Evolution of scale-free random graphs: Potts model formulation, Nuclear Physics B 696 (3), 351 (2004).
  * [4] P. W. Kasteleyn and C. M. Fortuin, Phase Transitions in Lattice Systems with Random Local Properties, J. Phys. Soc. Japan 26, 11 (1969)
  * [5] P. W. Kasteleyn and C. M. Fortuin, On the random-cluster model: I. Introduction and relation to other models. Physica 57, 536 (1972).
  * [6] M. R. Giri, M. J. Stephen, and G. S. Grest, Spin models and cluster distributions for bond and site percolation models, Phys. Rev. B 16, 4971 (1977)
  * [7] F. Y. Wu, Percolation and the Potts model, J Stat Phys 18, 115 (1978).
  * [8] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64, 026118 (2001).
  * [9] G. Bianconi and A.-L. Barabási, Bose-Einstein Condensation in Complex Networks, Phys. Rev. Lett. 86, 5632 (2001)
  * [10] S. G. Brush, History of the Lenz-Ising Model, Rev. Mod. Phys. 39, 883 (1967).
  * [11] G. Bianconi, Mean field solution of the Ising model on a Barabási–Albert network, Physics Letters A 303, 166 (2002).
  * [12] S. H. Lee, M. Ha, H. Jeong, J. D. Noh, and H. Park, Critical behavior of the Ising model in annealed scale-free networks, Phys. Rev. E 80, 051127 (2009).
  * [13] F. Battiston, A. Cairoli, V. Nicosia, A. Baule, and V. Latora, Interplay between consensus and coherence in a model of interacting opinions, Physica D 323, 12 (2016).
  * [14] C. P. Herrero, Ising model in small-world networks, Phys. Rev. E 65, 066110 (2002).
  * [15] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Ising model on networks with an arbitrary distribution of connections, Phys. Rev. E 66, 016104 (2002).
  * [16] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Potts model on complex networks, Eur. Phys. J. B 38, 177 (2004).
  * [17] S. Redner, Reality-inspired voter models: A mini-review, Comptes Rendus Physique, 20, 275 (2019).
  * [18] P. Clifford and A. Sudbury, A Model for Spatial Conflict, Biometrika 60, 581 (1973).
  * [19] R. A. Holley and T. M. Liggett, Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model, Ann. Probab. 3, 643 (1975).
  * [20] J. T. Cox and G. David, Occupation Time Limit Theorems for the Voter Model, Ann. Probab. 11, 876 (1983).
  * [21] T. M. Liggett, Interacting Particle Systems (Springer New York, NY, 1985).
  * [22] T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, (Springer Berlin, Heidelberg, 1999).
  * [23] T. M. Liggett, Coexistence in Threshold Voter Models, Ann. Probab. 22, 764 (1994).
  * [24] T. M. Liggett, Stochastic Models of Interacting Systems, Ann. Probab. 25, 1 (1997).
  * [25] A. Halu, K. Zhao, A. Baronchelli, and G. Bianconi, Connect and win: The role of social networks in political elections, Europhys. Lett. 102, 16002 (2013).
  * [26] N. Masuda, Voter model on the two-clique graph, Phys. Rev. E 90, 012802 (2014).
  * [27] M. Diakonova, M. San Miguel, and V. M. Eguíluz, Absorbing and shattered fragmentation transitions in multilayer coevolution, Phys. Rev. E 89, 062818 (2014).
  * [28] M. Diakonova, V. Nicosia, V. Latora, and M. San Miguel, Irreducibility of multilayer network dynamics: the case of the voter model, New J. Phys. 18, 023010 (2016).
  * [29] A. Chmiel and K. Sznajd-Weron, Phase transitions in the $q$-voter model with noise on a duplex clique, Phys. Rev. E 92, 052812 (2015).
  * [30] A. Chmiel, J. Sienkiewicz, A. Fronczak, and P. Fronczak, A Veritable Zoology of Successive Phase Transitions in the Asymmetric q-Voter Model on Multiplex Networks, Entropy 22, 1018 (2020).
  * [31] C. Castellano, Miguel A. Muñoz, and Romualdo Pastor-Satorras, Non-linear $q$-voter model, Phys. Rev. E 80, 041129 (2009).
  * [32] A. R. Vieira and C. Anteneodo, Threshold $q$-voter model, Phys. Rev. E 97, 052106 (2018).
  * [33] R. Durrett, Richard, H. Kesten, Random walks, Brownian motion, and interacting particle systems, (Birkhäuser Boston, MA, 1991)
  * [34] J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
  * [35] I. V. G. Oliveira, C. Wang, G. Dong, R. Du, C. E. Fiore, H. E. Stanley, and A. L. M. Vilela, Entropy Production on Cooperative Opinion Dynamics, arXiv:2311.05803.
  * [36] J. Ashkin and E. Teller, Statistics of Two-Dimensional Lattices with Four Components, Phys. Rev. 64, 178 (1943).
  * [37] M. Kohmoto, M. den Nijs, and L. P. Kadanoff, Hamiltonian studies of the $d=2$ Ashkin-Teller model, Phys. Rev. B 24, 5229 (1981)
  * [38] S. Jang, J. S. Lee, S. Hwang, and B. Kahng, Ashkin-Teller model and diverse opinion phase transitions on multiplex networks, Phys. Rev. E 92, 022110 (2015).
  * [39] C. H. Kim, M. Jo, J. S. Lee, G. Bianconi, and B. Kahng, Link overlap influences opinion dynamics on multiplex networks of Ashkin-Teller spins, Phys. Rev. E 104, 064304 (2021).
  * [40] D. Lee, S. Choi, M. Stippinger, J. Kertész, and B. Kahng, Hybrid phase transition into an absorbing state: Percolation and avalanches, Phys. Rev. E, 93, 042109 (2016).
  * [41] A. Blais, S. L. St‐Vincent, Personality traits, political attitudes and the propensity to vote, Euro. J. Polit. R. bf 50, 395 (2011).
  * [42] A. Gallego, D. Oberski, Personality and Political Participation: The Mediation Hypothesis, Polit. Behav. 34, 425 (2012).
  * [43] R. Tamura and N. Kawashima, First-Order Transition to Incommensurate Phase with Broken Lattice Rotation Symmetry in Frustrated Heisenberg Model, J. Phys. Soc. Jpn. 77, 103002 (2008).
  * [44] R. Tamura, S. Tanaka and N Kawashima, Phase transition in Potts model with hidden states, Prog. Theor. Phys. 124, 381 (2010).
  * [45] S. Okumura, H. Kawamura, T. Okubo, and Y. Motome, Novel spin-liquid states in the frustrated Heisenberg antiferromagnet on the honeycomb lattice, J. Phys. Soc. Jpn. 79, 114705 (2010).
  * [46] S. Tanaka, R. Tamura, and N. Kawashima, Phase Transition of Generalized Ferromagnetic Potts Model - Effect of hidden States -, J. Phys.: Conf. Ser. 297, 012022 (2011).
  * [47] E. M. Stoudenmire, S. Trebst, and L. Balents, Quadrupolar correlations and spin freezing in $S=1$ triangular lattice antiferromagnets, Phys. Rev. B 79, 214436 (2009).
  * [48] M. Krasnytska, P Sarkanych, B Berche, Y. Holovatch, and R. Kenna, Marginal dimensions of the Potts model with hidden states, J. Phys. A: Math. Theor. 49, 255001 (2016).
  * [49] J. Wajnflasz, Study of the low spin and high spin transition in octahedral transition ion complexes, J. Phys. Status Solidi 40, 537 (1970).
  * [50] S. Doniach, Thermodynamic fluctuations in phospholipid bilayers, J. Chem. Phys. 68, 4912 (1978).
  * [51] M. Sweeny, Monte Carlo study of weighted percolation clusters relevant to the Potts models, Phys. Rev. B. 27, 4445 (1983).
  * [52] T. A. Larsson, Cluster-weighted percolation-a novel approach to the Potts model, J. Phys. A: Math. Gen. 19, 2383 (1986).
  * [53] Y.S. Cho, J. S. Lee, H.J. Herrmann, and B. Kahng, Hybrid percolation transition in cluster merging processes: Continuously varying exponents, Phys. Rev. Lett. 116, 025701 (2016).
  * [54] K. Choi, D. Lee, Y. S. Cho, J. C. Thiele, H. J. Herrmann, and B. Kahng, Critical phenomena of a hybrid phase transition in cluster merging dynamics, Phys. Rev. E 96, 042148 (2017).
  * [55] B. Jhun, M. Jo, and B. Kahng, Simplicial SI. model in scale-free uniform hypergraph, J. Stat. Mech, 123207 (2019).
  * [56] B. Jhun, M. Jo, and B. Kahng, Quantum contact process on scale-free networks, Chaos Solitons & Fractals 160, 112262 (2022).
  * [57] M. Mobilia, Does a single zealot affect an infinite group of voters?, Phys. Rev. Lett. 91, 028701 (2003).
  * [58] M. Mobilia, A. Petersen, and S. Redner. On the role of zealotry in the voter model, J. Stat. Mech, 08029 (2007).
  * [59] M. Mobilia, Nonlinear q-voter model with inflexible zealots, Phys. Rev. E 92, 012803 (2015).
  * [60] A. Mellor, M. Mobilia, and R. K. P. Zia. Heterogeneous out-of-equilibrium nonlinear q-voter model with zealotry, Phys. Rev. E 95, 012014 (2017).
  * [61] Khalil, Nagi, M. S. Miguel, and R. Toral. Zealots in the mean-field noisy voter model, Phys. Rev. E 97, 012310 (2018).
  * [62] Khalil, Nagi, and T. Galla. Zealots in multistate noisy voter models, Phys. Rev. E 103, 012311 (2021).
  * [63] A. Carro, R. Toral, and M. S. Miguel, The noisy voter model on complex networks. Sci Rep 6, 24775 (2016).
  * [64] A. F. Peralta, A. Carro, M. S. Miguel, and R. Toral, Stochastic pair approximation treatment of the noisy voter model, New J. Phys. 20, 103045 (2018).
  * [65] O. Artime, A. Carro, A. F. Peralta, J. J. Ramasco, M. S. Miguel, and R. Toral, Herding and idiosyncratic choices: Nonlinearity and aging-induced transitions in the noisy voter model, Comptes Rendus Physique 20, 262 (2019).
  * [66] D. Lee, M. Jo, and B. Kahng, Critical behavior of $k$-core percolation: Numerical studies, Phys. Rev. E 94, 062307 (2016).
  * [67] J. Park, S. Yi, and B. Kahng, Interevent time distribution, burst, and hybrid percolation transition, Chaos 29, 091102 (2019).
  * [68] H. Choi, Y. S. Cho, R. D’Souza, and J. Kertész, and B. Kahng, Unified framework for hybrid percolation transitions based on microscopic dynamics, arXiv:2307.03584.
  * [69] W. Choi, D. Lee, and B. Kahng, Critical behavior of a two-contagion model with multiple seeds, Phys. Rev. E 95, 062115 (2017).