Gibbs measures and free energies of Ising–Vannimenus model on the Cayley tree

Abstract

In this paper, we consider the Ising-Vannimenus model on a Cayley tree for order two with competing nearest-neighbor and prolonged next-nearest neighbor interactions. We stress that the mentioned model was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical approach for the considered model. We find certain conditions for the existence of Gibbs measures corresponding to the model, which allowed to establish the existence of the phase transition. Moreover, the free energies and entropies, associated with translation invariant Gibbs measures, are calculated.Mathematics Subject Classification: 82B26, 46S10, 12J12, 39A70, 47H10, 60K35.

Gibbs measures of an Ising model with competing interactions on the triangular chandelier-lattice

“…If we take J SL = 0, then the equation is the same as the dynamical system in [28]. The recurrence equations obtained in the present paper totally differ from [1,28,34,39]. Note that for the Ising model associated with the Hamiltonian (3.4) on the chandelier lattices of order k, in contrast to the symmetry of arbitrary order Cayley tree [6,39], if k > 3, then the chandelier lattice of order k is not symmetry.…”

“…In [34,43], the authors have presented, for the Ising model on the Cayley tree, some explicit formulae of the free energies (and entropies) according to boundary conditions (b.c.). By applying the general formulae to various known boundary conditions on arbitrary order chandelier-lattices, we plan to obtain some explicit formula of free energy and relative entropy corresponding to the boundary conditions in our future work.…”

“…Previously, researchers frequently used memory of length 1 over a Cayley tree to study Gibbs measures [31]. In [1,21,22,34], the authors have studied Gibbs measures with a memory of length 2 for generalized ANNNI models on a Cayley tree of order 2 by means of a vector valued function (see [35] for details). In [1,21], the next generalizations are considered.…”

“…These authors have defined Gibbs measures or Gibbs states with a memory of length 2 (on spin-configurations σ) for generalized ANNNI models on Cayley trees of order 2. In [34], the authors have obtained some rigorous results to propose a measure-theoretical approach for the Ising-Vannimenus model.…”

On ground states and phase transition for λ-model with the competing Potts interactions on Cayley trees

“…Note that number of the configurations ϕ (11) i,j , |i − j| = 1, i, j ∈ Φ is four. Theorem is proved.…”