Ferromagnetic quantum phase transition in an itinerant three-dimensional system

The non-analytic behavior of the spin susceptibility both away and near the quantum critical point signals the breakdown of the Hertz-Millis scenario for a ferromagnetic quantum phase transition in itinerant systems. It is believed that in both 2D and 3D $\chi_{s}$ increases as a function of the magnetic field ($H)$ or momentum ($q),$ which indicates a tendency to either first order transition or ordering at finite $q$. We show that the 3D case is different from the 2D one. Away from the 3D critical point, the non-analytic part of $\chi _{s}$ can be of either sign, depending on microscopic parameters. The non-analyticity in 3D arises from two physically distinct processes: excitations of a single and three particle-hole pairs. Both processes contribute a max\{$H^{2},q^{2}\}\ln \max \{H^{2},q^{2}\}$ term to $\chi _{s}$, but the signs of these contributions are opposite. The single-pair process leads to an increase of $\chi _{s}$ with $H,q$ whereas the three pair one corresponds to a decrease. In the paramagnon model, the three pair contribution always wins sufficiently close to the Stoner instability. We also discuss the behavior of $\chi _{s}$ in the immediate vicinity of the quantum critical point within the spin-fermion model.