On phase transitions of the Potts model with three competing interactions on Cayley tree
In the present paper we study a phase transition problem for the Potts model with three competing interactions, the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order two. We prove that for some parameter values of the model there is...
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irk-123456789-1199772017-06-11T03:02:54Z On phase transitions of the Potts model with three competing interactions on Cayley tree Akin, H. Temir, S. In the present paper we study a phase transition problem for the Potts model with three competing interactions, the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order two. We prove that for some parameter values of the model there is phase transition. We reduce the problem of describing by limiting Gibbs measures to the problem of solving a system of nonlinear functional equations. We extend the results obtained by Ganikhodjaev and Rozikov [Math. Phys. Anal. Geom., 2009, 12, No. 2, 141-156] on phase transition for the Ising model to the Potts model setting. У цiй статтi ми вивчаємо проблему фазового переходу для моделi Поттса з трьома конкуруючими взаємодiями, найближчих сусiдiв, наступних близьких сусiдiв i наступних за наступними близькими сусiдами,з ненульовим зовнiшнiм полем на деревi Келi другого порядку. Ми доводимо, що для деяких значень параметрiв моделi є фазовий перехiд. Ми приводимо проблему опису граничних мiр Гiббса до проблеми розв’язку системи нелiнiйних функцiональних рiвнянь. Ми розширюємо результати, отриманi Ганiходяєвим i Розiковим [Math. Phys. Anal. Geom., 2009, 12, No 2, 141–156] для фазового переходу в моделi Iзiнга,на випадок моделi Поттса. 2011 Article On phase transitions of the Potts model with three competing interactions on Cayley tree / H. Akin, S. Temir // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23003:1-11. — Бібліогр.: 20 назв. — англ. 1607-324X PACS: 05.50.+q, 64.60.-i, 64.60.De, 75.10.Hk DOI:10.5488/CMP.14.23003 arXiv:math/0703699 http://dspace.nbuv.gov.ua/handle/123456789/119977 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In the present paper we study a phase transition problem for the Potts model with three competing interactions, the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order two. We prove that for some parameter values of the model there is phase transition. We reduce the problem of describing by limiting Gibbs measures to the problem of solving a system of nonlinear functional equations. We extend the results obtained by Ganikhodjaev and Rozikov [Math. Phys. Anal. Geom., 2009, 12, No. 2, 141-156] on phase transition for the Ising model to the Potts model setting. |
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Akin, H. Temir, S. |
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Akin, H. Temir, S. On phase transitions of the Potts model with three competing interactions on Cayley tree Condensed Matter Physics |
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Akin, H. Temir, S. |
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Akin, H. |
| title |
On phase transitions of the Potts model with three competing interactions on Cayley tree |
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On phase transitions of the Potts model with three competing interactions on Cayley tree |
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On phase transitions of the Potts model with three competing interactions on Cayley tree |
| title_fullStr |
On phase transitions of the Potts model with three competing interactions on Cayley tree |
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On phase transitions of the Potts model with three competing interactions on Cayley tree |
| title_sort |
on phase transitions of the potts model with three competing interactions on cayley tree |
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Інститут фізики конденсованих систем НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/119977 |
| citation_txt |
On phase transitions of the Potts model with three competing interactions on Cayley tree / H. Akin, S. Temir // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23003:1-11. — Бібліогр.: 20 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT akinh onphasetransitionsofthepottsmodelwiththreecompetinginteractionsoncayleytree AT temirs onphasetransitionsofthepottsmodelwiththreecompetinginteractionsoncayleytree |
| first_indexed |
2025-07-08T17:00:43Z |
| last_indexed |
2025-07-08T17:00:43Z |
| _version_ |
1837098907193049088 |
| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 2, 23003: 1–11
DOI: 10.5488/CMP.14.23003
http://www.icmp.lviv.ua/journal
On phase transitions of the Potts model with three
competing interactions on Cayley tree
H. Akin1∗, S. Temir2†
1 Department of Mathematics, Faculty of Education, Zirve University, 27260 Gaziantep, Turkey
2 Department of Mathematics, Arts and Science Faculty Harran University, 63200 Sanliurfa, Turkey
Received November 15, 2010, in final form March 22, 2011
In the present paper we study a phase transition problem for the Potts model with three competing interactions,
the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley
tree of order two. We prove that for some parameter values of the model there is phase transition. We reduce
the problem of describing by limiting Gibbs measures to the problem of solving a system of nonlinear functional
equations. We extend the results obtained by Ganikhodjaev and Rozikov [Math. Phys. Anal. Geom., 2009, 12,
No. 2, 141–156] on phase transition for the Ising model to the Potts model setting.
Key words: phase transition, Potts model, competing interactions, Gibbs measure
PACS: 05.50.+q, 64.60.-i, 64.60.De, 75.10.Hk
1. Introduction
It is well known that lattice spin system is a large class of systems considered in statistical
mechanic. It is also well known that the structure of the lattice plays an important role in examining
the spin systems. The Potts model, which was introduced by Potts (in 1952) as a generalization
of the Ising model, has drawn an increased attention to its theoretical and experimental aspects
in recent years [1]. The idea originated from the representation of the Ising model as interacting
spins which can be either parallel or antiparallel.
We consider a semi-infinite Cayley tree Jk of order k > 2, i.e., a graph without cycles with
(k+1) edges issuing from each vertex except x0 and with k edges issuing from the vertex x0, which
is called the tree root. We generalize the problem in [2] for the Potts model with three competing
interactions and non-zero external field. This problem has been investigated for the Potts model
with only two competing interactions [2, 3].
There are many approaches to the derivation of an equation or a system of equations describing
the limiting Gibbs measure for the lattice models on Cayley tree (see [1, 4–9] for details). One of
these approaches is based on recursive equations for partition functions [1]. Another approach is
Markov random fields (see [10, 11]). Naturally, both approaches lead to the same equation [10].
In this paper we employ the recursive equations for partition functions. We study the problem
of phase transition for the Potts model with three competing interactions, the nearest neighbors,
the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order
two. In [12], it has been proved that there is a phase transition for some parameter values of the
Ising model. We generalize this problem for the Potts model. In order to construct the Hamilto-
nian equation, we consider the generalized Kroneker’s δ function which well agrees with theory of
quadratic stochastic operators.
The paper is organized as follows. In section 2 we present basic definitions and notations. In
section 3 we present the recursive equations for partition functions. Section 4 contains the solutions
of the system of nonlinear equations. These solutions describe the presence or the absence of a phase
transition. Finally, the conclusions are presented in section 5.
∗E-mail: hasanakin69@gmail.com
†E-mail: temirseyit@harran.edu.tr
c© H. Akin, S. Temir, 2011 23003-1
http://dx.doi.org/10.5488/CMP.14.23003
http://www.icmp.lviv.ua/journal
H. Akin, S. Temir
2. Preliminaries
Cayley tree. The Cayley tree Γk (see [5]) of order k > 1 is an infinite tree, i.e., a graph without
cycles, from each vertex of which exactly k + 1 edges issue. Let Γk = (V, L, i), where V is the set
of vertices of Γk, L is the set of edges of Γk and i is the incidence function associating each edge
ℓ ∈ L with its end points x, y ∈ V. If i(ℓ) = {x, y}, then x and y are called the nearest neighboring
vertices and we write ℓ = 〈x, y〉. For x, y ∈ V , the distance d(x, y) on Cayley tree is defined by the
formula
d(x, y) = min{d|x = x0 , x1 , x2 , . . . , xd−1 , xd = y ∈ V such that the pairs
〈x0 , x1〉, . . . , 〈xd−1 , xd〉 are neighboring vertices}.
For the fixed x0 ∈ V we have
Wn =
{
x ∈ V : d(x0, x) = n
}
,
Vn =
{
x ∈ V : d(x0, x) 6 n
}
,
Ln = {ℓ = 〈x, y〉 ∈ L : x, y ∈ Vn} .
A collection of the pairs 〈x0 , x1〉, . . . , 〈xd−1 , y〉 is called a path from x to y. We write x < y if the
path from x0 to y goes through x. We call the vertex y a direct successor of x, if y > x and x, y
are the nearest neighbors. The set of direct successors of x is denoted by S(x), i.e.,
S(x) = {y ∈ Wn+1 : d(x, y) = 1} , x ∈ Wn .
It is clear that for any vertex x 6= x0, x has k direct successors and x0 has k + 1. The vertices x
and y are called the second neighbor which is denoted by 〉x, y〈 , if there exists a vertex z ∈ V such
that x, z and y, z are the nearest neighbors.
The second neighbors are called one-level neighbors, if vertices x and y belong to Wn for some
n, that is, if they are located on the same level. We will consider only one-level second neighbors.
Three vertices x, y and z are called a triple of neighbors, which is denoted by 〈x, y, z〉, if 〈x, y〉 and
〈y, z〉 are nearest neighbors and x 6= z.
For neighbors triple 〈x, y, z〉 we consider the following Kroneker’s symbol
δσ(x)σ(y)σ(z) =
1, if σ(x) = σ(y) = σ(z),
1/2, if σ(x) = σ(y) 6= σ(z) or σ(x) 6= σ(y) = σ(z),
0, otherwise,
(2.1)
where x, z ∈ Wn and y ∈ Wn−1 for some n. We can write (2.1) as follows:
δσ(x)σ(y)σ(z) =
1
2
(
δσ(x)σ(y) + δσ(y)σ(z)
)
.
The generalized Kroneker’s function in (2.1) which corresponds to the quadratic stochastic
operator is an identity transformation [14].
Model. In the Potts model, the spin variables σ(x) take their values on a discrete set Φ =
{1, 2, . . . , q}, q > 2 which are associated with each vertex of the tree Jk (see [15] for details).
The Potts model with three competing interactions and external field is defined by the following
Hamiltonian
H(σ) = −J
∑
〈x,y〉
δσ(x)σ(y) − J1
∑
〉x,y〈
δσ(x)σ(y) − J2
∑
〈x,y,z〉
δσ(x)σ(y)σ(z) − h
∑
x∈V
δ0σ(x) , (2.2)
where the first sum ranges within all nearest neighbors, the second sum ranges within all the second
neighbors, the third sum ranges within all triples of neighbors and the spin variables σ(x) may
take the values 1, 2, 3. J , J1 , J2 ∈ R are coupling constants and h is external field.
In [15], the authors have already obtained the exact solutions of the Potts model with com-
peting ternary and binary interactions and external field on Cayley tree described by means of
the Hamiltonian (2.2), where h = 0 and J1 = 0. In this paper, we assume that each parameter
J, J1, J2, h is different from zero.
23003-2
Phase transitions for the Potts model
3. Recursive equations for Partition Functions
Let Λ be a finite subset of V. We will denote the restriction of σ to Λ by σ(Λ). Let σ̄(V\Λ) be
a fixed boundary configuration. The total energy of σ(Λ) under condition σ̄(V\Λ) is defined as
HΛ(σ(Λ)|σ̄(V \Λ)) = − J
∑
〈x,y〉:x,y∈Λ
δσ(x)σ(y) − J
∑
〈x,y〉:x∈Λ,y /∈Λ
δσ(x)σ(y)
− J1
∑
〉x,y〈:x,y∈Λ
δσ(x)σ(y) − J1
∑
〉x,y〈:x∈Λ,y /∈Λ
δσ(x)σ(y)
− J2
∑
〈x,y,z〉:x,y,z∈Λ
δσ(x)σ(y)σ(z) − J2
∑
〈x,y,z〉:x,z /∈Λ,y∈Λ
δσ(x)σ(y)σ(z)
− h
∑
x∈Λ
δ0σ(x). (3.1)
Then, partition function ZΛ (σ̄(V \ Λ)) in volume Λ under boundary condition σ̄(V\Λ) is defined
as
ZΛ(σ̄(V \Λ)) =
∑
σ(Λ)∈Ω(Λ)
exp [−βHΛ(σ(Λ)|σ̄(V \Λ))] , (3.2)
where Ω(Λ) is the set of all configuration in volume Λ, and β = 1/kT is the inverse temperature.
Instead of the configurations σ(Vn) and the partition functions ZVn
in volume Vn we use
notations σn and Z(n), respectively [10, 15].
It is well known that the Gibbs measure is a probability measure used in many problems of
probability theory and of statistical mechanics [13]. The Gibbs measure of a configuration σΛ is
defined by
µ(σΛ) =
exp [−βH(σΛ)]
ZΛ(σ)
,
where ZΛ(σ) is the partition function.
As usual, one can introduce the notion of the Gibbs measure (phase) of the Potts model with
a competing interaction on the Cayley tree (see [6, 7, 13] for details).
Let us decompose the partition function Z(n) into the following summands
Z(n) =
q
∑
i=1
Z
(n)
i ,
where
Z
(n)
i =
∑
σn∈Ω(Vn):σ(x0)=i
exp [−βHVn
(σn|σ̄(V \Vn))] . (3.3)
Now the partition function is iteratively calculated by moving from the boundary towards to the
interior of the Cayley tree of order two. The partial partition function of a branch on n generations,
where the innermost site is in state i, is denoted by Z
(n)
i .
Let S(x0) = {x1, x2}, σ(x0) = i, σ(x1) = j and σ(x2) = m. For a given n, let us define the
partial partition functions of the next generation by
Z
(n)
i =
3
∑
j,m=1
exp [βJ(δij + δim) + βJ1δjm + βJ2δjim + hδ1i]Z
(n−1)
j Z(n−1)
m ,
where i = 1, 2, 3.
23003-3
H. Akin, S. Temir
Let θ = exp(βJ), θ1 = exp(βJ1), θ2 = exp(βJ2) and θ3 = exp(βh). From (3.1) and (3.2), we
can calculate the partial partition functions as follows:
Z
(n)
1 = θ3
{
θ2θ1θ2
(
Z
(n−1)
1
)2
+ 2θ
√
θ2
(
Z
(n−1)
1 Z
(n−1)
2 + Z
(n−1)
1 Z
(n−1)
3
)
+ θ1
[
(
Z
(n−1)
2
)2
+
(
Z
(n−1)
3
)2
]
+ 2Z
(n−1)
2 Z
(n−1)
3
}
,
Z
(n)
2 = θ1
(
Z
(n−1)
1
)2
+ 2θ
√
θ2Z
(n−1)
2
(
Z
(n−1)
1 + Z
(n−1)
3
)
+ 2Z
(n−1)
1 Z
(n−1)
3 + θ2θ1θ2
(
Z
(n−1)
2
)2
+ θ1
(
Z
(n−1)
3
)2
,
Z
(n)
3 = θ1
(
Z
(n−1)
1
)2
+ 2θ
√
θ2Z
(n−1)
3
(
Z
(n−1)
1 + Z
(n−1)
2
)
+ 2Z
(n−1)
1 Z
(n−1)
2 + θ2θ1θ2
(
Z
(n−1)
3
)2
+ θ1
(
Z
(n−1)
2
)2
.
After replacement un = Z
(n)
2 /Z
(n)
1 and vn = Z
(n)
3 /Z
(n)
1 we have the following system of recurrent
equations
θ3un =
θ1 + 2θ
√
θ2un−1(1 + vn−1) + 2vn−1 + θ2θ1θ2u
2
n−1 + θ1v
2
n−1
θ2θ1θ2 + 2θ
√
θ2(un−1 + vn−1) + θ1(u2
n−1 + v2n−1) + 2un−1vn−1
,
θ3vn =
θ1 + 2un−1 + 2θ
√
θ2vn−1(1 + un−1) + θ1u
2
n−1 + θ2θ1θ2v
2
n−1
θ2θ1θ2 + 2θ
√
θ2(un−1 + vn−1) + θ1(u2
n−1 + v2n−1) + 2un−1vn−1
. (3.4)
We define the transformation
F = (F1 , F2) : R
2 → R
2 (3.5)
with un = F1(un−1 , vn−1) and vn = F2(un−1 , vn−1). The fixed points of the function (3.5) w =
F(w), where w = (un , vn), describe translation-invariant Gibbs measures (phases) of the Potts
model defined by the Hamiltonian (2.2).
The recursive equations (3.4) can be written as wn = F(wn−1), n > 0. In the theory of
dynamical systems, wn is called trajectory of the initial point w1 under the action of the mapping
F. So, the asymptotic behavior of Z(n) for n → ∞ can be determined by value of limn wn .
If u = limn un and v = limn vn then we have the following equations;
θ3u =
θ1 + 2θ
√
θ2u(1 + v) + 2v + θ2θ1θ2u
2 + θ1v
2
θ2θ1θ2 + 2θ
√
θ2(u+ v) + θ1(u2 + v2) + 2uv
,
θ3v =
θ1 + 2u+ 2θ
√
θ2v(1 + u) + θ1u
2 + θ2θ1θ2v
2
θ2θ1θ2 + 2θ
√
θ2(u+ v) + θ1(u2 + v2) + 2uv
. (3.6)
To describe phases (Gibbs measures) of a given Hamiltonian on a Cayley tree, one has a
correspondence between Gibbs measures and a collection of vectors {hx , x ∈ V }, which satisfy
a non-linear equation. The recursive equations (3.6) considered in this paper describe a vector
function {(un , vn), n ∈ N} which is a particular case of the above mentioned function hx obtained
as hx = un if x ∈ Wn i.e., depends only on the number of the generation set to which belongs x
but not on x itself (see for example, [12, 15]).
The solutions of the system (3.6) describe the translation-invariant Gibbs measures [6]. The
number of the solutions of the equations (3.6) depends on the parameters β = 1/kT , θ, θ1 , θ2 ,
θ3 and h. The phase transition usually occurs for low temperature. It is possible to find an exact
value of T ∗ such that a phase transition occurs for all T < T ∗ where T ∗ is called a critical value
of temperature.
An attractive fixed point of a function f is a fixed point u0 of f such that for any value of u in a
domain that is close enough to u0 , the iterated function sequence u, f(u), f(f(u)), f(f(f(u))), . . .
converges to u0 . An attractive fixed point is said to be a stable fixed point if it is also Lyapunov
stable (see [18] for details).
23003-4
Phase transitions for the Potts model
4. Translation-invariant Gibbs measures
As stated in [6], for a given potential V we have exactly one Gibbs state with potential V if
the graph is finite. For some potentials there may be more than one Gibbs state if the graph is not
finite. When we have more than one Gibbs state, then we say that there exists a phase transition
for a given potential V . In other words, this occurrence of non-uniqueness of a Gibbs measure can
be interpreted as a phase transition.
In this section we determine whether any phase transition occurs or not by solving the sys-
tem (3.6). If there is more than one positive solution for the equations (3.6), then there is one Gibbs
measure [16] corresponding to each of these solutions. If there is more than one Gibbs measure,
then it is said that a phase transition occurs for this model (2.2) [12, 15].
4.1. First case
Assume that u = v, then some solutions of the system (3.6) can be found from equation
θ3u =
(
θ̃2θ1 + 2θ̃ + θ1
)
u2 + 2
(
θ̃ + 1
)
u+ θ1
2 (θ1 + 1)u2 + 4θ̃u+ θ̃2θ1
:= f(u), (4.1)
where θ̃ = θ
√
θ2 .
In this case, in order to obtain the solutions of the system (3.6), we should analyze the equation
in (4.1).
1. When we take the first derivative of f(u), we see that if θ̃ > 1, that is θ2 > 1/θ2, then f(u)
is increasing and if θ̃ < 1, f(u) is decreasing for u > 0;
2. When we take the second derivative of f(u), we release that if θ1 > θ∗1 , where
θ∗1 =
θ̃2 + θ̃ + 1 +
√
9θ̃4 + 26θ̃3 + 35θ̃2 + 50θ̃+ 33
(
θ̃2 + 2
)(
θ̃ + 1
) ,
then there is an inflection point u∗ > 0 such that f ′′(u) > 0 for 0 < u < u∗ and f ′′(u) < 0
for u > u∗ (see [6, Proposition 10.7] for details).
Theorem 1 [19, Theorem 5.1] Suppose that f ′′ exists on an interval I.
(i) If f ′′(u) > 0 on I, then the graph of f is concave upward on I.
(ii) If f ′′(u) < 0 on I, then the graph of f is concave downward on I.
Thus we can generalize the Lemma proved in [2] to the model (2.2) as follows.
Lemma 1 Assume that the quartic polynomial g in (4.2) has two positive real roots. Let θ̃ > 1,
θ2 > 1/θ2, that is the equation f in (4.1) has a positive inflection point. Then, there exist
η1(θ, θ1 , θ2) and η2(θ, θ1 , θ2) with 0 < η1(θ, θ1 , θ2) < η2(θ, θ1 , θ2) such that the equation (4.1)
has three positive roots u
(1)
∗ < u
(2)
∗ < u
(3)
∗ if η1(θ, θ1 , θ2) < θ3 < η2(θ, θ1 , θ2); has two solutions if
either θ3 = η1(θ, θ1 , θ2) or θ3 = η2(θ, θ1 , θ2) and has a single solution in other cases. In fact
ηi(θ, θ1 , θ2) =
1
u∗
i
f(u∗
i ),
where u∗
1 , u
∗
2 are the solutions of equation uf ′(u) = f(u).
Proof follows from properties 1) and 2) of function f . As mentioned in [6, Proposition 10.7], for
the equation (4.1) there is more than one solution if and only if there is more than one solution to
uf ′(u) = f(u), which is equivalent to the following equation
g(u) = Au4 +Bu3 + Cu2 +Du+ E = 0, (4.2)
23003-5
H. Akin, S. Temir
where A = 2 (θ1 + 1) (θ̃2θ1 + 2θ̃ + θ1) > 0, B = 8(θ̃ + 1) (θ1 + 1) > 0, D = 8θ̃θ1 > 0, E = θ̃2θ1 > 0
and
C = −
(
θ̃4 + θ̃2 − 6
)
θ21 −
(
2θ̃3 − 6
)
θ1 + 8θ̃2.
Let us determine where the graph of g given in (4.2) is concave upward and concave downward.
Here, we have
g′(u) = 4Au3 + 3Bu2 + 2Cu+D.
We now have g′′(u) = 12Au2 + 6Bu+ 2C. Assume that
3B2 − 8AC > 0. (4.3)
From Theorem 1, g is concave downward on interval
I =
(
−3B −
√
9B2 − 24AC
12A
,
−3B +
√
9B2 − 24AC
12A
)
and g is concave upward on interval R \ I. Notice that at the point
(
−3B +
√
9B2 − 24AC
12A
, g
(
−3B +
√
9B2 − 24AC
12A
))
,
the graph chances from concave downward to concave upward. Thus,
(
−3B +
√
9B2 − 24AC
12A
, g
(
−3B +
√
9B2 − 24AC
12A
))
is one of the inflection points of g.
Figure 1. The region satisfying the inequal-
ity (4.4).
It is clear that g(0) = E > 0 and
limu→∞ g(u) = ∞. Since g is a continuous
function, we can conclude that if g(u) < 0 on
J ⊂ R
+, then g has two positive roots under
condition (4.3).
Due to Descartes’ Rule of Signs [17], the
maximum number of positive real roots of the
quartic polynomial g in (4.2) can be found by
counting the number of sign changes in the
equation (4.2). Thus, the equation (4.2) has
no positive roots if C > 0, and the polyno-
mial (4.2) has at most two positive roots u∗
1 ,
u∗
2 if C < 0. In this case, if θ̃ >
√
2 and
(2θ̃3 − 6)2 + 32θ̃2(θ̃4 + θ̃2 − 6) < 0 then the
coefficient C is negative.
Due to
0 = Au4 +Bu3 + Cu2 +Du+ E > Au4 + Cu2 + E > 2
√
Au4E + Cu2 = (2
√
AE + C)u2
from this we get 2
√
AE + C 6 0 i.e. C 6 −2
√
AE, this simple calculation shows that a necessary
condition to have positive roots of the equation (4.2) is C < −2
√
AE. Thus, we have the following
inequality:
8θ̃ + 8θ̃2 + 6θ1 − 2θ̃3θ1 + 6θ21 − θ̃2θ21 − θ̃4θ21 + 2
√
θ̃2θ1(1 + θ1)
(
2θ̃ + θ1 + θ̃2θ1
)
< 0. (4.4)
Let us define the set H =
{
(θ1, θ̃) : C < −2
√
AE
}
(see figure 1).
Then, we have the following Corollary.
23003-6
Phase transitions for the Potts model
Corollary 1 If θ2 > 1/θ2, 3B2 − 8AC > 0, (θ1 , θ̃) ∈ H and η1(θ, θ1 , θ2) < θ3 < η2(θ, θ1 , θ2)
then the function f given in (4.1) has three fixed points. Thus, for the model (2.2) there are three
translation-invariant Gibbs measures µ1 , µ2 , µ3 indicating a phase transition.
It is easy to show that these three measures correspond to boundary conditions σ̄(V \ Vn) ≡ i,
i = 1, 2, 3.
In figure 1 we show the region with three positive roots. For parameters θ̃, θ1 in phase transition
region the equation (4.1) has three positive roots, where two of them are stable and the third one
is unstable.
4.1.1. Illustrative example
Using the Mathematica software, we have manipulated the function f/θ3 given in (4.1) for some
parameters θ, θ1 , θ2 and θ3. In figure 2, for θ = 0.2, θ1 = 64, θ2 = 2.2 and θ3 = 1.6 we have only
one fixed point. Thus, in this case the model (2.2) has no phase transition.
If we take as θ = 11.6, θ1 = 28.1, θ2 = 10.1, θ3 = 12.2, then we have two fixed points. As shown
figure 2 (c), if we take as θ = 10.3, θ1 = 6.49, θ2 = 6.9 and θ3 = 6.4, then we have three fixed
points, where two of them are stable and the third one is unstable.
Figure 2. Graphs of the function f/θ3 given in (4.1) (a) for θ = 0.2, θ1 = 64, θ2 = 2.2 and
θ3 = 1.6; (b) θ = 11.6, θ1 = 28.1, θ2 = 10.1, θ3 = 12.2; (c) θ = 10.3, θ1 = 6.49, θ2 = 6.9 and
θ3 = 6.4, respectively.
Thus, in order to study the phase transition for the model (2.2) we have clarified the role of θ,
θ1 , θ2 and θ3.
4.2. Second case
Now, let us find the other solutions of the system (3.6). Assume that u 6= v. Subtracting the
second equation of (3.6) from the first we have
θ3(u − v) =
2
(
θ̃ − 1
)
(u − v) + θ1
(
θ̃2 − 1
)
(
u2 − v2
)
θ̃2θ1 + 2θ̃(u+ v) + 2uv + θ1 (u2 + v2)
. (4.5)
After canceling to (u− v) two sides of the equation (4.5) we can obtain the following equation;
t =
θ1θ2s
2 +
[
2θ2θ̃ − θ1
(
θ̃2 − 1
)]
s+ θ̃2θ1θ2 − 2
(
θ̃ − 1
)
2θ3(θ1 − 1)
, (4.6)
where u+ v = s and uv = t.
After dividing the first equation of (3.6) to the second and simplifying we have
t =
θ1s
2 + 2s+ θ1
θ̃2θ1 + θ1 − 2θ̃
. (4.7)
23003-7
H. Akin, S. Temir
From (4.6) and (4.7) we obtain the following equality;
θ1
(
θ̃ + 1
)
s+ 2
θ3
=
θ1
[
θ1
(
θ̃ + 1
)
− 2
]
s2 + 2
[
θ1
(
θ̃2 + θ̃ − 2
)
− 2
(
θ̃ + 1
)]
s
θ1
(
θ̃2 + 1
)
− 2θ̃
+
θ1
[
θ1
(
θ̃3 + θ̃2 + 2θ̃ + 2
)
− 2
(
θ̃2 + θ̃ + 1
)]
θ1
(
θ̃2 + 1
)
− 2θ̃
. (4.8)
Let us consider a system of equations
u+ v = s,
uv = t. (4.9)
From (4.9), we can write v = s− u and u(s− u) = t, thus we have
u2 − su+ t = 0. (4.10)
To have two different real roots of the second-order equation (4.10), inequality s2 − 4t > 0 should
be valid. Then the equations (4.9) have solutions, if t < s2/4.
Thus, from (4.7) and from the inequality t < s2/4 we can get the following equation
[
θ1
(
θ̃2 − 3
)
− 2θ̃
]
s2 − 8s− 4θ1 > 0. (4.11)
Then, there is
s∗ =
4 +
√
16 + 4θ1
[
θ1
(
θ̃2 − 3
)
− 2θ̃
]
θ1θ̃2 − 3θ1 − 2θ̃
> 0,
such that for all s > s∗ the inequality t < s2/4 is valid.
One can show that the inequality (4.11) is satisfied, if
(
θ̃2 − 3
)
θ21 − 2θ̃θ1 + 4 < 0 (4.12)
and θ1(θ̃
2 − 3)− 2θ̃ > 0. Thus, for θ̃ >
√
3 and
0 < θ1 <
θ̃ +
√
12− 3θ̃2
θ̃2 − 3
(4.13)
the inequality (4.12) is valid.
In this case, the system of the equations (4.9) has solutions.
Assume that all coefficients of the equation (4.8) are positive. With the help of the elementary
analysis we obtain the following three cases:
1. If
θ3 <
2
[
θ1
(
θ̃2 + 1
)
− 2θ̃
]
θ1
[
θ1
(
θ̃3 + θ̃2 + 2θ̃ + 2
)
− 2
(
θ̃2 + θ̃ + 1
)] , (4.14)
then the equation (4.8) has only one positive root.
2. If
θ3 >
2
[
θ1
(
θ̃2 + 1
)
− 2θ̃
]
θ1
[
θ1
(
θ̃3 + θ̃2 + 2θ̃ + 2
)
− 2
(
θ̃2 + θ̃ + 1
)] (4.15)
there is k0 such that the line y − 2/θ3 = k0s is a tangent for the parabola on the right-hand
side of (4.8) and then for θ3 < θ1(θ̃ + 1)/k0 the equation (4.8) has two positive roots and
only one of them is greater than s∗.
23003-8
Phase transitions for the Potts model
3. The equation (4.8) has two positive roots, when
θ >
√
3
θ2
, 0 < θ1 <
θ̃ +
√
12− 3θ̃2
θ̃2 − 3
, θ1 >
2θ̃
θ̃2 − 3
and
2
[
θ1
(
θ̃2 + 1
)
− 2θ̃
]
θ1
[
θ1
(
θ̃3 + θ̃2 + 2θ̃ + 2
)
− 2
(
θ̃2 + θ̃ + 1
)] < θ3 <
θ1
(
θ̃ + 1
)
k0
, (4.16)
where k0 is defined as above.
4.2.1. Illustrative example
Let us give an illustrative example. With the help of the Mathematica software, we have manip-
ulated the line on the left-hand side and the parabola on the right-hand side of the equation (4.8)
for some parameters θ, θ1 , θ2 and θ3. We have found the points of intersection of the parabola and
the line given in (4.8).
In figure 3 (a), the inequality (4.14) is satisfied, thus we have only one positive root. In fig-
ure 3 (b), the line y− 2/θ3 = θ1(θ̃ + 1)s/θ3 is a tangent for the parabola (4.8). In figure 3 (c), the
inequality (4.16) is satisfied, we have two positive solutions. Thus, in order to obtain the solutions
of the system (4.9), we have clarified the role of the parameters θ, θ1 , θ2 and θ3 .
Figure 3. Graphs of the parabola and the line given in (4.8) a) for θ = 1.1, θ1 = 2.9, θ2 = 1.8
and θ3 = 1.1; b) θ = 27.5, θ1 = 4.5, θ2 = 5.9 and θ3 = 27.5; c) θ = 6.42, θ1 = 19, θ2 = 50 and
θ3 = 19, respectively.
Recalling the results for the equations (3.6), we get the following result:
if (θ, θ1 , θ2) ∈ M and K < θ3 < L, where
M =
(θ, θ1 , θ2) : θ >
√
3
θ2
, θ1 >
2θ̃
(
θ̃2 − 3
) , 0 < θ1 <
θ̃ +
√
12− 3θ̃2
θ̃2 − 3
∩H,
K = max
η1 ,
2
[
θ1
(
θ̃2 + 1
)
− 2θ̃
]
θ1
[
θ1
(
θ̃3 + θ̃2 + 2θ̃ + 2
)
− 2
(
θ̃2 + θ̃ + 1
)]
,
L = min
η2 ,
θ1
(
θ̃ + 1
)
k0
(4.17)
then the system (3.6) has five positive solutions, three of which are obtained from the first case and
the other two solutions arise in the second case. Similar to [2] and [20], a more detailed analysis
shows that only three solutions are stable.
Combining the first and the second cases, we have proved the following theorem.
Theorem 2 Assume that the conditions (4.17) are satisfied, then for the model (2.2) there are
five translation-invariant Gibbs measures indicating a phase transition.
23003-9
H. Akin, S. Temir
Remark 1 In a similar way we can study the solutions of a new system of nonlinear equations by
using Kroneker’s symbol δ that has the form
δσ(x)σ(y)σ(z) =
{
1, if σ(x) = σ(y) = σ(z),
0, otherwise.
5. Conclusion
In this paper we have exactly solved the Potts model on a Cayley tree, the Hamiltonian of
which contains three competing interactions, the nearest neighbors, the second neighbors, triples
of neighbors and the external field. Namely, we have calculated the critical curve such that there
is a phase transition above it, and a single Gibbs state is found elsewhere. We have clarified the
role of the coupling constants J , J1 , J2 and external field h to study the existence problem of a
phase transition. We have considered transformation F = (F1 ;F2) : R
2 → R
2 and proved that its
fixed points describe the translation-invariant Gibbs measures of our model. We have seen that for
some values J , J1 , J2 and h, the phase transition occurs.
In [12], an Ising model with four competing interactions (external field, nearest neighbor, second
neighbors and triples of neighbors) on the Cayley tree of order two has been solved. The authors [12]
have also constructed numerous non-periodic extreme Gibbs measures. The investigation of the
periodic Gibbs measures for the Potts model (2.2) is planned to be the subject of forthcoming
publications.
Acknowledgement
The authors would like to thank Prof. Dr. N.N. Ganikhodjaev and Prof. Dr. Utkir Rozikov
for valuable support and suggestions. We are also grateful to anonymous referees whose comments
greatly improved the paper.
23003-10
Phase transitions for the Potts model
References
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doi:10.1007/s10955-009-9869-z.
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doi:10.1063/1.1781747.
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ics, 1, Providence, R. I., Amer. Math. Soc., 1980.
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doi:10.1007/s11040-009-9056-0.
13. Georgii H.O., Gibbs Measures and Phase Transitions, De Gruyter studies in Mathematics, 9. Berlin,
New York, 1988.
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doi:10.1007/BF01017186.
15. Ganikhodjaev N.N., Temir S., Akin H., CUBO Math. J., 2005, 7, 37–48.
16. Mukhamedov F., Rozikov, U., J. Stat. Phys., 2005, 119, 427–446; doi:10.1007/s10955-004-2056-3.
17. Hall H.S., Knight S.R., Higher Algebra: A Sequel to Elementary Algebra for Schools. Macmillan,
London, 1950.
18. Devaney R.L., An Introduction to Chaotic Dynamical Systems. Benjamin/Cummings, Menlo Park,
CA, 1986.
19. Smith R.T., Minton R.B., Calculus. 2 ed., McGraw-Hill Science/Engineering/Math, 2002.
20. Ganikhodjaev N.N., Teor. Mat. Fiz., 1990, 85, 163–175 (in Russion); [Engl. transl.: Theor. Math.
Phys., 1990, 85, 1125–1134; doi:10.1007/BF01086840].
До фазових переходiв моделi Поттса з трьома
конкуруючими взаємодiями на деревi Келi
Г. Акiн1, С. Темiр 2
1 Унiверситет Зiрве, 27260 Газiантеп, Туреччина
2 Унiверситет Гарран, 63200 Санлiурфа, Туреччина
У цiй статтi ми вивчаємо проблему фазового переходу для моделi Поттса з трьома конкуруючими
взаємодiями, найближчих сусiдiв, наступних близьких сусiдiв i наступних за наступними близьки-
ми сусiдами, з ненульовим зовнiшнiм полем на деревi Келi другого порядку. Ми доводимо, що для
деяких значень параметрiв моделi є фазовий перехiд. Ми приводимо проблему опису граничних
мiр Гiббса до проблеми розв’язку системи нелiнiйних функцiональних рiвнянь. Ми розширюємо ре-
зультати, отриманi Ганiходяєвим i Розiковим [Math. Phys. Anal. Geom., 2009, 12, No 2, 141–156] для
фазового переходу в моделi Iзiнга, на випадок моделi Поттса.
Ключовi слова: фазовий перехiд, модель Поттса, конкуруючi взаємодiї, мiра Гiббса
23003-11
http://dx.doi.org/10.1103/RevModPhys.54.235
http://dx.doi.org/10.1007/s10955-009-9869-z
http://dx.doi.org/10.1103/PhysRevB.34.7975
http://dx.doi.org/10.1214/aop/1176996347
http://dx.doi.org/10.1063/1.1781747
http://dx.doi.org/10.1007/s11040-009-9056-0
http://dx.doi.org/10.1007/BF01017186
http://dx.doi.org/10.1007/s10955-004-2056-3
http://dx.doi.org/10.1007/BF01086840
Introduction
Preliminaries
Recursive equations for Partition Functions
Translation-invariant Gibbs measures
First case
Illustrative example
Second case
Illustrative example
Conclusion
|