Ground states and Gibbs measures for the sos model with competing interactions on a Cayley tree of order two

Ground states and Gibbs measures for the SOS model with competing interactions on a Cayley tree of order two
¹M. M. Rahmatullae, ²B.U. Abraev
¹Institute of mathematics after named V.I.Romanovsky: E-mail: [e-mail]
²Chirchik State Pedagogical University: E-mail: [e-mail]
Let be the Cayley tree of order , i.e., an infinite tree such that exactly edges are incident to each vertex. Here is the set of vertices and is the set of edges of Let denote the free product of cyclic groups of order 2 with generators i.e., let (see [1]). For an arbitrary vertex we put
where is the distance between and in the Cayley tree, i.e., the number of edges of the path between and For each let denote the set of direct successors of i.e., if then For each let denote the set of all neighbors of i.e. The set is a singleton. Let denote the (unique) element of this set. Let us assume that the spin values belong to the set A function is called configuration on The set of all configurations coincides with the set
The Hamiltonian of the model SOS model with competing interactions has the form:
(1)
where We consider the relative Hamiltonian describing the energy differences of the two configurations and
(2)
where
Let be the set of all unit balls with vertices in i.e. A restriction of a configuration to the ball is a bounded configuration and it is denoted by We define the energy of the configuration on by the following formula
(3)
where
We consider the case Let It is easy to see that for , where


Definition 2. The configuration is called the ground state for the Hamiltonian (1) if for any .
Let It is easy to check that





and
Definition 3. [4] Let be the complete set of all ground states of the Hamiltonian (1). A ball is said to be an improper ball of the configuration if for any . The union of the improper balls of a configuration is called the boundary of the configuration and denoted by .
Definition 4. [4] The relative Hamiltonian defined in (2) with the set of ground states satisfies the Peierls condition if for any and any configuration coinciding almost everywhere with , where is a positive constant which does not depend on and is the number of unit balls in .
Theorem 2. If then the Peierls condition is satisfied.
Let The following Lemmas can be proved very similarly to corresponding lemmas of [2], [3] and [4].
Lemma 4. Assume Let be a fixed contour and Then where and , -temperature.
Lemma 5. For any we have uniform convergence in

The following theorem is true
Theorem 3. If then for all sufficiently large there are at least three Gibbs measure for the model (1) on Cayley tree of order two.
References
1. Rozikov U.A., Gibbs measures on Cayley trees. World scientific. 2013.
2. Rozikov U. A., A contructite Description of Grond States and Gibbs Measures for Ising Model with two step interations on Cayley tree. Journal of Statistical Physics. Vol. 122, N2, 2006, pp. 217-235.3.
3. Botirov G.I., Rozikov U.A., Potts model with competing interactions on the Cayley tree: The contour method, Theor. Math. Phys., 153(1): 2007, pp. 1423-1433.
4. Rozikov U.A., A Contour Method on Cayley Trees, Journal of Statistical Physics, 2008, 130, pp. 801-813.