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Brazilian Journal of Physics

Monte Carlo study of the anisotropic three-dimensional Heisenberg model in a crystal field

Rodrigo Teixeira Santos Freire, J. A. Plascak, B. V. Costa

Published Jun 1, 2004• 7 citations
Publisher

Abstract

We study the phase diagram of the three-dimensional classical ferromagnetic Heisenberg model with an easy-plane crystalline anisotropy and an easy-axis exchange anisotropy through Monte Carlo simulations. We employ the Metropolis algorithm together with single-histogram techniques in order to characterize the transitions in each region of the phase diagram. Our results reveal, besides the disordered phase, the existence of Ising-like and XY-like ordered phases which are separated by a first-order transition line.

DOI
10.1590/s0103-97332004000300022

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We study the phase diagram of the three-dimensional classical ferromagnetic Heisenberg model with an easy-plane crystalline anisotropy and an easy-axis exchange anisotropy through Monte Carlo simulations. We employ the Metropolis algorithm together with single-histogram techniques in order to characterize the transitions in each region of the phase diagram. Our results reveal, besides the disordered phase, the existence of Ising-like and XY-like ordered phases which are separated by a first-order transition line.
Abstract In this work we use numerical Monte Carlo techniques to study the phase diagram of the three‐dimensional classical Heisenberg anti‐ferromagnetic (AF) model with single‐ion anisotropy model with an uniform magnetic field applied along the easy axis. Early experimental as well as theoretical calculations have shown that the phase diagram of the model is quite rich. We simulated the model on a simple cubic lattices using the Wang–Landau sampling and Metropolis algorithm. Finite size scaling theory is used to obtain the phase diagram. At low field and low temperature the system exhibits AF order separated from a spin‐flop (SF) phase by a first‐order line. As temperature grows a paramagnetic phase is reached by crossing a second‐order line both from the AF phase and from the SF phase. The critical exponents are obtained for each transition line. Contrary to early works we find that the line separating the SF phase from the paramagnetic phase is in the Ising class of universality.
The bicritical properties of the three-dimensional classical anisotropic Heisenberg model in a crystal field are investigated through extensive Monte Carlo simulations on a simple cubic lattice, using Metropolis and Wolff algorithms. Field-mixing and multidimensional histogram techniques were employed in order to compute the probability distribution function of the extensive conjugate variables of interest and, using finite-size scaling analysis, the first-order transition line of the model was precisely located. The fourth-order cumulant of the order parameter was then calculated along this line and the bicritical point located with good precision from the cumulant crossings. The bicritical properties of this point were further investigated through the measurement of the universal probability distribution function of the order parameter. The results lead us to conclude that the studied bicritical point belongs in fact to the three-dimensional Heisenberg universality class.
Extensive Monte Carlo simulations are used to locate the bicritical point of the ferromagnetic anisotropic Heisenberg model on a simple cubic lattice in the presence of a crystal field. In order to achieve this goal, we employ a hybrid algorithm comprising Metropolis and Wolff algorithms along with field mixing. Multiple histogram techniques have also been used to obtain the probability distribution function of the conjugate extensive variables of interest. In conjunction with finite-size scaling analysis we computed the fourth-order cumulant of the order parameter along the first-order line and the corresponding bicritical point has been located from the crossings of the cumulants.
The three-dimensional anisotropic classical XY ferromagnet has been investigated by extensive Monte Carlo simulation using the Metropolis single spin flip algorithm. The magnetisation (M) and the susceptibility (χ) are measured and studied as functions of the temperature of the system. For constant anisotropy, the ferro-para phase transition has been found to take place at a higher temperature than that observed in the isotropic case. The system gets ordered at higher temperatures for higher values of the strength of anisotropy. The opposite scenario is observed in the case of random anisotropy. For all three different kinds of statistical distributions (uniform, Gaussian, and bimodal) of random anisotropy, the system gets ordered at lower temperatures for higher values of the width of the distribution of anisotropy. We have provided the phase boundaries in the case of random anisotropy. The critical exponents for the scaling laws M∼L−βν and χ∼Lγν are estimated through the finite size analysis.
We study the ±J three-dimensional (3D) Ising model with a spatially uniaxial anisotropic bond randomness on the simple cubic lattice. The ±J random exchange is applied on the xy planes, whereas, in the z direction, only a ferromagnetic exchange is used. After sketching the phase diagram and comparing it with the corresponding isotropic case, the system is studied at the ferromagnetic-paramagnetic transition line using parallel tempering and a convenient concentration of antiferromagnetic bonds (p(z)=0;p(xy)=0.176). The numerical data clearly point out a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3D random Ising model. The smooth finite-size behavior of the effective exponents, describing the peaks of the logarithmic derivatives of the order parameter, provides an accurate estimate of the critical exponent 1/ν=1.463(3), and a collapse analysis of magnetization data gives an estimate of β/ν=0.516(7). These results are in agreement with previous papers and, in particular, with those of the isotropic ±J three-dimensional Ising model at the ferromagnetic-paramagnetic transition line, indicating the irrelevance of the introduced anisotropy.
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.