Anisotropy of a cubic ferromagnet at criticality

Critical fluctuations change the effective anisotropy of cubic ferromagnet near the Curie point. If the crystal undergoes phase transition into orthorhombic phase and the initial anisotropy is not too strong, reduced anisotropy of nonlinear susceptibility acquires at $T_c$ the universal value ${\delta }_4^{*}=\frac{2v^{*}}{3(u^{*}+v^{*})}$ where $u^{*}$ and $v^{*}$ are coordinates of the cubic fixed point on the flow diagram of renormalization group equations. In the paper, the critical value of the reduced anisotropy is estimated within the pseudo-$\epsilon$ expansion approach. The six-loop pseudo-$\epsilon$ expansions for $u^{*}$, $v^{*}$, and ${\delta }_4^{*}$ are derived for the arbitrary spin dimensionality $n$. For cubic crystals ($n=3$) higher-order coefficients of the pseudo-$\epsilon$ expansions obtained turn out to be so small that use of simple Padé approximants yields reliable numerical results. Padé resummation of the pseudo-$\epsilon$ series for $u^{*}$, $v^{*}$, and ${\delta }_4^{*}$ leads to the estimate ${\delta }_4^{*}=0.079\pm 0.006$, indicating that detection of the anisotropic critical behavior of cubic ferromagnets in physical and computer experiments is certainly possible.

Figure 1

The cubic fixed point coordinate $v^{*}$ as a function of $n$ obtained by means of Padé resummation of $\tau$-series (11). Three curves given by the diagonal [2/2] and near-diagonal approximants [3/2], [2/3] for $v^{*}/\tau$, i.e., with insignificant factor $\tau$ omitted, are shown. The values of marginal spin dimensionality $n_c$ found by the analysis of six-loop 3D RG series [13] (2.89) and extracted from the $\epsilon$ expansion [13, 17] (2.87, 2.855) and from the pseudo-$\epsilon$ expansion [19] (2.862) for $n_c$ are marked with diamonds. The diamond on the right vertical axis shows the value of $v^{*}$ the six-loop 3D RG analysis yields. The dashed line is the curve $v^{*}\left(n\right)$ given by the powers series in $(n-n_c)$, which arrives at 0.163 when the spin dimensionality approaches the physical value $n=3$.