Néel Temperature of Quasi-Low-Dimensional Heisenberg Antiferromagnets

The Néel temperature $T_{\mathrm{N}}$ of quasi-one- and quasi-two-dimensional antiferromagnetic Heisenberg models on a cubic lattice is calculated by Monte Carlo simulations as a function of interchain (interlayer) to intrachain (intralayer) coupling $J^{\prime }/J$ down to $J^{\prime }/J\simeq {10}^{-3}$. We find that $T_{\mathrm{N}}$ obeys a modified random-phase approximationlike relation for small $J^{\prime }/J$ with an effective universal renormalized coordination number, independent of the size of the spin. Empirical formulas describing $T_{\mathrm{N}}$ for a wide range of $J^{\prime }$ and useful for the analysis of experimental measurements are presented.

Figure 1

$J^{\prime }$ dependence of $T_{\mathrm{N}}/JS(S+1)$ for the Q1D systems. The error bar of each point is much smaller than the symbol size. The dashed curve passing through the $S=\infty$ data is obtained by Eq. (2) with $k_1=0.7$ and Eq. (4) for ${\chi }_{\mathrm{s}}$. The solid curve denotes the proposed empirical formula (8). The others are results of various approximations discussed in the text.

Figure 2

$J^{\prime }$ dependence of $\zeta (J^{\prime })/z_1$ for the Q1D systems. In all cases $\zeta (J^{\prime })/z_1$ approaches a constant ($\simeq 0.695$), denoted by the dotted line, at small $J^{\prime }/J$. The error bar of each point is smaller than the symbol size.

Figure 3

$J^{\prime }$ dependence of $T_{\mathrm{N}}/JS(S+1)$ for the Q2D systems. The open symbols denote our numerical results for the $S=1/2$ (circles), $S=1$ (squares), and $S=\infty$ (triangles) models. The error bar of each point is much smaller than the symbol size. The dashed (RPA) and dotted (modified RPA) curves for $S=1/2$ are obtained from Eqs. (1, 2) with $k_2=1$ and $0.65$, respectively. The solid curves denote the proposed empirical formula (9), while the curve passing through the $S=\infty$ data is simply a guide for the eye. The inset shows the same data on a linear scale.

Figure 4

$J^{\prime }$ dependence of $\zeta (J^{\prime })/z_2$ for the Q2D systems. In all cases $\zeta (J^{\prime })/z_2$ approaches a constant ($\simeq 0.65$), denoted by the dotted line, at small $J^{\prime }/J$. The error bar of each point is smaller than the symbol size unless given explicitly.