Ferromagnetism of $UGe_2$

Magnetism of $UGe_2$ is due to the magnetic ordered moments of $5f$ uranium electrons. The strong spin-orbit coupling splits them into two groups. The magnetization is investigated in terms of two vector fields ${\bf M}_{1i}$ and ${\bf M}_{2i}$ which identify the local orientation of the magnetization of the two groups of $f$ electrons. Renormalized spin-wave theory, which accounts for the magnon-magnon interaction, and its extension are developed to describe two ferromagnetic phases in the system: low temperature large moment phase $0<T<T^{*}$ (FM2), where all $5f$ electrons contribute the ordered ferromagnetic moment, and high temperature low-moment phase $T^{*}<T<T_C$ (FM1), where $f$ electrons are partially ordered. Both phases are strictly ferromagnetic in accordance with experiment. The magnetization as a function of temperature is calculated. The anomalous temperature dependence of the ordered moment, known from the experiments with $UGe_2$, is very well reproduced theoretically. Below $T_x$ ($T^*$ in the present paper) the ferromagnetic moment increases in an anomalous way. The new understanding of the anomalous $FM2\to FM1$ transition, as a result of the magnetic order of two well separated groups of $f$ electrons, yields the key to an understanding of the ferromagnetism and transport properties in these compounds.