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Physica Scripta

Uniaxial Anisotropy Effects in the Ising Model: an Exactly Soluble Model

L.L. Gonçalves

Published Sep 1, 1985• 106 citations
Publisher

Abstract

The effect of the uniaxial anisotropy in an Ising model consisting of mixed spins of magnitudes s and 1/2, on the honeycomb lattice, is studied. The exact solution is determined by mapping the model onto an effective spin 1/2 Ising model on the triangular lattice. The transition temperature is obtained as a function of the parameters and it is shown that for s integer there is no long range order for the parameter values in a determined region which is independent of the value of s. The critical exponents are obtained and it is shown that they are the same as those of the two-dimensional spin 1/2 Ising model.

DOI
10.1088/0031-8949/32/3/012

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The effect of the uniaxial anisotropy in an Ising model consisting of mixed spins of magnitudes s and 1/2, on the honeycomb lattice, is studied. The exact solution is determined by mapping the model onto an effective spin 1/2 Ising model on the triangular lattice. The transition temperature is obtained as a function of the parameters and it is shown that for s integer there is no long range order for the parameter values in a determined region which is independent of the value of s. The critical exponents are obtained and it is shown that they are the same as those of the two-dimensional spin 1/2 Ising model.
An Ising model consisting of mixed spins of magnitudes s and 1/2, on the honeycomb lattice is considered. The spins of a given magnitude occupy one of the two equivalent sublattices of the honeycomb lattice and the next-nearest-neighbour interactions are allowed between the spins 1/2 only. The exact solution of the problem, in the presence of uniaxial anisotropy, is obtained by mapping the model onto an effective spin 1/2 Ising model on a triangular lattice. The transition temperature as a function of the parameters of the Hamiltonian is explicitly determined when the second neighbour interaction is allowed in one direction or two directions of the sublattice occupied by the spins 1/2. It is shown that in the first case, frustration effects are the dominant mechanism for suppression of the ground state long range order, irrespective of s being integer or half integer. The important feature in the second case is the appearance of three transition temperatures for some values of the parameters, and this will imply in a reentrant critical surface. Explicit results are worked out for s = 1 and s = 3/2, which present the main features of the model. By using a general result for decorated lattices the critical exponents are determined for arbitrary s .
The Ising model on the honeycomb lattice with spins of magnitudes 1/2 and s with random exchange constant and random uniaxial anisotropy is considered. The two cases are studied separately, and in both cases binary annealed distributions of the random parameters are considered. An exact solution is obtained by mapping the model onto the uniform spin 1/2 Ising model on the triangular lattice, and the transition temperature is determined as a function of the parameters for arbitrary s. Explicit results are presented for s=1 and s=3/2, and it is shown that some critical parameters, surprisingly, are not universal.
The mixed-spin-(1∕2,SB,SC)(1∕2,{S}_{\mathrm{B}},{S}_{\mathrm{C}})(1∕2,SB​,SC​) Ising model on a decorated square lattice with two different kinds of decorating spins SB{S}_{\mathrm{B}}SB​ and SC{S}_{\mathrm{C}}SC​ $({S}_{\mathrm{B}}\ensuremath{\ne}{S}_{\mathrm{C}})$ placed on its horizontal and vertical bonds, respectively, is exactly solved by establishing a precise mapping relationship with the corresponding spin-1∕21∕21∕2 Ising model on an anisotropic square (rectangular) lattice. The effect of uniaxial single-ion anisotropy acting on both types of decorating spins SB{S}_{\mathrm{B}}SB​ and SC{S}_{\mathrm{C}}SC​ is examined, in particular. If decorating spins SB{S}_{\mathrm{B}}SB​ and SC{S}_{\mathrm{C}}SC​ are integer and half-odd-integer, respectively, or if the reverse is the case, the model under investigation displays a very peculiar critical behavior that had bearing on the spontaneously ordered ``quasi-one-dimensional'' spin system, which appears as a result of the single-ion anisotropy strengthening. We have found convincing evidence that this remarkable spontaneous ordering virtually arises even though all integer-valued decorating spins tend toward their ``nonmagnetic'' spin state S=0S=0S=0 and the system becomes disordered only upon further increase of the single-ion anisotropy. The single-ion anisotropy parameter is also at an origin of various temperature dependences of the total magnetization when imposing the pure ferrimagnetic or the mixed ferro-ferrimagnetic character of the spin arrangement.
The effect of uniaxial single-ion anisotropy on magnetic and critical properties of the mixed spin-1/2 and spin-S (S > 1/2) Ising model on a three-coordinated Bethe lattice is rigorously examined with the help of star-triangle transformation and exact recursion relations.In particular, our attention is focused on the ferrimagnetic version of the model, which exhibits diverse temperature dependences of the total and both sublattice magnetizations.It is shown that the critical behavior of the mixed-spin Ising model on the Bethe lattice basically depends on whether the quantum spin number S is integer or half-odd-integer.
The diagrammatical approach to the exact solution of two-dimensional Ising model, developed for the system with spin 1/2 and 1, is extended to be applicable to the Ising model of spin S greater than 1/2. The exact solution for the system on the square lattice and other lattices with non-crossing interactions is obtained only when the spin is half an odd integer, and the interaction between two spins s j and s k is expressed as - J j k φ j ( s j )φ k ( s k ) in terms of one paramater J j k and odd functions φ j ( s ) and φ k ( s ) of s satisfying |φ j ( s )|=|φ k ( s )|=1 for s =- S , - S +1,· ··, and S . The exact solution on the honeycomb lattice is obtained for all S , for the cases when the interactions divided by the temperature are expressed in terms of S +1/2 or S parameters.
We propose a new approach which permits us to obtain exact results for the mixed spin-1/2 and Spin- S ( S > 1/2) Ising models with single ion anisotropy Δ on the square lattice. We drive an explicit expression for the critical temperature for arbitrary values of S. We determine the exact phase diagrams for different values of S , and we show that there is no tricritical point. For S integer, there is no long range order when the anisotropy exceeds a critical value which is independent of S . Furthermore, the exact Ising transition temperature T C is always recovered, for any values of S, in the limit of Δ → − ∞ . These exact results are based on a conjecture which extend the analyticity of the n-spin-1/2 correlations functions (except at T = T C (Ising)) for any finite number n, to n → ∞ . This is confirmed by our calculations since our results are practically similar to those obtained by Monte Carlo simulations for S = 1, 2 and 3/2.