Anisotropy of a Cubic Ferromagnet at Criticality

TL;DR

This paper estimates the universal anisotropy value at the critical point of cubic ferromagnets using high-order pseudo-epsilon expansions and Padé resummation, suggesting experimental detectability.

Contribution

The paper derives six-loop pseudo-epsilon expansions for critical anisotropy parameters in cubic ferromagnets for arbitrary spin dimensionality, providing reliable numerical estimates.

Findings

Estimated universal anisotropy value $oxed{oxed{ ext{0.079} ext{ with } ext{0.006} ext{ uncertainty}}}$.

Showed that higher-order coefficients are small, enabling accurate Padé approximations.

Indicated that anisotropic critical behavior is detectable in experiments.

Abstract

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 $\delta_4^* = {{2v^*} \over {3(u^* + v^*)}}$ where $u^*$ and $v^*$ -- 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 Pad\'e approximants yields reliable numerical results. Pad\'e 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.