Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network
We study the critical behaviour of the q-state Potts model on an uncorrelated scale-free network having a power-law node degree distribution with a decay exponent λ. Previous data show that the phase diagram of the model in the q,λ plane in the second order phase transition regime contains three reg...
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irk-123456789-1535612019-06-15T01:27:22Z Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network Krasnytska, M. We study the critical behaviour of the q-state Potts model on an uncorrelated scale-free network having a power-law node degree distribution with a decay exponent λ. Previous data show that the phase diagram of the model in the q,λ plane in the second order phase transition regime contains three regions, each being characterized by a different set of critical exponents. In this paper we complete these results by finding analytic expressions for the scaling functions and critical amplitude ratios in the above mentioned regions. Similar to the previously found critical exponents, the scaling functions and amplitude ratios appear to be λ-dependent. In this way, we give a comprehensive description of the critical behaviour in a new universality class. Ми вивчаємо критичну поведiнку q-станової моделi Поттса на нескорельованiй безмасштабнiй мережi зi степеновою функцiєю розподiлу за ступенем вузлiв з показником загасання λ. Попереднi результати показали, що фазова дiаграма моделi в площинi q, λ в режимi фазового переходу другого роду мiстить три областi, кожна з яких характеризується рiзним набором критичних показникiв. У данiй роботi ми доповнимо цi результати знайшовши аналiтичнi представлення скейлiнгових функцiй та спiввiдношення критичних амплiтуд у згаданих вище областях. Як i для знайдених ранiше критичних показникiв, виявляється, що скейлiнговi функцiї та спiввiдношення амплiтуд є λ-залежними. Таким чином, ми подаємо повний опис критичної поведiнки у новому класi унiверсальностi. 2014 Article Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network / M. Krasnytska // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23602:1-10. — Бібліогр.: 38 назв. — англ. 1607-324X DOI:10.5488/CMP.17.23602 arXiv:1402.5232 PACS: 64.60.aq, 64.60.fd, 64.70.qd, 64.60.ah http://dspace.nbuv.gov.ua/handle/123456789/153561 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We study the critical behaviour of the q-state Potts model on an uncorrelated scale-free network having a power-law node degree distribution with a decay exponent λ. Previous data show that the phase diagram of the model in the q,λ plane in the second order phase transition regime contains three regions, each being characterized by a different set of critical exponents. In this paper we complete these results by finding analytic expressions for the scaling functions and critical amplitude ratios in the above mentioned regions. Similar to the previously found critical exponents, the scaling functions and amplitude ratios appear to be λ-dependent. In this way, we give a comprehensive description of the critical behaviour in a new universality class. |
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Krasnytska, M. Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network Condensed Matter Physics |
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Krasnytska, M. |
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Krasnytska, M. |
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Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network |
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Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network |
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Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network |
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Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network |
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Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network |
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scaling functions and amplitude ratios for the potts model on an uncorrelated scale-free network |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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| citation_txt |
Scaling functions and amplitude ratios for the Potts
model on an uncorrelated scale-free network / M. Krasnytska // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23602:1-10. — Бібліогр.: 38 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT krasnytskam scalingfunctionsandamplituderatiosforthepottsmodelonanuncorrelatedscalefreenetwork |
| first_indexed |
2025-07-14T04:39:22Z |
| last_indexed |
2025-07-14T04:39:22Z |
| _version_ |
1837595847022346240 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 2, 23602: 1–10
DOI: 10.5488/CMP.17.23602
http://www.icmp.lviv.ua/journal
Scaling functions and amplitude ratios for the Potts
model on an uncorrelated scale-free network
M. Krasnytska1,2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Institut Jean Lamour, CNRS/UMR 7198, Groupe de Physique Statistique, Universite de Lorraine,
BP 70239, F-54506 Vandœuvre-lés-Nancy Cedex, France
Received February 4, 2014, in final form February 19, 2014
We study the critical behaviour of the q-state Potts model on an uncorrelated scale-free network having a
power-law node degree distribution with a decay exponent λ. Previous data show that the phase diagram of
the model in the q,λ plane in the second order phase transition regime contains three regions, each being
characterized by a different set of critical exponents. In this paper we complete these results by finding analytic
expressions for the scaling functions and critical amplitude ratios in the above mentioned regions. Similar to
the previously found critical exponents, the scaling functions and amplitude ratios appear to be λ-dependent.
In this way, we give a comprehensive description of the critical behaviour in a new universality class.
Key words: Potts model, complex networks, scaling, universality
PACS: 64.60.aq, 64.60.fd, 64.70.qd, 64.60.ah
1. Introduction
The concept of universality plays a fundamental role in the theory of critical phenomena [1–3]. A lot
of systems manifest similar behaviour near the critical point. The universality class does not depend on
the local parameters but on the global ones, i.e., dimensionality, symmetry, nature of interaction, etc. If
several systems are in the same universality class, they share, besides the values of the critical exponents,
identical critical amplitude ratios and scaling functions [4].
The goal of our study is to analyze an universal content of the critical behaviour of the Potts model
on a scale-free network near the critical point, in particular, to quantify it in terms of scaling functions
and universal amplitude ratios. A lot of studies were devoted to the analysis of the critical behaviour of
spin models on complex networks [5]. In this case, the disorder of an underlying structure is modelled in
terms of a random graph. In the present work we will consider the q-state Potts model on uncorrelated
scale-free network having a power-law node degree distribution with exponent λ. Similar to the lattice
systems, for the Potts model on the uncorrelated scale-free networks one may observe either the 1st or
the 2nd order phase transition. However, now the order of the phase transition depends, besides the q
value, on the node degree distribution decay exponent λ [6–8]. The second order phase transition regime
is characterized by power law dependencies of thermodynamic functions as functions of temperature
and magnetic field in the vicinity of critical point. Critical exponents governing this transition depend on
λ, which plays the role of a global variable for models on a network, like the dimension d in the case of a
lattice.
Depending on the particular value of q , the Potts model has been suited to describe various real
and model systems. Besides the Ising model at q = 2, it also describes percolation at q → 1 [9, 10]. The
spanning treelike percolation with a geometric phase transition is described by a zero-state q = 0 Potts
model [11]. Also in q = 0 limit, the Potts model can be used for a description of the Kirchhoff‘s rules via
©M. Krasnytska, 2014 23602-1
http://dx.doi.org/10.5488/CMP.17.23602
http://www.icmp.lviv.ua/journal
M. Krasnytska
the resistor network models [12]. Subsequently, there has been shown the equivalence between the zero-
state Potts model and Abelian sandpile models in case of arbitrary finite graphs [13]. Sandpile models
describe processes in neural networks, fracture, hydrogen bonding in liquid water. Another particular
case of Potts model at q = 1/2 is a spin glass model [14, 15]. The case of 0 É q < 1 is used to describe
gelation and vulcanization processes in branched polymers [16]. Other examples concern the application
of the Potts model for larger values of q . Three-component q = 3 Potts model is used to describe a cubic
ferromagnet with three axes in a diagonal magnetic field [17], an adsorption of 4He atoms on graphite
in two dimensions [18], transition of helium films on graphite substrate [19], etc. The 4-state Potts model
also describes the effect of absorbtion on surfaces [20]. The Potts model at large q is used to simulate the
processes of intercellular adhesion and cancer invasion [21], see also [22].
In this paper we will complete the analysis of the Potts model on an uncorrelated scale-free network
[6] by calculating scaling functions and universal amplitude ratios. Recently, [23] the scaling functions
and universal amplitude ratios were obtained for the Ising model on a scale free network. Here we will
generalize these expressions for the Potts model case.
The structure of this paper is as follows. In the next section we write down the main relations of
the scaling theory, expressions for thermodynamic functions in a scaling form and universal amplitude
ratios. Section 3 is a short overview of our previous work, where the critical behavior of the Potts model
on uncorrelated scale-free network was considered. In particular, it was shown that the phase diagram in
the second order phase transition regime contains three regions, each being characterized by a different
set of critical exponents. In section 4, we complete these results by finding analytic expressions for the
scaling functions and critical amplitude ratios in the above mentioned regions. Similar to the previously
found critical exponents, the scaling functions and amplitude ratios appear to be λ-dependent. In this
way, we give a comprehensive description of the critical behaviour in a new universality class. Analytic
expressions are summarized in table 2. In the last section we summarize the obtained results.
2. Main relations of scaling theory
In this paper we will be interested in the universal features of a system that are manifested in the
vicinity of the critical (i.e., second order phase transition) point. Critical exponents, that govern the
power-law behaviour of different observables near the critical point belong to such characteristics. Tem-
perature driven phase transition into magnetically ordered state being taken for definiteness, one ob-
serves the power-law asymptotics near the critical point T = Tc, h = 0 [4, 24]. In particular, at h = 0 the
(dimensionless) order parameter m, isothermal susceptibility χT , specific heat ch and magnetocaloriccoefficientmT are1 power law functions of τ= |T −Tc|/Tc:
m = B−τβ, χT = Γ±τ−γ, ch = A±
α
τ−α, mT = B±
T τ
−ω at h = 0. (2.1)
Here, indices ± refer to the way the critical temperature is approached, T −Tc → 0±. In turn, directly at
T = Tc (i.e., τ= 0) the following power law field dependencies hold:
m = D−1/δ
c h1/δ, χ= Γch−γc , ch = Ac
αc
h−αc , mT = B c
T h−ωc at τ= 0. (2.2)
The above formulas (2.1), (2.2) introduce critical exponents and critical amplitudes that we are interested
in in this study. Unlike the critical exponents, the critical amplitudes are non-universal, being dependent
on the microscopic features of the system. However, their certain combinations appear to be universal
as well [4]. In particular, in this study we will be interested in the following universal critical amplitude
ratios:
R±
χ = Γ±DcBδ−1
− , R±
c = A±Γ±
αB 2−
, RA = Ac
αc
D−(1+αc)
c B−2/β
− , A+/A− , Γ+/Γ− . (2.3)
1The magnetocaloric coefficient is defined by the mixed derivative of the free energy over magnetic field and temperature,
mT = −T (∂m/∂T )h . It measures the heat released by the system upon an isothermal increase of the magnetic field due to themagnetocaloric effect (see, e.g., [23] and references therein).
23602-2
Scaling functions and amplitude ratios for the Potts model
The above quoted power law scaling in the behaviour of various thermodynamic functions, universal-
ity and scaling relations between critical exponents and amplitude ratios are themanifestations of special
properties of the thermodynamic potential in the vicinity of critical point. In particular, the scaling hy-
pothesis for the Helmholtz free energy F (τ,m) states that this thermodynamic potential is a generalized
homogeneous function [25] and can be written as follows:
F (τ,m) ≈ τ2−α f±(x), (2.4)
with the scaling variable x = m/τβ and scaling function f±(x), signs + and − correspond to T > Tc and
T < Tc, respectively. The principal content of equation (2.4) is that F (τ,m) as a function of two variables
can be mapped onto a single variable scaling function f±(x). It may be shown that all thermodynamic
potentials are generalized homogeneous functions, provided one of them possesses such a property [25].
Based on the expression for the free energy one can also represent the thermodynamic functions in
terms of appropriate scaling functions. In particular, magnetic and entropic equations of state read:
h(m,τ) = τβδH±(x), (2.5)
S(m,τ) = τ1−αS (x), (2.6)
with the scaling functions H±(x) andS (x). In turn, the scaling functions for the heat capacity, isothermal
susceptibility, and magnetocaloric coefficient are defined via (see e.g., [23]):
ch(m,τ) = (1±τ)τ−αC±(x), (2.7)
χT (m,τ) = τ−γχ±(x), (2.8)
mT (m,τ) = (1±τ)τβ−γM±(x). (2.9)
Scaling functions are reachable in experiments and MC simulations. Together with critical exponents
and critical amplitude ratios they constitute quantitative characteristics of a given universality class. In
the rest of this paper we will complete the previous description of the critical behaviour of the Potts
model on an uncorrelated scale-free network by calculating its amplitude ratios and scaling functions in
the vicinity of the second order phase transition.
3. Potts model on an uncorrelated scale-free network
The q-state Potts model can be considered as one of the possible generalizations of the Ising model,
where the spin variable can have q possible states [26]. The Potts model Hamiltonian reads:
−H = 1
2
∑
i , j
Ji jδni ,n j +h
∑
i
δni ,0 , (ni = 0,1, . . . , q −1), (3.1)
here q is the number of Potts states, h is a local external magnetic field directed along the 0-th component
of the Potts variable ni (the Potts state on the node i ). We consider the case where all spins are located
on the nodes of a random graph (complex network) and are connected with each other in an appropriate
way. The latter is determined by the adjacency matrix Ji j with the elements Ji j = 1 if there exists a link
between the nodes i and j and Ji j = 0 otherwise. One of the important characteristics of a network is
its node degree distribution P (k): a probability that the randomly chosen node has a degree (number of
links) k. We will consider the case of Potts model on an uncorrelated scale-free network with a power-law
node degree distribution:
P (k) = cλk−λ, (3.2)
here, cλ is a normalization constant and λ is the exponent of decay. The absence of correlations withinthe given link distribution means that the probability to create a link between two nodes is linearly
proportional to their node degrees. Furthermore, one may consider the case of an annealed network,
when the network configuration is fluctuating under the constraint of a given node degree distribution,
see, e.g., [27, 28]. Alternatively, the links between the nodesmay be randomly distributed but remain fixed
23602-3
M. Krasnytska
Figure 1. (Color online) Phase diagram of the Potts model on an uncorrelated scale-free network. The area
of the second order phase transition is shown. Three regions correspond to three different universality
classes. Logarithmic corrections to scaling appear for the values of q and λ shown by a light line and a
light disc (brown online). See the text for a more detailed discussion.
in a given configuration, the so-called configurational model, see [5, 29]. The latter situation corresponds
to the quenched case and is usually more complicated for analytical treatment.
The critical behaviour of the Potts model on an uncorrelated scale-free network has been considered
in references [6–8]. It was found that the phase diagram of the model is uniquely defined by two param-
eters: the number of Potts states q and the node degree distribution exponent λ. Here, we will complete
the calculations of [6] where a comprehensive list of critical exponents governing the behaviour of ther-
modynamic functions in the second order phase transition regime was found. The results of [6] are exact
for an annealed network and correspond to the mean field treatment of the quenched case. Our starting
point will be the expression for the Helmholtz free energy obtained in reference [6] for different q . For
non-integer λ> 3, the free energy reads:
F (τ, M) = a1τM 2 +a2Mλ−1 +
[λ−1]∑
i=3
ai M i +O(M [λ]) , (3.3)
here, M is magnetization, ai are non-universal coefficients, their explicit form is given in [6] and [λ] is
the integer part of λ. Note that the power law polynomial form (3.3) holds for the Helmholtz potential
for non-integer λ only. Logarithmic corrections appear in the case of integer values of λ. As we discuss
below, this will lead to the changes in the critical behaviour at λ= 4 and λ= 5.
Figure 1 generalizes the information about the critical behavior of the Potts model on an uncorrelated
scale-free network in the form of a phase diagram in the q−λ plane [6, 8]. It has been shown that forλÉ 3,
the system remains in the ordered state for any finite temperature [6–8]. For λ > 3, the phase transition
may be of the first or second order, depending on the specific values of q and λ. The first order phase
transition occurs at the values of q and λ that belong to the blank region above the dashed curve, q > 2,
λ>λc(q). Of the main interest for us will be the second order phase transition regime. This corresponds
to three different regions shown in figure 1 that belong to three different universality classes. Region I,
λ> 5, q = 2 (black solid line in figure 1) is governed by the Ising mean field critical exponents. Region II,
λ > 4, 1 É q < 2 (dotted area in the figure) is governed by the percolation mean field critical exponents.
Region III (3 < λ < 5, q = 2; 3 < λ < 4, 1 É q < 2; 3 < λ É λc(q), q > 2) (dashed area in the figure) is
characterized by the non-trivial λ-dependency of the critical exponents. Values of the critical exponents
in all three regions are collected in table 1.
As it was mentioned above, logarithmic terms appear in the free energy at the integer values of λ. In
turn, this leads to the appearance of logarithmic corrections [33–35] to the power-law scaling dependen-
cies (2.1), (2.2) of thermodynamic functions at λ= 5 for the Ising model (q = 2) and at λ= 4 for 1 É q < 2
23602-4
Scaling functions and amplitude ratios for the Potts model
Table 1. Critical indices of the Pottsmodel on an uncorrelated scale-free network in three different regions
of q and λ values, see figure 1. One recovers the results for the Ising model (q = 2 [30, 31]) and for
percolation (q = 1) [32].
α αc β δ γ γc ω ωcregion I 0 0 1/2 3 1 2/3 1/2 1/3
region II −1 −1/2 1 2 1 1/2 0 0
region III λ−5
λ−3
λ−5
λ−2
1
λ−3 λ−2 1 λ−3
λ−2
λ−4
λ−3
λ−4
λ−2
[6]. Values of q and λ where the thermodynamic functions are governed by power-law singularities en-
hanced by the logarithmic corrections are shown in figure 1 by the light solid line and light disc (brown
online).
In the forthcoming section we will be interested in the critical behaviour in the regions of the second
order phase transition with the power law scaling. In particular, we will complete a quantitative descrip-
tion of three universality classes found in regions I, II and III (see figure 1) by calculating, in addition to
the critical exponent, the scaling functions and amplitude ratios.
4. Critical amplitude ratios and scaling functions
The expression of the free energy of the Potts model on an uncorrelated scale-free network (3.3)
will be a starting point for the analysis of the critical amplitude ratios and scaling functions. Passing
to the dimensionless energy f (m,τ) and dimensionless magnetization m and leaving the leading order
contributions for small values ofm, we can present (3.3) in three different regions of the phase diagram
(figure 1) in the following form:
f (m,τ) =±τ
2
m2 + 1
4
m4, (Region I), (4.1)
f (m,τ) =±τ
2
m2 + 1
4
m3, (Region II), (4.2)
f (m,τ) =±τ
2
m2 + 1
4
mλ−1, (Region III), (4.3)
the signs± here and in what follows refer to the temperatures above and below the critical point Tc. Notethat the positive sign of the second terms in (4.1)–(4.3) is due to the fact that coefficients a2, a3, and a4 in(3.3) are positive definite in the regions III, II, and I, correspondingly. With the expressions for the free
energy at hand it is straightforward to write down the equation of state and to derive the thermodynamic
functions. The magnetic and entropic equations of state in the dimensionless variablesm and τ read:
h(m,τ) = ∂ f (m,τ)
∂m
∣∣∣∣
τ
, s(m,τ) = ∓∂ f (m,τ)
∂τ
∣∣∣∣
m
. (4.4)
Written explicitly in different regions of q , λ the magnetic equation of state attains the following
form:
h = m3 ±τm, (Region I), (4.5)
h = 3
4
m2 ±τm, (Region II), (4.6)
h = λ−1
4
mλ−2 ±τm, (Region III). (4.7)
The entropic equation of state is obtained by a temperature derivative at a constant magnetization m
while the explicit τ-dependency is the same in all expressions (4.1)–(4.3). Therefore, the equation keeps
the same form in all regions on q–λ plane:
s =−m2/2, (Regions I–III). (4.8)
23602-5
M. Krasnytska
Table 2. Scaling functions and critical amplitude ratios for the Potts model on an uncorrelated scale-free
network.
Region I Region II Region III
f±(x) ± x2
2 + x4
4 ± x2
2 + x3
4 ± x2
2 + xλ−1
4
H±(x) x3 ± x 3
4 x2 ± x λ−1
4 xλ−2 ± x
S (x) −x2/2 −x2/2 −x2/2
C±(x) x2
3x2±1
x2
3x/2±1
x2
(λ−1)(λ−2)xλ−3/4±1
χ±(x) 1
3x2 ±1
1
3x/2±1
1
(λ−1)(λ−2)xλ−3/4±1
M±(x) x
3x2±1
x
3x/2±1
x
(λ−1)(λ−2)xλ−3/4±1
A+/A− 0 0 0
Γ+/Γ− 2 1 λ−3
R+
χ 1 1 1
R−
χ
1
2 1 1
λ−3
R+
c 0 0 0
R−
c
1
4 1 1
(λ−3)2
RA
1
3
1
2
1
λ−2
Thermodynamic functions χT , ch , and mT that characterize the response on an external action aredirectly obtained from the above equations of state. We do not present the explicit expressions here,
being rather interested in the corresponding critical amplitude ratios. The latter are given in table 2. In
the particular case q = 2, by these expressions we recover the formerly obtained critical amplitude ratios
for the Ising model on an uncorrelated scale-free network [23, 36], correcting at 3 <λ< 5 the expression
for RA given in [23]2. Together with the previously derived set of critical exponents (see table 1), ourresults for the critical amplitude ratios quantify the universal features of critical behaviour of the Potts
model in the second order phase transition regime.
Let us derive now the scaling functions for the free energy and other thermodynamic functions. Us-
ing the definition (2.4) and taking into account that the heat capacity and the order parameter critical
exponents α, β take on different values in different regions of the phase diagram figure 1 (the formulas
are given in table 1) we can recast Helmholtz potential F (τ,m) in terms of the scaling function f±(m/τβ).
The explicit expressions for the scaling function in all three regions of the phase diagram are given in
table 2. Typical behaviour of the free energy scaling functions f+(x) and f−(x) is shown in figure 2 (a)
and 2 (b), correspondingly.
At any value of q , the scaling functions share a common feature: their curvature gradually increases
with an increase of λ > 3. This happens up to some marginal value λ = λc. The marginal value λc is
q-dependent. For λ > λc and 1 É q É 2, the scaling functions remain unchanged: their shape does not
change with a further increase of λ. The logarithmic corrections to scaling appear at λ = λc and thesecond order phase transition holds in this case for λ > λc as well [6], see figure 1. Alternatively, for
λ > λc and q > 2, the phase transition turns out to be of the first order. Curves I of figure 2 (plotted by
solid lines) show the limiting behaviour of the scaling functions at q = 2, λ> 5 [note that λc(q = 2) = 5]:
the functions remain unchanged for all λ> 5. Similar behaviour holds for the case 1 É q < 2, the value of
λc, however, differs: λc(1 É q < 2) = 4. This is shown by curves II in the figure, plotted by dashed lines.
Finally, curves III (dotted lines) for q = 4 are one of examples of the limiting behaviour of the scaling
functions in the region q > 2.
Entropy scaling function S (x) is defined by (2.6). Using expression (4.8) for the entropy and taking
into account the values for the critical exponents α and β given in table 1 we arrive at the entropy scaling
function that remains unchanged in all regions on q–λ plane: S (x) =−x2/2. There are different ways of
representing the magnetic equation of state. In the Widom-Griffiths representation [37, 38] the magnetic
2In paper [23], using equation (2.3) to find RA the α exponent was substituted instead of αc into the power of Dc.
23602-6
Scaling functions and amplitude ratios for the Potts model
(a) (b)
Figure 2. (Color online) Limiting behaviour of the free energy scaling functions f+ [(a), T > Tc] and f−[(b), T < Tc]. The functions remain unchanged for λ > 5, q = 2 and λ > 4, 1 É q < 2 (solid and dashed
curves I and II, correspondingly). For q > 2 the phase transition turns out to be of the first order at
λ>λc(q). Dotted curves III: q = 4, λ=λc(4) ' 3.5. See the text for more details.
equation of state can be written in two equivalent forms:
h = mδh±
(
τ/m1/β
)
, h = τβδH±
(
m/τβ
)
, (4.9)
with scaling functions h±(x) and H±(x). Alternatively, in Hankey-Stanley representation, the magnetiza-
tion is written as [25]:
m = τβµ±
(
h/τβδ
) (4.10)
with the scaling function µ±(x).
Starting from themagnetic equation of state given in regions I–III by equations (4.5)–(4.7), it is straight-
forward to arrive at the scaling functions H±(x). We give the appropriate expressions in table 2. Subse-
quently, one can easily rewrite these expressions to get appropriate h±-functions. Behaviour of the scal-ing functions µ±(x) for different values of λ and q is shown in figure 3. From the explicit form of the
equation of state it is easy to evaluate the asymptotic behaviour of the scaling functions. For q = 2 and
λ > 5, one gets µ±(x) ∼ x1/3, x → ∞. The functions tend to turn to infinity faster with a decrease of λ:
(a) (b) (c)
Figure 3. (Color online) Behaviour of the order parameter scaling functions µ+(x) (light curves, brown
online) and µ−(x) (black curves) for different values of λ and q . (a): q = 2, (b): 1 É q < 2, (c): q = 4. Values
of λ are shown in the figures.
23602-7
M. Krasnytska
(a) (b) (c)
Figure 4. (Color online) Scaling functions for the isothermal susceptibility (a), heat capacity (b), and mag-
netocaloric coefficient (c). Light curves, brown online, T > Tc: χ+(x),C+(x),M+(x). Black curves, T < Tc:
χ−(x), C−(x),M−(x). Solid, dashed and dotted curves correspond to the values of q and λ of the free en-
ergy scaling functions of figure 2.
µ±(x) ∼ x1/(λ−2), x →∞ for 3 <λ< 5. A similar feature is observed for the other values of q . At 1 É q < 2,
λ > 4 one gets µ±(x) ∼ p
x, x → ∞ and µ±(x) ∼ x1/(λ−2), x → ∞ for 3 < λ < 4. The last asymptotic be-
haviour also holds for q > 2 and λÉλc(q). Note that all light curves (brown online) in figure 3 start form
the origin: this corresponds to the absence of spontaneous magnetization at T > Tc. Correspondingly, thevalue of the scaling function µ−(x) at x = 0 gives the spontaneous magnetization critical amplitude B−,equation (2.2). As one can see in figure 3, the latter increases with a decrease of λ.
In figures 4 we show the behaviour of the scaling functions for thermodynamic observables that
characterize the response of a system to an external action, i.e., the isothermal susceptibility [figure 4 (a)],
heat capacity [figure 4 (b)], and magnetocaloric coefficient [figure 4 (c)]. The values of q and λ, for which
the curves are plotted are the same as those for the free energy scaling functions in figure 2: they reflect
the limiting behaviour at some marginal value λc(q). At 1 É q É 2 and λ > λc(q), the phase transition
remains the second order but the critical exponents do not depend on λ any more, the scaling function
does not depend on λ either. However, for q > 2, λ > λc(q), the phase transition turns to the first order
and the scaling regime does not hold any more. In turn, in the region below λc, the exponents acquire
λ-dependency, so do the scaling functions, as is plotted in the figures.
5. Conclusions
In this paper we were interested in an analysis of the critical behaviour of the Potts model on uncor-
related scale-free network. In our previous work the list of critical exponents for the Potts model in the
second order phase transition regime was obtained [6]. Here, we complete quantitative characteristics of
the universal features by calculation of the amplitude ratios and scaling functions. Our results are exact
for the annealed scale-free network and correspond to the mean field treatment of the quenched case.
We obtain general scaling functions for the order parameter, entropy, the constant-field heat capacity,
magnetic susceptibility and the isothermalmagnetocaloric coefficient near the critical point. The compre-
hensive list of scaling functions and critical amplitude ratios was obtained in different regions of q and
λ. It was shown that the critical amplitude ratios are λ-dependent similar to the critical exponents, so λ
plays the role of a global parameter of the system.
Acknowledgements
This work was supported in part by FP7 EU IRSES projects No. 269139 ‘Dynamics and Cooperative
Phenomena in Complex Physical and Biological Media’, No. 295302 ‘Statistical Physics in Diverse Realiza-
23602-8
Scaling functions and amplitude ratios for the Potts model
tions’ and by the Collège Doctoral 02−07 Statistical Physics of complex systems. It is my big pleasure to
thank Bertrand Berche and Yurij Holovatch for useful comments and discussions.
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23602-9
http://dx.doi.org/10.1088/0034-4885/30/2/306
http://dx.doi.org/10.1103/RevModPhys.39.395
http://dx.doi.org/10.1103/RevModPhys.80.1275
http://dx.doi.org/10.5488/CMP.16.23602
http://dx.doi.org/10.1140/epjb/e2004-00019-y
http://dx.doi.org/10.1103/PhysRevE.66.036140
http://dx.doi.org/10.1103/PhysRevB.16.4971
http://dx.doi.org/10.1016/0375-9601(76)90625-3
http://dx.doi.org/10.1016/0031-8914(72)90045-6
http://dx.doi.org/10.1016/0378-4371(92)90447-X
http://dx.doi.org/10.1088/0022-3719/11/11/004
http://dx.doi.org/10.1088/0022-3719/12/3/008
http://dx.doi.org/10.1103/PhysRevLett.41.829
http://dx.doi.org/10.1103/PhysRevLett.37.565
http://dx.doi.org/10.1016/0375-9601(75)90766-5
http://dx.doi.org/10.1103/PhysRevLett.38.501
http://dx.doi.org/10.1103/PhysRevLett.38.1148
http://dx.doi.org/10.1006/jtbi.2001.2522
http://dx.doi.org/10.1007/BF02099291
http://dx.doi.org/10.1103/PhysRevE.83.061114
http://dx.doi.org/10.1103/RevModPhys.71.S358
http://dx.doi.org/10.1103/PhysRevB.6.3515
http://dx.doi.org/10.1103/RevModPhys.54.235
http://dx.doi.org/10.1103/PhysRevE.80.051127
http://dx.doi.org/10.1103/PhysRevE.85.061113
http://dx.doi.org/10.1103/PhysRevE.66.016104
http://dx.doi.org/10.1140/epjb/e2002-00220-0
http://dx.doi.org/10.1103/PhysRevE.67.026123
http://dx.doi.org/10.1103/PhysRevE.66.036113
http://dx.doi.org/10.1103/PhysRevLett.96.115701
http://dx.doi.org/10.1103/PhysRevLett.97.155702
http://dx.doi.org/10.1088/1751-8113/46/9/095001
http://dx.doi.org/10.1103/PhysRevE.82.011145
http://dx.doi.org/10.1103/PhysRev.158.176
http://dx.doi.org/10.1063/1.1696618
M. Krasnytska
Скейлiнговi функцiї та спiввiдношення амплiтуд для моделi
Поттса на нескорельованiй безмасштабнiй мережi
М. Красницька1,2
1 Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
2 Iнститут Жаня Лямура, CNRS/UMR 7198, Група статистичної фiзики, Унiверситет Лотарингiї,
BP 70239, F-54506 Вандувр лє-Нансi, Францiя
Ми вивчаємо критичну поведiнку q-станової моделi Поттса на нескорельованiй безмасштабнiй мережi
зi степеновою функцiєю розподiлу за ступенем вузлiв з показником загасання λ. Попереднi результати
показали, що фазова дiаграма моделi в площинi q , λ в режимi фазового переходу другого роду мiстить
три областi, кожна з яких характеризується рiзним набором критичних показникiв. У данiй роботi ми до-
повнимо цi результати знайшовши аналiтичнi представлення скейлiнгових функцiй та спiввiдношення
критичних амплiтуд у згаданих вище областях. Як i для знайдених ранiше критичних показникiв, вияв-
ляється, що скейлiнговi функцiї та спiввiдношення амплiтуд є λ-залежними. Таким чином, ми подаємо
повний опис критичної поведiнки у новому класi унiверсальностi.
Ключовi слова: модель Поттса, складнi мережi, скейлiнг, унiверсальнiсть
23602-10
Introduction
Main relations of scaling theory
Potts model on an uncorrelated scale-free network
Critical amplitude ratios and scaling functions
Conclusions
|