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 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.


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 J k of order k 2, i.e., a graph without cycles with (k + 1) edges issuing from each vertex except x 0 and with k edges issuing from the vertex x 0 , 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][5][6][7][8][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 Hamiltonian 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.

Preliminaries
Cayley tree. The Cayley tree Γ k (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 Γ k = (V, L, i), where V is the set of vertices of Γ k , L is the set of edges of Γ k 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 A collection of the pairs x 0 , x 1 , . . . , x d−1 , y is called a path from x to y. We write x < y if the path from x 0 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., It is clear that for any vertex x = x 0 , x has k direct successors and x 0 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 W n 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 = z.
For neighbors triple x, y, z we consider the following Kroneker's symbol where x, z ∈ W n and y ∈ W n−1 for some n. We can write (2.1) as follows: 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 J k (see [15] for details).
The Potts model with three competing interactions and external field is defined by the following Hamiltonian 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, J 1 , J 2 ∈ R are coupling constants and h is external field. In [15], the authors have already obtained the exact solutions of the Potts model with competing ternary and binary interactions and external field on Cayley tree described by means of the Hamiltonian (2.2), where h = 0 and J 1 = 0. In this paper, we assume that each parameter J, J 1 , J 2 , h is different from zero.

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). Letσ(V\Λ) be a fixed boundary configuration. The total energy of σ(Λ) under conditionσ(V\Λ) is defined as x,y,z :x,y,z∈Λ 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(V n ) and the partition functions Z Vn in volume V n we use notations σ n 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 σ Λ is defined by where Z Λ (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 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 For a given n, let us define the partial partition functions of the next generation by where i = 1, 2, 3.

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Let θ = exp(βJ), θ 1 = exp(βJ 1 ), θ 2 = exp(βJ 2 ) and θ 3 = exp(βh). From (3.1) and (3.2), we can calculate the partial partition functions as follows: we have the following system of recurrent equations We define the transformation  (3.4) can be written as w n = F(w n−1 ), n > 0. In the theory of dynamical systems, w n is called trajectory of the initial point w 1 under the action of the mapping F. So, the asymptotic behavior of Z (n) for n → ∞ can be determined by value of lim n w n .
If u = lim n u n and v = lim n v n then we have the following equations; 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 {h x , x ∈ V }, which satisfy a non-linear equation. The recursive equations (3.6) considered in this paper describe a vector function {(u n , v n ), n ∈ N} which is a particular case of the above mentioned function h x obtained as h x = u n if x ∈ W n 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, θ 1 , θ 2 , θ 3 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 u 0 of f such that for any value of u in a domain that is close enough to u 0 , the iterated function sequence u, f (u), f (f (u)), f (f (f (u))), . . . converges to u 0 . An attractive fixed point is said to be a stable fixed point if it is also Lyapunov stable (see [18] for details).

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 system (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].

First case
Assume that u = v, then some solutions of the system (3.6) can be found from equation In this case, in order to obtain the solutions of the system (3.6), we should analyze the equation in (4.1).
We now have g ′′ (u) = 12Au 2 + 6Bu + 2C. Assume that From Theorem 1, g is concave downward on interval and g is concave upward on interval R \ I. Notice that at the point the graph chances from concave downward to concave upward. Thus, is one of the inflection points of g. It is clear that g(0) = E > 0 and lim u→∞ 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 polynomial (4.2) has at most two positive roots u * 1 , u * 2 if C < 0. In this case, ifθ > √ 2 and (2θ 3 − 6) 2 + 32θ 2 (θ 4 +θ 2 − 6) < 0 then the coefficient C is negative.
Due to 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: Let us define the set H = (θ 1 ,θ) : C < −2 √ AE (see figure 1). Then, we have the following Corollary.
It is easy to show that these three measures correspond to boundary conditionsσ(V \ V n ) ≡ i, i = 1, 2, 3.
In figure 1 we show the region with three positive roots. For parametersθ, θ 1 in phase transition region the equation (4.1) has three positive roots, where two of them are stable and the third one is unstable.

Second case
Now, let us find the other solutions of the system (3.6). Assume that u = v. Subtracting the second equation of (3.6) from the first we have After canceling to (u − v) two sides of the equation (4.5) we can obtain the following equation; where u + v = s and uv = t.

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From (4.6) and (4.7) we obtain the following equality; Let us consider a system of equations From (4.9), we can write v = s − u and u(s − u) = t, thus we have To have two different real roots of the second-order equation (4.10), inequality s 2 − 4t > 0 should be valid. Then the equations (4.9) have solutions, if t < s 2 /4. Thus, from (4.7) and from the inequality t < s 2 /4 we can get the following equation Then, there is such that for all s > s * the inequality t < s 2 /4 is valid.
One can show that the inequality (4.11) is satisfied, if and θ 1 (θ 2 − 3) − 2θ > 0. Thus, forθ > √ 3 and 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 then the equation (4.8) has only one positive root.

If
there is k 0 such that the line y − 2/θ 3 = k 0 s is a tangent for the parabola on the right-hand side of (4.8) and then for θ 3 < θ 1 (θ + 1)/k 0 the equation (4.8) has two positive roots and only one of them is greater than s * .
3. The equation (4.8) has two positive roots, when where k 0 is defined as above.

Illustrative example
Let us give an illustrative example. With the help of the Mathematica software, we have manipulated the line on the left-hand side and the parabola on the right-hand side of the equation (4.8) for some parameters θ, θ 1 , θ 2 and θ 3 . 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 figure 3 (b), the line y − 2/θ 3 = θ 1 (θ + 1)s/θ 3 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, θ 1 , θ 2 and θ 3 . Recalling the results for the equations (3.6), we get the following result: if (θ, θ 1 , θ 2 ) ∈ M and K < θ 3 < L, where 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.

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, J 1 , J 2 and external field h to study the existence problem of a phase transition. We have considered transformation F = (F 1 ; F 2 ) : 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, J 1 , J 2 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.