Precritical anomalous scaling and magnetization temperature dependence in cubic ferromagnetic crystals

Recent developments in spintronics have drawn renewed attention to the spin dynamics of cubic ferromagnetic crystals EuO and EuS. These ferromagnets have the simplest possible magnetic structure, making them the most suitable systems for testing various theoretical models of magnetic materials. A commonly used Weiss mean-field approximation (MFA) provides only a qualitative description of the magnetization temperature dependence $M(T)$. We develop a consistent theory for $M(T)$ based on the perturbation diagrammatic technique for spin operators. Our theory is in excellent quantitative agreement with the experimental dependence of $M(T)$ for EuO and EuS throughout the entire temperature range from $T=0$ to Curie temperature ${T}_{\mathrm{C}}$. In particular, our theoretical dependence $M(T)$ demonstrates a scaling behavior $M(T)\ensuremath{\propto}{({T}_{\mathrm{C}}\ensuremath{-}T)}^{{\ensuremath{\beta}}_{*}}$ with the scaling index ${\ensuremath{\beta}}_{*}\ensuremath{\approx}1/3$ in a wide range of temperatures, in agreement with the experimentally observed apparent scaling in EuO and EuS. The scaling behavior with ${\ensuremath{\beta}}_{*}\ensuremath{\approx}1/3$ is manifested in the temperature range $T\ensuremath{\lesssim}{T}_{\mathrm{C}}$ where corrections to the magnetization due to its fluctuations $\ensuremath{\delta}M(T)\ensuremath{\lesssim}M(T)$. To distinguish it from the narrow ``critical'' range $T\ensuremath{\approx}{T}_{\mathrm{C}}$ with $\ensuremath{\delta}M(T)>M(T)$, we term this $T$-range ``precritical.'' The precritical corrections $\ensuremath{\delta}M(T)$ are still large enough to affect the $M(T)$ behavior. The index ${\ensuremath{\beta}}_{*}$ fundamentally differs from the ``normal'' scaling index ${\ensuremath{\beta}}_{\mathrm{MFA}}=1/2$ predicted by the MFA, which neglects the magnetization fluctuations. We refer to the apparent magnetization scaling with ${\ensuremath{\beta}}_{*}\ensuremath{\approx}1/3$, which emerges in our theory, as ``precritical anomalous.''