On the sos model with one-level competing interactions on a binary tree

Ключевые слова: дерево Кэли, модель SOS, конкурирующие взаимодействия, меры Гиббса.
IKKINCHI TARTIBLI KELI DARAXTI USTIDA BIR-QAVAT RAQOBATLASHUVCHI O’ZARO TA’SIRLI SOS MODELI
Abstrakt: Ikkinchi tartibli Keli daraxtida yaqin-qo’shnilar va bir-qavat raqobatlashuvchi o’zaro ta’sirli SOS modeli qaralgan. Bu model uchun har bir musbat yechimi Gibbs o’lchovlariga mos keladigan funksional tenglamalar sistemasi hosil qilingan.
Kalit so’zlar: Keli daraxti, SOS modeli, raqobatlashuvchi o’zaro ta’sir, Gibbs o’lchovlari.
In this work, we consider a SOS (solid-on-solid) model with nearest-neigbour and one-level second competing interactions on the Cayley tree of order two. We obtain a system of functional equations for this model, which each solution of the system corresponds to a limiting Gibbs measure.
The Cayley tree of order is an infinite tree, i.e., a cycles-free graph such that from each vertex of which issues exactly edges. We denote by the set of the vertices of tree and by the set of edges of tree. The distance on this tree, denoted by , is defined as the number of nearest-neighbour pairs of the minimal path between the vertices and (where path is a collection of nearest neighbour pairs, two consecutive pairs sharing at least a given vertex). For a fixed called the root, we set

and the set of direct successors of is denoted by

We observe that, for any vertex has direct successors and has The vertices and are called second neighbors which is denoted by if there exists a vertex such that and are nearest neighbors. We consider a semi-infinite Cayley of order i.e. a cycles-free graph with edges issuing from each except and with edges issuing from the vertex which is called the root. According to well-known theorems, this can be reconstituted as a Cayley tree [1]. The second neighbors is called one-level neighbors, if vertices and belong to for some , that is if they situated on the same level. We will consider only one-level second neigbours. In the SOS model, spin variables take their values on a discrete set which are associated with each vertex of the tree. The SOS model with competing two binary interactions is defined by the following Hamiltonian:

where the sum in the first term ranges all nearest neighbors, second sum ranges all one-level second neighbors, and (see Fig.1).
There are several approaches to derive equation or system equations describing limiting Gibbs measure for lattice models on Cayley tree [2]. One approach is based on properties of Markov random fields on Bethe lattices [2]. Another approach is based on recursive equations for partition functions (for example [3,4]). Naturally, both approaches lead to the same equation (see [2]). Since the second approach more suitable for models with competing interactions, in this work we follow this approach.

Fig. 1. The Cayley tree of order two with nearest-neighbour (–––) and one-level second neighbour (- - - -) interactions.
Let be a finite subset of . Denote by the restriction of to and let be a fixed boundary configuration. The total energy of under condition is defined as

Then partition function in volume boundary condition is defined as

where is the set of all configurations in volume and is the inverse of temperature.
We consider the configuration and the partitions functions in volume and for the sake of simplicity, we denote them as and , respectively. The partitions functions can be written as follows:
(1)

where
(2)

We will restrict ourselves to the case and
Denote

Let If and then from (1) and (2) we have

After denoting we have the system of recurrent equations:

If and then
(3)