Bicritical universality of the anisotropic Heisenberg model in a crystal field

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.

Figure 1

Phase diagram for the anisotropic Heisenberg model in a crystal field in a cubic lattice $\left(L=14\right)$. Lines are only guides to the eyes. The continuous lines indicate second-order phase transitions and the dashed line stands for first-order phase transitions. Error bars are of the size or smaller than symbol sizes.

Figure 2

Area under each of the peaks as a function of the parameter $s$ $\left(L=20\right)$. This plot is obtained once the peak heights are equalized, so that the symmetrization of the probability distribution is related to value of $s$ where the area for both peaks coincide.

Figure 3

Probability distribution function for the field-mixing extensive conjugate variable $Q-sE$ $\left(L=20\right)$. The distributions displayed refer to two different points along the first-order line. Error bars are of the size or smaller than symbol sizes. Distributions normalized to unit norm.

Figure 4

Finite-size scaling analysis for two different points along the first-order line. The line is the best fit to the data points. Error bars are smaller than symbol sizes.

Figure 5

Fourth-order magnetization cumulant for the three-dimensional anisotropic Heisenberg model in a crystal field. Lines are only guides to the eyes. Error bars are of the size or smaller than symbol sizes.

Figure 6

Fourth-order cumulant for the $z$ component of the magnetization for the three-dimensional anisotropic Heisenberg model in a crystal field. Lines are only guides to the eyes. Error bars not shown are of the size or smaller than symbol sizes. Only some points are depicted for $L=40$ and $L=44$ in order to show that the crossing at the vicinity of $D=3.9555$ is in fact spurious.

Figure 7

Fourth-order magnetization cumulant for the three-dimensional anisotropic Heisenberg model in a crystal field. Data obtained through histogram reweighting from a simulation in the vicinity of the bicritical point.

Figure 8

Fourth-order cumulant for the $z$ component of the magnetization for the three-dimensional anisotropic Heisenberg model in a crystal field. Data obtained through histogram reweighting from a simulation in the vicinity of the bicritical point.

Figure 9

First-order transition line on the D-T plane, obtained through the procedure described in the text. The bicritical point is represented by a full circle with error bars. The line is the best fit to the data. Error bars when not shown are smaller than symbol sizes.

Figure 10

Bicritical universal probability distribution function for the magnetization $\left(L\le 40\right)$. The distribution is scaled to unit norm and variance. Lines are only guides to the eyes. Error bars are smaller than symbol sizes. Note the discrepancy with the universal PDF for the three-dimensional Heisenberg model. Compare this graph with Fig. 11.

Figure 11

Bicritical universal probability distribution function for the magnetization $\left(L\ge 60\right)$. The distribution is scaled to unit norm and variance. Error bars are smaller than symbol sizes. Observe, for system sizes large enough, the collapse with the universal PDF for the three-dimensional Heisenberg model. Compare this graph with Fig. 10.

Figure 12

Individual collapse of the PDF for a given lattice size over the universal PDF for the three-dimensional Heisenberg model. The collapse for each lattice size occurs at a different value of $T$ and $D$. Lines are only guides to the eyes.

Figure 13

Finite-size scaling analysis for the bicritical temperature. The line is the best fit to the data.

Figure 14

Finite-size scaling analysis of the standard deviation of the magnetization PDF at the bicritical point. The line is the best fit to the data. Error bars are smaller than symbol sizes.

Figure 15

Finite-size scaling analysis for the magnetization at the bicritical point. The line is the best fit to the data. Error bars are smaller than symbol sizes.

Figure 16

Finite-size scaling analysis for the true $\left(\langle m\rangle =0\right)$ magnetic susceptibility at the bicritical point. The line is the best fit to the data. Error bars are smaller than symbol sizes.

Figure 17

Finite-size scaling analysis for the logarithmic derivative of $\langle m\rangle$ and $\langle m^2\rangle$. The line is the best fit to the data. Error bars are smaller than symbol sizes.

Figure 18

Finite-size scaling analysis for the magnetization at the bicritical point (two different estimates used). The line is the best fit to the data. Error bars are smaller than symbol sizes. Notice that only larger lattice sizes were used for this analysis.