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Cluster variation method in the analysis of spin-1 models

Carla Buzano, L. R. Evangelista

Published Jan 1, 1991• 0 citations
OpenAlex

Abstract

The cluster variation method, in its recent formulation using the Moebius inversion, is applied to the study of the critical properties of the spin-1 BlumeCapel model. In order to stress the various steps of the approximation a face-centered-cubic lattice is considered and the free energy of the system is written in several degrees of approximations: single site (one-body), pair (two-body), triangle (three-body), tetrahedron (four-body). The case of the pair approximation is discussed in detail and the resulting phase diagram is compared with that obtained using other methods. The behavior of the dipolar and quadrupolar order parameters and of the pair-correlation functions is examined in three typical cases: second order transition, first order transition, no transitions.

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THREE- AND FOUR-BODY CORRELATION FUNCTIONS FOR THE BLUME-CAPEL MODEL

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The Cluster Variation Method (CVM), in its recent formulation using the Moebius Inversion, is applied to study the critical properties of the face-centered-cubic Blume-Capel model in the triangle and tetrahedron approximations. We develop a procedure which preserves the characteristics of Kikuchi’s Natural Iteration Method, moreover reducing considerably the number of involved variables. The resulting phase diagram is compared with that obtained from lower orders of CVM approximation (single-site and pair) and from other methods (series expansions, Monte Carlo, …). The tetrahedron approximation shows a good agreement with these high-precision methods, in particular for the location of the tricritical point. The behavior of order parameters and multisite (up to four-body) correlation functions is determined for all significant regions of the phase space.

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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

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.

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Quantum spin-1 anisotropic ferromagnetic Heisenberg model in a crystal field: A variational approach

D. C. Carvalho, J. A. Plascak, L. M. Castro

A variational approach based on Bogoliubov inequality for the free energy is employed in order to treat the quantum spin-1 anisotropic ferromagnetic Heisenberg model in the presence of a crystal field. Within the Bogoliubov scheme an improved pair approximation has been used. The temperature-dependent thermodynamic functions have been obtained and provide much better results than the previous simple mean-field scheme. In one dimension, which is still nonintegrable for quantum spin-1, we get the exact results in the classical limit, or near-exact results in the quantum case, for the free energy, magnetization, and quadrupole moment, as well for the transition temperature. In two and three dimensions the corresponding global phase diagrams have been obtained as a function of the parameters of the Hamiltonian. First-order transition lines, second-order transition lines, tricritical and tetracritical points, and critical endpoints have been located through the analysis of the minimum of the Helmholtz free energy and a Landau-like expansion in the approximated free energy. Only first-order quantum transitions have been found at zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capel, and Ising models, have been analyzed and compared to previous results obtained from other analytical approaches as well as from Monte Carlo simulations.

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Cluster-variation–Padé-approximant method for the simple cubic Ising model

Alessandro Pelizzola

The cluster-variation-Pade-approximant method is a recently proposed tool, based on the extrapolation of low- and high-temperature results obtained with the cluster-variation method, for the determination of critical parameters in Ising-like models. Here the method is applied to the three-dimensional simple cubic Ising model, and new results, obtained with an 18-site basic cluster, are reported. Other techniques for extracting nonclassical critical exponents are also applied and their results compared with those by the cluster-variation-Pade-approximant method.

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Cluster variation method, Padé approximants, and critical behavior

Alessandro Pelizzola

In the present paper we show how nonclassical, quite accurate, critical exponents can be extracted in a very simple way from the Pad\'e analysis of the results obtained by mean-field-like approximation schemes, and in particular by the cluster variation method. We study the critical behavior of the Ising model on several lattices (quadratic, triangular, simple cubic and face centered cubic) and two problems of surface critical behavior. Both unbiased and biased approximants are used, and results are in very good agreement with the exact or numerical ones.

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The mixed spin-3/2 and spin-2 Blume-Capel model by renormalization group approach

Karima El Kihel, Hasnae Saadi, Rachid Aharrouch +4 more

Purpose The authors investigate the magnetic properties of a mixed spin-3/2 and spin-2 Blume-Capel model on square and cubic lattices with two different single-ion anisotropies. Design/methodology/approach To study the critical behavior of a mixed spin-3/2 and spin-2 system, the authors have used a real space renormalization group approximation and specifically the Migdal-Kadanoff technique. The authors give the phase diagrams for two different cases: (1) on the (Δ/|J|, 1/|J|) plane with ΔA = ΔB = Δ, and (2) on the (ΔA/|J|, 1/|J|) and (ΔB/|J|, 1/|J|) planes for selected values of ΔB/|J| and ΔA/|J|, respectively. Findings The phase diagrams obtained show that the system exhibits both second- and first-order phase transitions as well as tricritical points for some values of the anisotropies. Moreover, using the variation of the free energy and its derivative at low temperatures, the authors have seen the appearance of first-order transitions at very low temperatures. Originality/value Few investigations of mixed spin-3/2 and spin-2 systems with crystal field have been realized. For this reason, the authors use the renormalization group approach to complete the work done on these systems. In absence of an exact solution, this contributes to the synthesis of the approximation results on mixed spins models.

Multidiscipline Modeling in Materials and Structures•Jul 26, 2021•3 citations
The cluster variation method, in its recent formulation using the Moebius inversion, is applied to the study of the critical properties of the spin-1 BlumeCapel model. In order to stress the various steps of the approximation a face-centered-cubic lattice is considered and the free energy of the system is written in several degrees of approximations: single site (one-body), pair (two-body), triangle (three-body), tetrahedron (four-body). The case of the pair approximation is discussed in detail and the resulting phase diagram is compared with that obtained using other methods. The behavior of the dipolar and quadrupolar order parameters and of the pair-correlation functions is examined in three typical cases: second order transition, first order transition, no transitions.
The Cluster Variation Method (CVM), in its recent formulation using the Moebius Inversion, is applied to study the critical properties of the face-centered-cubic Blume-Capel model in the triangle and tetrahedron approximations. We develop a procedure which preserves the characteristics of Kikuchi’s Natural Iteration Method, moreover reducing considerably the number of involved variables. The resulting phase diagram is compared with that obtained from lower orders of CVM approximation (single-site and pair) and from other methods (series expansions, Monte Carlo, …). The tetrahedron approximation shows a good agreement with these high-precision methods, in particular for the location of the tricritical point. The behavior of order parameters and multisite (up to four-body) correlation functions is determined for all significant regions of the phase space.
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〉, $〈{S}_{i}{S}_{{i}^{\ensuremath{'}}}〉$, $〈{S}_{i}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$, $〈S_{i}^{}{}_{}{}^{2}S_{{i}^{\ensuremath{'}}}^{}{}_{}{}^{2}〉$ (iii, ${i}^{\ensuremath{'}}$ 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 variational approach based on Bogoliubov inequality for the free energy is employed in order to treat the quantum spin-1 anisotropic ferromagnetic Heisenberg model in the presence of a crystal field. Within the Bogoliubov scheme an improved pair approximation has been used. The temperature-dependent thermodynamic functions have been obtained and provide much better results than the previous simple mean-field scheme. In one dimension, which is still nonintegrable for quantum spin-1, we get the exact results in the classical limit, or near-exact results in the quantum case, for the free energy, magnetization, and quadrupole moment, as well for the transition temperature. In two and three dimensions the corresponding global phase diagrams have been obtained as a function of the parameters of the Hamiltonian. First-order transition lines, second-order transition lines, tricritical and tetracritical points, and critical endpoints have been located through the analysis of the minimum of the Helmholtz free energy and a Landau-like expansion in the approximated free energy. Only first-order quantum transitions have been found at zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capel, and Ising models, have been analyzed and compared to previous results obtained from other analytical approaches as well as from Monte Carlo simulations.
The cluster-variation-Pade-approximant method is a recently proposed tool, based on the extrapolation of low- and high-temperature results obtained with the cluster-variation method, for the determination of critical parameters in Ising-like models. Here the method is applied to the three-dimensional simple cubic Ising model, and new results, obtained with an 18-site basic cluster, are reported. Other techniques for extracting nonclassical critical exponents are also applied and their results compared with those by the cluster-variation-Pade-approximant method.
In the present paper we show how nonclassical, quite accurate, critical exponents can be extracted in a very simple way from the Pad\'e analysis of the results obtained by mean-field-like approximation schemes, and in particular by the cluster variation method. We study the critical behavior of the Ising model on several lattices (quadratic, triangular, simple cubic and face centered cubic) and two problems of surface critical behavior. Both unbiased and biased approximants are used, and results are in very good agreement with the exact or numerical ones.
Purpose The authors investigate the magnetic properties of a mixed spin-3/2 and spin-2 Blume-Capel model on square and cubic lattices with two different single-ion anisotropies. Design/methodology/approach To study the critical behavior of a mixed spin-3/2 and spin-2 system, the authors have used a real space renormalization group approximation and specifically the Migdal-Kadanoff technique. The authors give the phase diagrams for two different cases: (1) on the (Δ/|J|, 1/|J|) plane with ΔA = ΔB = Δ, and (2) on the (ΔA/|J|, 1/|J|) and (ΔB/|J|, 1/|J|) planes for selected values of ΔB/|J| and ΔA/|J|, respectively. Findings The phase diagrams obtained show that the system exhibits both second- and first-order phase transitions as well as tricritical points for some values of the anisotropies. Moreover, using the variation of the free energy and its derivative at low temperatures, the authors have seen the appearance of first-order transitions at very low temperatures. Originality/value Few investigations of mixed spin-3/2 and spin-2 systems with crystal field have been realized. For this reason, the authors use the renormalization group approach to complete the work done on these systems. In absence of an exact solution, this contributes to the synthesis of the approximation results on mixed spins models.