arXiv subjects
Publications and source records attributed to M. M. Rahmatullaev.
We consider the Ising model with competing interactions and a nonzero external field on the Cayley tree of order two. We describe ground states and verify the Peierls condition for the model. Using a contour argument we show the existence of two different Gibbs measures.
We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values $0,1,\ldots,m,$ $m\geq2,$ and nonzero external field, on a Cayley tree of degree $k$ (with $k+1$ neighbors). We are aiming to extend the results of \cite{rs} where the SOS model is studied with (mainly) three spin values and zero external field. The SOS model can be treated as a natural generalization of the Ising model (obtained for $m=1$). We mainly assume that $m=2$ (three spin values) and study translation-invariant (TI) and splitting (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) For $m=2$, in the antiferromagnet (AFM) case, a TISGM is unique for all temperatures with an external field. In the ferromagnetic (FM) case, for $m=2,$ the number of TISGMs varies with the temperature and the external field: this gives an interesting example of phase transition. Our second result gives a classification of all TISGMs of the Three-State SOS-Model on the Cayley tree of degree two with the presence of an external field. We show uniqueness in the case of antiferromagnetic interactions and the existence of up to seven TISGMs in the case of ferromagnetic interactions, where the number of phases depends on the interaction strength and external field.
In this paper, we consider the Potts-SOS model where the spin takes values in the set $\{0, 1, 2\}$ on the Cayley tree of order two. We describe all the translation-invariant splitting Gibbs measures for this model in some conditions. Moreover, we investigate whether these Gibbs measures are extremal or non-extremal in the set of all Gibbs measures.
We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values $0,1,2,...,m, m\geq2$ and non zero external field, on a Cayley tree of order $k$. In the case $k=2, m=2$, we describe translation-invariant ground states for the SOS model with a translation-invariant external field. Some periodic ground states for the SOS model with periodic external field are described.
In this paper for the Potts-SOS model on a Cayley tree under some conditions the existence of at least one periodic (non translation-invariant) Gibbs measure is proved.
We consider Potts model, with competing interactions and countable spin values $Φ=\{0,1,\dots \}$ on a Cayley tree of order three. We study periodic ground states for this model.
We consider translation-invariant splitting Gibbs measures (TISGMs) for the $q$-state Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is $2^{q}-1$. In this paper for each TISGM $μ$ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with $μ$.
For the Potts model on the Cayley tree, some explicit formulae of the free energies and entropies (according to vector-valued boundary conditions (BCs)) are obtained. They include translation-invariant, periodic, Dobrushin-like BCs, as well as those corresponding to weakly periodic Gibbs measures.
Recently by Rozikov an Ising model with competing interactions and spin values $\pm 1$, on a Cayley tree of order $k\geq 1$ has been considered and the ground states of the model are described. In this paper we describe some weak periodic ground states of the model.