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arXiv:cond-mat/0611254 · Chinese Journal of Physics

Phase diagram of the spin-1 XXZ Heisenberg ferromagnet with a single-ion anisotropy

Jozef Strečka, J. Dely, Lucia Čanová

Published Nov 9, 2006• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.cond-mat/0611254

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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.
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.
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.
We have studied the effects of the random transverse single-ion anisotropy in the phase diagram and in the thermodynamic properties of the spin−1 Blume–Capel quantum model, which has been used the approach the mean-field theory based on the Bogoliubov inequality for the Gibbs free energy. The phase diagrams were obtained in the temperature τ=kBT∕zJ versus single-ion anisotropy δx=Dx∕zJ plane for several values of δy=Dy∕zJ, p and q. The results show that there is a tricritical behavior only in the range of δy given by −0.048≤δy<0.5. This behavior is shown again by the curves of the magnetization m as a function of temperature τ with corresponding values of δy, p and q.
The phase diagram at T = 0 of a family of spin-1 quantum Ising-Heisenberg models is studied using a mean-field approximation. The model Hamiltonian is characterized by single ion anisotropy terms D μ , (μ = x, y, z ) and by two positive parameters η x , η y measuring the degree of anisotropy of the interaction. They allow to describe a variety of models: among them the spin-1 Ising-Blume-Capel model (η x = η y = 0), and the isotropic Heisenberg model (η x = η y = 1). Both dipolar and quadrupolar order parameters are used to derive the phase diagram in the parameter space (η x , η y , D μ ). The resulting phase diagram is rather complex as it exhibits first and second order critical transitions as well as multicritical points.
Using density-matrix renormalization group calculations, ground-state properties of the spin-1 Heisenberg chain with exchange and single-ion anisotropies in an external field are studied. Our findings confirm and refine recent numerical and analytic results by Sengupta and Batista [Phys. Rev. Lett. 99, 217205 (2007)] on the same model. In particular, we present evidence for two types of biconical (or supersolid) and for two types of spin-flop (or superfluid) structures for chains of finite length. Basic features of the quantum phase diagram may be interpreted qualitatively in the framework of classical spin models.