Mixed-spin Ising model on the Bethe lattice

Abstract

We obtain the phase diagram of a ferromagnetic mixed Ising system, consisting of spin--, and spin-5 variables, on a Bethe lattice of coordination number z, with nearest-neighbor exchange interactions and single-ion terms. The problem is formulated as a discrete nonlinear map. There is a tricritical point for S integer and z~5. In the infinite-coordination-number limit, we regain the results of an exact calculation for a Curie-Weiss version of the model.

Exact results of a mixed spin-1/2 and spin-S Ising model on a bathroom tile (4–8) lattice: Effect of uniaxial single-ion anisotropy

“…It is worth mentioning that in this way generalized mapping transformations were recently employed to obtain exact results of the mixed-spin Ising models on the honeycomb lattice [4] as well as on some decorated planar lattices [5]. To the best of our knowledge, these are the only mixed-spin planar Ising models with generally known exact solutions except several mixed-spin Ising models on the Bethe (Cayley tree) lattices studied within a discrete non-linear map [6] and an approach based on exact recursion equations [7]. Among the remarkable models for which a precise solution is restricted to a certain subspace of interaction parameters only, one should further mention the mixed-spin Ising model on the Union-Jack (centered square) lattice treated within the mapping onto a symmetric eight-vertex model [8].…”

“…The purpose of this work is to provide an exact formulation of the mixed spin-1/2 and spin-S (S ≥ 1) Ising model on the bathroom tile (4)(5)(6)(7)(8) lattice and to establish accurate phase diagrams of this system for several values of the quantum spin number S. Exact results for the system under consideration are obtained by applying the generalized star-triangle mapping transformation constituting an exact correspondence with an effective spin-1/2 Ising model on the Shastry-Sutherland (orthogonal-dimer) lattice (further abbreviated as SSL), which has recently been solved by the present author elsewhere [9]. The obtained phase diagrams will also be compared with those of another exactly soluble three-coordinated planar Ising model, namely, on the mixedspin honeycomb lattice [4].…”

The mixed spin-1/2 and spin-1 Ising system on a two-layer Bethe lattice

“…The different results achieved here bear some resemblances with those reported in Refs. [12,50,57,58]. The present model can be extended to study spin transitions in spin-crossover materials.…”

“…One often relies on approximate methods such as the Bethe approximation [47,48] which yields results that are better than those obtained using the common mean field scheme [49]. Exact calculations are possible when the model is defined on the Bethe lattice [50]. On such a recursive graph, the behaviour of one spin, say the central spin, can lead to the full picture of the whole system.…”

Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagomé lattice

“…The effect of uniaxial and biaxial single-ion anisotropies has been precisely investigated on the mixed-spin honeycomb lattice [7] as well as on some decorated planar lattices [8]. With the exception of several mixed-spin models formulated on the Bethe (Cayley tree) lattices, which can be accurately treated within a discrete non-linear map [9] or exact recursion equations [10], these are the only mixed-spin planar Ising models with generally known exact solutions, yet.…”