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In this paper we investigate the Ising model on Z² with competing interactions. In this model we consider J1 as horizontal interactions and J2 as vertical interactions where J1,J2 > 0. We prove that this model can reach a phase transition. Onsager considered the case where horizontal interaction parameter J1 and vertical interaction parameter J2 are different. For any fixed J1 and J2, he showed that below a critical temperature Tc which depends on J1 and J2, phase transition occurs using some matrix transfer method. However in this paper we will prove the existence of phase transition using contours methods introduced by Sinai. We will show that there exists a ß0 > 0 such that for ß > ß0 there exist at least two limit Gibbs distribution which leads to the phenomena of phase transition. [ABSTRACT FROM AUTHOR]

In this paper we investigate the problem of phase transition for Ising model on a semi-infinite Cayley tree of third order with inhomogeneous left, middle and right parameters describing the strength of the interactions between left, middle and right nearest-neighbors respectively. We produce a recurrent equation that describes a limit behavior of the sequence conditional Gibbs measures and prove that if at least two of the parameters are positive one can reach phase transition. [ABSTRACT FROM AUTHOR]

In this paper we consider Ising models with competing interactions defined on semi-infinite Cayley tree of second order. Phase diagram of such models contain new modulated phases, in addition to the expected paramagnetic and ferromagnetic phase. The aim of this paper is to select phases of these models with competing interactions on Cayley tree which exhibit phase transitions and at the same time can be solved exactly. [ABSTRACT FROM AUTHOR]

In this research it has been investigated the spin compensation points of a mixed spin-3 and spin-7/2 Blume-Capel Ising system on a square lattice. Magnetic phase transitions, in a molecular based magnet, are tested by using mean field method (MFM) Equilibrium magnetizations can be accomplished by minimizing free energy, which is also a significant technique for locating and inducing spin compensation points. On principles of Bogoliubov inequality of the free energy, it has been investigated the second-order phase transitions in the (DB / |J| , KBTC / |J|), or (DA /|J|, KBTC / |J|) planes, respectively. Distinctive ferrimagnetic traits, in specific, have been demonstrated focusing on the negative numbers of the various single-ion anisotropies interacting on the A− atoms and B− atoms, respectively. [ABSTRACT FROM AUTHOR]

In this paper we consider Ising model on the semi-infinite Cayley tree of second order with left and right interaction and investigate the problem of phase transition for this model. [ABSTRACT FROM AUTHOR]

We investigate an Ising model with two restricted competing interactions (nearest neighbors, and one-level neighbors) on the Cayley tree of order 5. The translation Gibbs measures is considered for this model. Our result of the critical curve shows that the phase transition occurs in this model, further it confirms a particular case of a conjecture. [ABSTRACT FROM AUTHOR]