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Physical review. B, Condensed matter

Cluster-variation method applied in the pair approximation to the S=1 Ising ferromagnet having additional single-ion-type uniaxial anisotropy

Wai Man Ng, Jeremiah H. Barry

Published May 1, 1978• 47 citations
Publisher

Abstract

In the Weiss molecular-field approximation, the $S=1$ Ising-model ferromagnet possessing additional single-ion-type uniaxial anisotropy is known to exhibit (depending upon the ratio of anisotropy and interaction constants) both first-order (discontinuous) and second-order (continuous) magnetic phase transitions which join at a tricritical point. We further investigate the above model system in the spin-pair approximation using the cluster-variation method. Within this improved approximation, results are found for the temperature and anisotropy dependences of the thermal-average quantities $〈{S}_{i}〉$, $〈S_{i}^{}{}_{}{}^{2}〉$, $〈{S}_{i}{S}_{{i}^{\ensuremath{'}}}〉$, $〈{S}_{i}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$, $〈S_{i}^{}{}_{}{}^{2}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$ ($i$, ${i}^{\ensuremath{'}}$ are nearest-neighbor lattice sites), a better estimate for the location of the tricritical point is achieved, and the specific heat and initial paramagnetic susceptibility are calculated. To illustrate these results, the simple cubic lattice is used as an example of application.

DOI
10.1103/physrevb.17.3675

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In the Weiss molecular-field approximation, the S=1S=1S=1 Ising-model ferromagnet possessing additional single-ion-type uniaxial anisotropy is known to exhibit (depending upon the ratio of anisotropy and interaction constants) both first-order (discontinuous) and second-order (continuous) magnetic phase transitions which join at a tricritical point. We further investigate the above model system in the spin-pair approximation using the cluster-variation method. Within this improved approximation, results are found for the temperature and anisotropy dependences of the thermal-average quantities 〈Si〉〈{S}_{i}〉〈Si​〉, 〈Si2〉〈S_{i}^{}{}_{}{}^{2}〉〈Si​​2〉, 〈SiSi\ensuremath′〉〈{S}_{i}{S}_{{i}^{\ensuremath{'}}}〉〈Si​Si\ensuremath′​〉, 〈SiSi\ensuremath′2〉〈{S}_{i}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉〈Si​Si\ensuremath′​​2〉, 〈Si2Si\ensuremath′2〉〈S_{i}^{}{}_{}{}^{2}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉〈Si​​2Si\ensuremath′​​2〉 (iii, i\ensuremath′{i}^{\ensuremath{'}}i\ensuremath′ are nearest-neighbor lattice sites), a better estimate for the location of the tricritical point is achieved, and the specific heat and initial paramagnetic susceptibility are calculated. To illustrate these results, the simple cubic lattice is used as an example of application.
A finite cluster effective-field theory in which the single-site kinematic relations are treated exactly is applied to a study of the spin-one Ising ferromagnet on a honeycomb lattice. The Ising model studied contains both biquadratic exchange, J′ and single-ion anisotropy D in addition to the usual bilinear exchange interactions J. The second-order phase boundary is studied for a range of interaction strengths and the results compared to those of other methods. The dependence of the results on cluster size is investigated. The magnetization predicted by the four-site cluster theory for a selection of single-ion anisotropy strengths is also compared with exact results that have recently been deduced in the subspace J′/J=−[ln cosh(βJ)]/(βJ). In extending the theory to account for site dilution, a form of configurational averaging is adopted that is superior to that of earlier authors. The variation with site dilution of the tricritical point in the T-D plane is reported.
We propose a method to incorporate pair correlations of an Ising lattice system in molecular field theory to higher orders. The theory is applied to ferromagnetism, where a system of equations is obtained for the response function (here the susceptibility). We obtain important corrections on the Weiss model and the spherical model. This is illustrated by an explicit calculation of the critical temperature and the spurious phase, appearing for T>TcT>{T}_{c}T>Tc​, near the critical point in the case of a simple cubic lattice.
Phase diagram of the spin-1 quantum Heisenberg model with both exchange as well as single-ion anisotropy is constructed within the framework of pair approximation formulated as a variational procedure based on the Gibbs-Bogoliubov inequality. In this form adapted variational approach is used to obtain the results equivalent with the Oguchi's pair approximation. It is shown that the single-ion anisotropy induces a tricritical behaviour in the considered model system and a location of tricritical points is found in dependence on the exchange anisotropy strength.
The phase diagram of the Ising square ferromagnet with nearest-neighbor, next-nearest-neighbor, and four-spin (plaquette) interactions is investigated using the cluster variation method in the square approximation. A complete analysis in temperature is performed for the ferromagnetic, superantiferromagnetic, and frustrated region of the ground state. Particular attention is devoted to the role of the four-spin interaction. The most remarkable effect is the occurrence, in several regions of the interaction parameter space, of first-order transitions and sometimes reentrance phenomena. Critical and tricritical points are almost analytically determined.
We have considered the paramagnetic—ferromagnetic transition for the Ising system with ``infinite range'' interactions (i.e., each spin interacts equally with all other spins) and in the presence of a single-ion anisotropy term: H=−2jΣ lim i>i SizSjz+ΔΣ lim i Siz2−gμHΣ lim iSiz. Kittel and Shore1 have shown that this system (in the absence of the term in Δ) is exactly soluble, and leads rigourously, in the limit of an infinite number of particles, to the equations of the molecular-field approximation, and to a second-order phase transition. The single-ion term can cause the transition to become first order2,3 in that there is a discontinuity in the magnetization as a function of temperature. We have solved for the relevant thermodynamic parameters as a function of temperature for the case of ions with spin one. The development of the first-order transition as a function of the number of particles will be discussed. This work will be submitted for publication in the Physical Review.
We investigate the ground-state and finite-temperature properties of a mixed spin-1/2 and spin-1 Ising model on a decorated square (Lieb) lattice incorporating a uniaxial single-ion anisotropy and magnetic field. By employing the generalized decoration-iteration transformation, the model is mapped exactly onto an effective spin-1/2 Ising model on the square lattice characterized by an effective nearest-neighbor interaction and an effective field. The studied model consequently becomes exactly solvable even for finite values of the applied magnetic field whenever the effective field vanishes. The ground-state analysis reveals three distinct phases: ferrimagnetic phase (FRI), disordered phase (DP), and ferromagnetic (FM) phase. The ground-state boundary between FRI and DP phases gives rise to a dome-shaped surface of discontinuous thermal phase transitions, which is terminated by a line of Ising-type critical points associated with continuous thermal phase transitions. Both continuous and discontinuous thermal phase transitions belong to the exactly solvable parameter regime defined by a vanishing effective field in spite of the fact that the applied magnetic field is finite. Two consecutive discontinuous thermally-induced reentrant phase transitions DP-FRI-DP are identified in a narrow parameter region. The exact analytical predictions including reentrance, field- and thermally-driven phase transitions are independently verified by classical Monte Carlo simulations.