On the discontinuity of the specific heat of the Ising model on a scale-free network
We consider the Ising model on an annealed scale-free network with node-degree distribution characterized by a power-law decay P(K)∼ K-λ. It is well established that the model is characterized by classical mean-field exponents for λ > 5. In this note we show that the specific-heat discontinuity δ...
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irk-123456789-1557992019-06-18T01:29:39Z On the discontinuity of the specific heat of the Ising model on a scale-free network Krasnytska, M. Berche, B. Holovatch, Yu. Kenna, R. We consider the Ising model on an annealed scale-free network with node-degree distribution characterized by a power-law decay P(K)∼ K-λ. It is well established that the model is characterized by classical mean-field exponents for λ > 5. In this note we show that the specific-heat discontinuity δc_h at the critical point remains λ-dependent even for λ > 5: δch=3(λ-5)(λ-1)/[2(λ-3)²] and attains its mean-field value δch=3/2 only in the limit λ → ∞. We compare this behaviour with recent measurements of the d dependency of δch made for the Ising model on lattices with d > 4 [Lundow P.H., Markström K., Nucl. Phys. B, 2015, 895, 305]. Ми розглядаємо модель Iзiнга на вiдпаленiй безмасштабнiй мережi зi степенево-спадною функцiєю розподiлу вузлiв P(K ) ∼ K −λ. Вiдомо, що ця модель описується класичними критичними показниками середнього поля при λ > 5. Тут ми покажемо, що стрибок теплоємностi δch при критичнiй температурi залишається λ-залежним навiть для λ > 5: δch = 3(λ−5)(λ−1)/[2(λ−3)² ] i досягає свого середньопольового значення δch = 3/2 тiльки в границi λ → ∞. Ми порiвнюємо цю поведiнку iз недавнiми результатами залежностi δch вiд d для моделi Iзiнга на гратках з d > 4 [Lundow P.H., Markstr¨om K., Nucl. Phys. B, 2015, 895, 305]. 2015 Article On the discontinuity of the specific heat of the Ising model on a scale-free network / M. Krasnytska, B. Berche, Yu. Holovatch, R. Kenna // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 44601: 1–4. — Бібліогр.: 20 назв. — англ. 1607-324X PACS: 64.60.aq, 64.60.fd, 64.70.qd, 64.60.De DOI:10.5488/CMP.18.44601 arXiv:1510.06216 http://dspace.nbuv.gov.ua/handle/123456789/155799 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We consider the Ising model on an annealed scale-free network with node-degree distribution characterized by a power-law decay P(K)∼ K-λ. It is well established that the model is characterized by classical mean-field exponents for λ > 5. In this note we show that the specific-heat discontinuity δc_h at the critical point remains λ-dependent even for λ > 5: δch=3(λ-5)(λ-1)/[2(λ-3)²] and attains its mean-field value δch=3/2 only in the limit λ → ∞. We compare this behaviour with recent measurements of the d dependency of δch made for the Ising model on lattices with d > 4 [Lundow P.H., Markström K., Nucl. Phys. B, 2015, 895, 305]. |
| format |
Article |
| author |
Krasnytska, M. Berche, B. Holovatch, Yu. Kenna, R. |
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Krasnytska, M. Berche, B. Holovatch, Yu. Kenna, R. On the discontinuity of the specific heat of the Ising model on a scale-free network Condensed Matter Physics |
| author_facet |
Krasnytska, M. Berche, B. Holovatch, Yu. Kenna, R. |
| author_sort |
Krasnytska, M. |
| title |
On the discontinuity of the specific heat of the Ising model on a scale-free network |
| title_short |
On the discontinuity of the specific heat of the Ising model on a scale-free network |
| title_full |
On the discontinuity of the specific heat of the Ising model on a scale-free network |
| title_fullStr |
On the discontinuity of the specific heat of the Ising model on a scale-free network |
| title_full_unstemmed |
On the discontinuity of the specific heat of the Ising model on a scale-free network |
| title_sort |
on the discontinuity of the specific heat of the ising model on a scale-free network |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/155799 |
| citation_txt |
On the discontinuity of the specific heat of the Ising model on a scale-free network / M. Krasnytska, B. Berche, Yu. Holovatch, R. Kenna // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 44601: 1–4. — Бібліогр.: 20 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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| first_indexed |
2025-07-14T08:01:59Z |
| last_indexed |
2025-07-14T08:01:59Z |
| _version_ |
1837608594705481728 |
| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 4, 44601: 1–4
DOI: 10.5488/CMP.18.44601
http://www.icmp.lviv.ua/journal
Rapid communication
On the discontinuity of the specific heat of the Ising
model on a scale-free network
M. Krasnytska1,2,4, B. Berche2,4, Yu. Holovatch1,4, R. Kenna3,4
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Institut Jean Lamour, Université de Lorraine, F-54506 Vandœuvre les Nancy, France
3 Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom
4 Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry (L4)
Received October 21, 2015
We consider the Ising model on an annealed scale-free network with node-degree distribution characterized by
a power-law decay P(K ) ∼ K −λ. It is well established that the model is characterized by classical mean-field
exponents for λ> 5. In this note we show that the specific-heat discontinuity δch at the critical point remains
λ-dependent even for λ> 5: δch = 3(λ−5)(λ−1)/[2(λ−3)2 ] and attains its mean-field value δch = 3/2 only
in the limit λ→∞. We compare this behaviour with recent measurements of the d dependency of δch made
for the Ising model on lattices with d > 4 [Lundow P.H., Markström K., Nucl. Phys. B, 2015, 895, 305].
Key words: Ising model, scale-free networks, annealed network
PACS: 64.60.aq, 64.60.fd, 64.70.qd, 64.60.De
In the Ehrenfest classification, a second-order phase transition is manifest by a discontinuity of the
second derivative of the free energy at the transition temperature Tc [1]. However, derivatives takenwith
respect to different thermodynamic variables may demonstrate qualitatively different behaviour. For
magnetic systems, it is well known that the isothermal susceptibility χT and magnetocaloric coefficient
mT (a mixed derivative of the free energy with respect to magnetic field and temperature) are strongly
diverging quantities, whereas the specific heat ch often does not diverge at Tc. Considered in the mean-
field approximation, the first two quantities are singular at τ= |T −Tc|/Tc = 0: χT ∼ τ−γ, mT ∼ τ−ω with
γmfa = 1, ωmfa = 1/2. However, the third quantity displays a jump at Tc:
δch = ch (T → T −
c )−ch (T → T +
c ), (1)
with δcmfa
h
= 3/2 and hence ch ∼ τ−α with αmfa = 0.
For the Ising model in d dimensions, the singularity of the specific heat is d -dependent: the famous
Onsager solution [2] predicted ch(d = 2) ∼ lnτ (a weak singularity with α = 0) while α(d = 3) ≃ 0.109(4)
[3] and α attains its mean-field value in dimensions higher than the upper critical value, α(d > 4) = 0.
Strictly at d = 4, the scaling is affected by the logarithmic correction [4]
ch ∼ τ−α
mfa
(lnτ)α̂. (2)
Since αmfa = 0 and the logarithmic correction-to-scaling exponent α̂= 1/3 is positive [4], the specific heat
of the Ising model diverges at d = 4.
Although the critical exponents attain their mean-field values above the upper critical dimension,
this is not the case for critical amplitudes. For d > 4, the latter determine the value of the specific heat
discontinuity in equation (1). As has been shown recently [5], δch for the Ising model at d > 4 remains a
d -dependent quantity that reaches the mean-field result only in the limit δch (d →∞) = 3/2. Inspired by
©M. Krasnytska, B. Berche, Yu. Holovatch, R. Kenna, 2015 44601-1
http://dx.doi.org/10.5488/CMP.18.44601
http://www.icmp.lviv.ua/journal
M. Krasnytska et al.
this observation, which was produced using Monte Carlo simulations for 5, 6, and 7-dimensional lattices
[5], in this note we analyze the behaviour of the specific-heat discontinuity of the Ising model on complex
networks. Recent interest in structures of numerous natural and man-made systems [6–9] lead, in partic-
ular, to the development of phase transition theory on complex networks [10]. Of particular interest are
scale-free networks, where the node-degree distribution is characterized by a power-law decay:
P (K ) = c/K λ . (3)
Here, P (K ) is the probability that the number of nearest neighbours of a node (node degree) is K and c is a
normalizing constant. It appears that many real-world complex networks (e.g., the internet, www, trans-
portation networks, social networks of communication between people and many others) are scale-free
[6–9]. In turn, studying properties of phase transitions on scale-free networks may also explain peculiar-
ities of processes occurring on such networks too. To give just two examples, the analysis of percolation
phenomena on scale-free networks is directly related to the stability of the network to random break-
downs or targeted attacks, whereas the onset of an ordered phase (e.g., ferromagnetic ordering in a spin
model on a network) may correspond to a unanimous opinion formation in a social network.
Here, the subject of our analysis is the Ising model on a complex scale-free network. In particular,
we will consider the behaviour of the specific heat on an annealed network. This has been widely used
to analyze properties of various spin models (see e.g., [11–13] and references therein). For annealed net-
works, the links fluctuate on the same time scale as the spin variables [11–13], therefore, the partition
function is averaged both with respect to the link distribution and the Boltzmann distribution. This is
achieved by assigning to each node i a hidden variable ki . In our particular case of a scale-free network,
the distribution of ki is given by (3) too. The probability of a link between any pair of nodes (i , j ) is cho-
sen to be proportional to the product ki k j of k-variables on these nodes. One can check that the expected
node-degree value is then E[Ki ] = ki . This choice leads to the Hamiltonian which, in the absence of an
external magnetic field, reads:
H =−
1
N〈k〉
∑
i> j
ki k j Si S j . (4)
Here, Si =±1 is a spin variable, the sum spans all pairs of N nodes and 〈k〉 =
∑N
i=1 ki /N .
The prominent feature of (4) is that the interaction term attains a separable form. In turn, this allows
for an exact representation of the partition function via e.g., Stratonovich-Hubbard transformation, as
it is usually done for the Ising model on a complete graph [14], see [15, 16] and references therein. It
is straightforward to get thermodynamic functions and, in particular, to arrive at the conclusion that
universal behaviour of the specific heat depends on the node-degree distribution exponent λ [17–20]1:
α= (λ−5)/(λ−3), 3<λÉ 5; α= 0, λ> 5. (5)
The negativity of the exponent α in the region 3 < λ < 5 means that δch = 0 there. Moreover, directly
at λ = 5 the logarithmic correction-to-scaling exponent governs the behaviour, similar as for lattices at
d = 4, see equation (2). However, in contrast to the lattice case, the value of the exponent for scale-free
networks is negative: α̂=−1 [17–20]. This means that δch = 0 at λ= 5 too.
Here, we are interested in the behaviour of the specific heat in the region λ > 5, where usual mean-
field results for the critical exponents hold. Keeping terms leading in N for the partition function, one
can represent it in the form (see [15, 16] for more details)
ZN (T ) =
+∞
∫
−∞
exp
{
N
[
−〈k〉x2
2
(T −Tc)−
〈k4〉x4
12
]}
dx , λ> 5, (6)
where Tc = 〈k2〉/〈k〉 and we have omitted a prefactor which is not important for our analysis.
Using the method of steepest descent one finds points of maxima (x⋆) of the function under integra-
tion at T > Tc (x⋆ = 0) and T < Tc (x⋆ = [−(3〈k〉/〈k4〉)(T −Tc)]1/2 ). The free energy reads:
f (T ) =
{
0, T > Tc,
−
3〈k〉2
4〈k4〉
T (T −Tc)2, T < Tc.
(7)
1The system remains ordered at any finite temperature for 2< λÉ 3.
44601-2
On the discontinuity of the specific heat of the Ising model on a scale-free network
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
6 6.5 7 7.5 8 8.5 9
δ
c
h
λ
d+1
Figure 1. The jump in the specific heat of the Ising model on lattices at d > 4 (squares, results of MC
simulations [5]) and on an annealed scale-free network for λ > 5, bold line equation (10). The thin line
shows classical mean-field value δch = 3/2. Although δch(λ → ∞) = δch(d → ∞) = 3/2, the functions
approach the mean-field limit from below and from above.
Correspondingly, for the specific heat one obtains
ch =
{
0, T > Tc,
−
9〈k〉2
2〈k4〉
T 2 +
6〈k〉2
2〈k4〉
T Tc, T < Tc.
(8)
The jump of the specific heat at Tc is defined by the ratio
δch =
3〈k2〉2
2〈k4〉
. (9)
Substituting the averages calculated with the distribution (3) we obtain
δch =
3(λ−5)(λ−1)
2(λ−3)2
, λ> 5. (10)
In the limit of large λ this delivers δch = 3/2, which coincides with the corresponding value on a complete
graph.
It is well known that Ising model on an annealed scale-free network is characterized by classical
mean-field exponents at λ> 5. As we have shown in this note, the mean-field behaviour does not concern
the specific heat jump δch at λ > 5. The jump remains λ-dependent and reaches the mean-field value
δch = 3/2 only in the limit λ → ∞. The function δch (λ) is shown in figure 1. Similar effect has been
observed for the Ising model on lattices at d > 4. We show the results of MC simulations of d = 5,6,7-
dimensional lattices [5] in the figure too. Note, that although δch(λ→∞) = δch (d →∞) = 3/2, the func-
tions approach the mean-field limit from below and from above. Another essential difference between
the behaviour of δch in the Ising model on scale-free networks and on lattices is observed directly at the
upper critical values of λ and of d , respectively. While α = 0 in both cases, the overall behaviour of ch
remains singular on lattices at d = 4 (logarithmic singularity, α̂ = 1/3) whereas α̂ = −1 for networks at
λ = 5 and hence δch = 0. This last case provides an example where the logarithmic correction to scaling
leads to smoothing of behaviour of the thermodynamic function at Tc.
Acknowledgements
This work was supported in part by FP7 EU IRSES projects No. 295302 “Statistical Physics in Diverse
Realizations”, No. 612707 “Dynamics of and in Complex Systems”, No. 612669 “Structure and Evolution
of Complex Systems with Applications in Physics and Life Sciences”, and by the Doctoral College for the
Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry (L4). We thank Yuri Kozitsky for
fruitful discussions. M.K. is grateful to Klas Markström for useful correspondence which initiated this
study.
44601-3
M. Krasnytska et al.
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Стрибок теплоємностi моделi Iзiнга на безмасштабнiй
мережi
М. Красницька1,2,4, Б. Берш2,4, Ю. Головач1,4, Р. Кенна3,4
1 Iнститут фiзики конденсованих систем НАН України,
вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
2 Iнститут Ж. Лямура, Унiверситет Лотарингiї, F-54506 Вандувр лє Нансi, Францiя
3 Центр прикладної математики, Унiверситет Ковентрi, Ковентрi CV1 5FB, Англiя
4 Коледж докторантiв статистичної фiзики складних систем, Ляйпцiг-Лотарингiя-Львiв-Ковентрi (L4)
Ми розглядаємо модель Iзiнга на вiдпаленiй безмасштабнiй мережi зi степенево-спадною функцiєю роз-
подiлу вузлiв P(K ) ∼ K −λ. Вiдомо, що ця модель описується класичними критичними показниками сере-
днього поля при λ> 5. Тут ми покажемо, що стрибок теплоємностi δch при критичнiй температурi зали-
шається λ-залежним навiть для λ> 5: δch = 3(λ−5)(λ−1)/[2(λ−3)2 ] i досягає свого середньопольового
значення δch = 3/2 тiльки в границi λ → ∞. Ми порiвнюємо цю поведiнку iз недавнiми результатами
залежностi δch вiд d для моделi Iзiнга на гратках з d > 4 [Lundow P.H., Markström K., Nucl. Phys. B, 2015,
895, 305].
Ключовi слова: модель Iзiнга, безмасштабна мережа, вiдпалена мережа
44601-4
http://dx.doi.org/10.1103/PhysRev.65.117
http://dx.doi.org/10.1088/0305-4470/31/40/006
http://dx.doi.org/10.1016/j.nuclphysb.2015.04.013
http://dx.doi.org/10.1103/RevModPhys.74.47
http://dx.doi.org/10.1103/RevModPhys.80.1275
http://dx.doi.org/10.1103/PhysRevE.80.051127
http://dx.doi.org/10.1103/PhysRevE.85.061113
http://arxiv.org/abs/1509.07327
http://dx.doi.org/10.1209/0295-5075/111/60009
http://arxiv.org/abs/1510.00534
http://dx.doi.org/10.1140/epjb/e2002-00220-0
http://dx.doi.org/10.1140/epjb/e2004-00019-y
http://dx.doi.org/10.1103/PhysRevE.80.011108
http://dx.doi.org/10.1103/PhysRevE.83.061114
|