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Two Potts Models on Cayley Tree of Arbitrary Order

On a Cayley tree of arbitrary order k we consider two different Potts models with competing nearest-neighbour interactions J1 and next-nearest-neighbour interactions Jp and Jo, where coupling Jp corresponds to interaction of spins belonging to the same branch of the tree (prolonged) and coupling Jo corresponds to interaction of spins belonging to the same shell of the tree (one-level) and find for each model in addition to the expected paramagnetic, ferromagnetic and antiferromagnetic phases, an intermediate range of coupling values where the local magnetization has chaotic oscillatory glass-like behaviour. We also show that the ranges corresponding two models are different.


Introduction
Consideration of spin models with multispin interactions has proved to be fruitful in many fields of physics, ranging from the determination of phase diagrams in metallic alloys and exhibition of new types of phase transition, to site percolation. Systems exhibiting spatially modulated structures, commensurate or incommensurate with the underlying lattice, are of current interest in condensed matter physics [1]. Among the idealized systems for modulated ordering, the axial next-nearest-neighbour Ising (ANNNI) model, originally introduced by Elliot [2] to describe the sinusoidal magnetic structure of Erbium, and the chiral Potts model, introduced by Ostlund [3] and Huse [4] in connection with monolayers adsorbed on rectangular substrates, have been studied extensively by a variety of techniques. A particularly interesting and powerful method is the study of modulated phases through the measure-preserving map generated by the meanfield equations, as applied by Bak [5] and Jensen and Bak [6] to the ANNNI model. The main drawback of the method lies in the fact that thermodynamic solutions correspond to stationary but unstable orbits. However, when these models are defined on a Cayley tree ( or on a the Bethe lattice; see [7] for terminology), as in the case of the Ising model with competing interactions examined by Vannimenus [8], it turns out that physically interesting solutions correspond to the attractors of the mapping. This simplifies the numerical work considerably, and detailed study of the whole phase diagram becomes feasible. Apart from the intrinsic interest attached to the study of models on trees, it is possible to argue that the results obtained on trees provide a useful guide to the more involved study of their counterparts on crystal lattices.
The ANNNI model, which consists of an Ising model with nearest-neighbour interactions augmented by competing next-nearest-neighbour couplings acting parallel to a single axis direction, is perhaps the simplest nontrivial model displaying a rich phase diagram with a Lifshitz point and many spatially modulated phases. There has been a considerable theoretical effort to obtain the structure of the global phase diagram of the ANNNI model in the T −p space, where T is temperature and p = −J 2 /J 1 is the ratio between the competing exchange interactions. On the basis of numerical mean-field calculations, Bak and von Boehm [9] suggested the existence of an infinite succession of commensurate phases, the so-called devil's staircase, at low temperatures. This mean-field picture has been supported by low-temperature series expansions performed by Fisher and Selke [11]. At the paramagnetic-modulated boundary analytic mean-field calculations show that the critical wave number varies continuously and vanishes at the Lifshitz point.
A phase diagram of a model describes a morphology of phases, stability of phases, transitions from one phase to another and corresponding transition lines. The Cayley tree is not a realistic lattice; however, its amazing topology makes the exact calculation of various quantities possible. For many problems the solution on a tree is much simpler than on a regular lattice and is equivalent to the standard Bethe-Peierls theory [20]. On the Cayley tree one can consider two type of next-nearest-neighbours: prolonged and one-level next-nearest-neighbours (definitions see below). In the case of the Ising model with competing nearest-neighbor interactions J and prolonged next-nearest-neighbour interactions J p Vannimenus [8] was able to find new modulated phases, in addition to the expected paramagnetic and ferromagnetic ones. From this result follows that Ising model with competing interactions on a Cayley tree is real interest since it has many similarities with models on periodic lattices. In fact, it has many common features with them, in particular the existence of a modulated phase, and shows no sign of pathological behavior -at least no more than mean-field theories of similar systems [8]. Moreover a detailed study of its properties was carried out with essentially exact results, using rather simple numerical methods.
This suggest that more complicated models should be studied on trees, with the hope to discover new phases or unusual types of behavior. The important point is that statistical mechanics on trees involve nonlinear recursion equations and are naturally connected to the rich world of dynamical systems, a world presently under intense investigation [8].
The Potts model (with q ≥ 2 spin values ) was introduced as a generalization of the Ising model [26]. At present the Potts model encompasses a number of problems in statistical physics and lattice theory (see for example [27]). A Potts model just as an Ising model with competing interactions has recently been studied extensively because of the appearance of nontrivial magnetic orderings ( see [8], [12], [13], [15], [17], [18], [19], [16], [21], [22], [23], [28], [29] and references therein). The Potts model with competing interactions J 1 and prolonged next-nearest neighbours interactions J p on Cayley tree of second order have been studied in detail in [19], and Potts model that include also interaction of one-level next-nearest-neighbour interaction J o have been studied in [28]. In this paper we define a single-trunk Cayley tree, produce the recursion equations for model with competing interactions on the Cayley tree and for the same model on the single-trunk Cayley tree, and show how to reduce the recursion equations on Cayley tree to the simpler recursion equations on the single-trunk Cayley tree. Note that this approach is mathematical justification only of the method developed in [8] and later generalized in [12]. We will consider also the symmetry group of the corresponding model [10]. The aim of this paper is to extend the results of [28] to the Potts model with competing interactions on a Cayley tree of arbitrary order k and to clarify the role of the order k. The paper has been organized in the following way. In Section 2, the model Hamiltonian is discussed. In Section 3, the recursion equations are defined. Section 4 is devoted to the discussion of the phase diagram features. Finally, the conclusions are given in Section 5.

Cayley Tree
A Cayley tree Γ k of order k ≥ 1 (or the Bethe lattice; see [7] for terminology) is an infinite tree, i.e., a graph without cycles with exactly k + 1 edges issuing from each vertex. Let denote the Cayley tree as Γ k = (V, Λ), where V is the set of vertices of Γ k , Λ is the set of edges of Γ k . Two vertices x and y, x, y ∈ V are called nearest-neighbors if there exists an edge l ∈ Λ connecting them, which is denoted by l =< x, y >. The distance d(x, y), x, y ∈ V , on the Cayley tree Γ k , is the number of edges in the shortest path from x to y. For a fixed x 0 ∈ V we set and L n denotes the set of edges in V n . The fixed vertex x 0 is called the 0-th level and the vertices in W n are called the n-th level. For the sake of simplicity we put |x| = d(x, x 0 ), x ∈ V . Two vertices x, y ∈ V are called the next-nearest-neighbours if d(x, y) = 2. The nextnearest-neighbour vertices x and y are called prolonged next-nearest-neighbours if |x| = |y| and is denoted by > x, y <. The next-nearest-neighbour vertices x, y ∈ V that are not prolonged are called one-level next-nearest-neighbours since |x| = |y| and are denoted by> x, 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 nearest neighbours. The set of the direct successors of x is denoted by S(x), i.e., if x ∈ W n , then We observe that for any vertex x = x 0 , x has k direct successors and x 0 has k + 1.
The collection S(x) = {y 1 , · · · , y k } we will call one-level k tuple of neighbours. Note that if k = 2 then S(x) is a one-level next-nearest-neighbours.
Below we will consider a semi-infinite Cayley tree Γ k + of k-th order, i.e. an infinite graph without cycles with k + 1 edges issuing from each vertex except for x 0 which has only k edges. In this case |S(x)| = k for any x ∈ V. Let < x 0 , x >= l ∈ L be an edge of semi-infinite Cayley tree Γ k + . Definition 2.1 The infinite subtree Γ k + (l) = (V l , L l ) is called a single-trunk Cayley tree, if from vertex x 0 a single edge l emanates and from any other vertex x ∈ V l , x = x 0 exactly k + 1 edges emanate.

The Models
Let Γ k + be a semi-finite Cayley tree of order k. In addition to nearest-neighbours interaction J 1 and prolonged next-nearest -neighbours interaction J p we consider also interactions of sites in the set S(x) = {y 1 , · · · , y k } of direct successors for any x ∈ V. Here one can introduce two type interactions: 1) jointly interaction of all sites in S(x), called one-level k-tuple interaction ; 2) binary interaction of any two sites in S(x) = {y 1 , · · · , y k }, called one-level next-nearest neighbours interaction. For the Potts model with spin values in Φ = {1, 2, 3}, i.e., the spin variables σ(x), x ∈ V assume the values {1, 2, 3}, the relevant Hamiltonians with competing nearest-neighbor interactions J 1 , prolonged next-nearest-neighbor interactions J p and one-level k-tuple neighbours or one-level next-nearest neighbors interaction J o have the forms:

1) Model with one-level k-tuple neighbours interactions
2) Model with one-level next-nearest neighbor interactions where J 0 , J p , J 1 ∈ R are coupling constant, δ in second and third sum in (1) and all sums in (2) is the usual Kronecker symbol and δ in first sum in (1) is the generalized Kronecker symbol, that is defined as follow: Note that for k = 2 generalized Kroneker's symbol coincides with usual one and one-level 2tuple of neighbours coincide with one-level next-nearest neighbours. These models recover that in [19] for k = 2 with J o = 0 and in [28] for k = 2 with J o = 0 and they coincide if J o = 0. The generalized Kroneker's symbol was introduced in [24]. The Ising model with one-level ktuple interactions was considered in [23] and Ising model with one-level next-nearest-neighbours interactions was considered in [30].

Conditional Gibbs measures with symmetries
Recall that in the case of the Ising model with competing interactions examined by Vannimenus [8], it turns out that physically interesting solutions correspond to the attractors of the recurrence relations, where initial conditions chosen randomly. The main drawback of the method lies in the fact that thermodynamic solutions correspond to stationary but unstable orbits. In this paper we produce recurrence relations fixing boundary configurationσ(V \ Λ). Let Λ be a finite subset of V. We will denote by σ(Λ) the restriction of a configuration σ : V → Φ = {1, 2, 3} to Λ. Letσ(V \ Λ) be a fixed boundary configuration. As usual, one can introduce the notions of total energy H(σ(Λ)|σ(V \ Λ)) of configuration σ(Λ) under boundary conditionσ(V \ Λ) and partition function Z Λ (σ(V \ Λ)) in volume Λ under boundary condition σ(V \ Λ) that is defined as where Ω(Λ) is the set of all configurations in volume Λ, k B is Boltzmann constant and T is the absolute temperature. Then conditional Gibbs measure µ Λ of a configuration σ(Λ) is defined as We consider the configuration σ(V n ), the partition function Z Vn (σ(V \ V n )) and conditional Gibbs measure µ Vn (σ(V n )|σ(V \ V n )) in volume V n and for brevity denote them as σ n , Z (n) and µ n respectively. Let S 3 be a group of all permutations of a set Φ = {1, 2, 3}. Definition 2.2 For any permutation π ∈ S 3 let us define transformation T π : Ω → Ω by the following way: for any σ ∈ Ω assume T π σ(x) = T π (σ(x)) for any x ∈ V. It is evident the following Proposition.  (1) and (2) are S 3 invariant, i.e., for any π ∈ S 3 we have H(T π σ) = H(σ) for any σ ∈ Ω. In order to produce the recurrence equations, we consider the relation of the partition function on V n+1 to the partition function on subsets of V n . Given the initial condition σ( with be a configuration on the set S = x 0 ∪ S(x 0 ) and Ω(S) be the set of all such configurations. Assume be the partition function on V n with fixed configuration σ S on V 1 . There are a priori 3 k+1 different partition functions Z (n) (σ S ) and the partition function Z (n) in volume V n can be written as follows: Lastly let Z (n) (i, j) be a partition function on the single-trunk Cayley tree V l n with the configuration (i, j) on an edge l =< x 0 , y >, where y ∈ W 1 and i, j = 1, 2, 3.

Basic Equations
In order to set up our basic equations in a recurrence scheme relating the partition function of an n-generation tree to the partition functions of its subsystems, we should take into account the partial partition functions for all the possible configurations of the spins in two successive generations. Below we produce the recurrent equations relating the partition functions Z (n) (i, j) for Hamiltonian considered on the single-trunk Cayley tree.

First Model
Firstly we consider Potts model (1) with competing prolong next-nearest-neighbour interactions, nearest-neighbour interactions and one level k-tuple neighbour interactions. We introduce 5 new variables u (n) 1 = q Z (n) (1, 1, 1, · · · , 1) = a Z (n) (1, 1) where k is the order of the Cayley tree. Let Z (n) (i 0 , i 1 , i 2 , · · · , i k ) be a partition function and let m 1 be the number of spins {1} and m 2 be the number of spins {2} on first level W 1 . Then we have One gets the following system of recurrent equations through direct calculation: The total partition function is given in terms of (u i ) by For discussing the phase diagram, the following choice of reduced variables is convenient (see (8)): As shown above, in the paramagnetic phase (high symmetry phase, i,e., symmetry with respect to S 3 ) we have u order parameter like y 1 , y 2 and y 3 . The relations (6) now have following form: where

Second Model
Now we consider Potts model (2) with competing prolong next-nearest-neighbour interactions, nearest-neighbour interactions and one level next nearest neighbour interactions. Over again, we select only five variables with introducing new variables, then we have In this case assume where A = a m 1 2 +m 2 2 +m 1 m 2 −k(m 1 +m 2 ) . Next, we produce the following recurrence system through a direct calculation: Then, total partition function is given in terms of (u i ) by Choosing the same reduced variables as above we produce following recurrent equations (11 ): and

The Average Magnetization
In order to investigate the phase diagram we have to look at local properties, namely the local magnetization or the magnetization of the root x 0 . The average magnetization m for the n-th generation is given by, Since the form of spins of the Potts model is unessential one can replace the set of spin values {1,2,3} by the centered set {-1,0,1} [31] and then we havẽ Below we will consider the average magnetizationm to study the wavevectors.
Average magnetization for first model is given bỹ and average magnetization for second model is given bỹ where M (i, j, x, y 1 , y 2 ,

Morphology of phase diagrams
It is convenient to know the broad features of the phase diagram before discussing the different transitions in more detail. Below we will consider phase diagram for k = 3. This can be achieved numerically in a straightforward fashion. The recursion relations (11) for the first model and (12) for the second model provide us the numerically exact phase diagram in (−J p /J 1 , J 0 /J 1 , T /J 1 ) space. Let β = −J p /J 1 , α = T /J 1 and γ = J 0 /J 1 , by first model and second model, for some fixed values of , starting from initial conditions with the parameters a = exp (kα) −1 γ , b = exp −α −1 β , c = exp α −1 and for second model's as with the parameters a = exp α −1 γ , b = exp −α −1 β , c = exp α −1 corresponding to boundary conditionσ ≡ 1, one iterates the recurrence relations (11) and (12), we observe their behaviour after a large number of iterations. In the simplest situation a fixed point (x * , y * 1 , y * 2 , y * 3 ) is reached. It corresponds to paramagnetic phase if y * 1 = 0, y * 2 = 0, y * 3 = 0 to a ferromagnetic phase if y * 1 , y * 2 , y * 3 = 0. The system may be periodic with period p, where case p = 2 correspond to antiferromagnetic phase and case p = 4 correspond to so-called antiphase that denoted < 2 > for compactness. Finally, the system may remain aperiodic. The distinction between a truly aperiodic case and one with a very long period is difficult to make numerically. Below we consider periodic phase with period p where p ≤ 4. All periodic phase with period p > 4 and aperiodic phase we will consider as modulated phase.
The resultant phase diagram on the plane (β, α) for both models with some fixed values of γ and k are shown in Figs. 1 and 3. (Here: P -paramagnetic phase, F -ferromagnetic phase, AFantiferromagnetic phase, P3 -phase with period 3, <2> -antiphase, and M -modulated phase) Looking on the fig. 5, one can see that the difference between both models occur in the case of k = 4 and γ = 10, as phase of period 3 and antiphase not exist in the third quadrant of our investigation area. (Figs. 5 (c) and (d)). Furthermore, in fig. 6, the phase diagrams were produced with order k = 10 and γ = ±5 give an additional in sizes of some phases. In Fig.  6(c), we found that a small phase of antiferromagnetic occur in second quadrant for the second model. Compare to first model, as in first quadrant for second model, we have more region of paramagnetic and modulated phases (see Fig. 6(c)). In addition, one can see in Fig. 6(d), as third quadrant of second model, same region of paramagnetic and modulated phases occur and increase as compare to the first model.
From the previously presented resultant phase diagrams, one can see that an investigation on the behaviour of the system (11) and (12) can be performed by applying the numerical approach. The systems were iterated in a large number before the detailed behaviour was studied and visualized on the (α, β) space. As been discussed, differences in the values of k results in differences and changes in the phase. Hence it is very difficult for us to study the behaviour of the systems analytically. Yet, the transition line separating the phases is suggested to be solved analytically. This can be done by linearizing both systems (11) and (12). Later on, the linearized systems and analytical justification of the phase diagrams will be discussed in more detailed.

Conclusion
In this paper, we study on relationship between two types of one-level neighbour interactions of Potts model on arbitrary order Cayley tree. One can consider for one-level neighbour interactions, k -tuple interactions and binary interactions. We study on the differences for both types of model on Cayley tree order 3 and order 4 and order 10.
We have found the phase diagram of Potts model (1) and (2) and show that there consist of six phases: ferromagnetic, paramagnetic, antiferromagnetic, period 3, antiphase and modulated phase. Moreover, we study on the difference of phase diagram for both types of model by fixing γ equal to ±0.1, ±1, ±5, and ±10 . From considerations above, we can see that role of k rather significant because there show the phase diagram quite different for each k in first and second model. Indeed the first and second model show us the difference, even though we only considering two different way in competing the one level interactions. We can say that this two model are differ to each other. They may give an unique significant to any physical phenomena occur. Further investigation can be consider to understand how this difference attribute appear.

Acknowledgments
This work was supported by Fundamental Research Grant Scheme FRGS10-022-0141.