Nonanalytic paramagnetic response of itinerant fermions away and near a ferromagnetic quantum phase transition

We study nonanalytic paramagnetic response of an interacting Fermi system both away and in the vicinity of a ferromagnetic quantum phase transition (QCP). Previous studies found that (i) the spin susceptibility chi scales linearly with either the temperature T or magnetic field H in the weak-coupling regime; (ii) the interaction in the Cooper channel affects this scaling via logarithmic renormalization of prefactors of the T, parallel to H parallel to terms, and may even reverse the signs of these terms at low enough energies. We show that Cooper renormalization becomes effective only at very low energies, which get even smaller near a QCP. However, even in the absence of such renormalization, generic (non-Cooper) higher-order processes may also inverse the sign of T, parallel to H parallel to scaling. We derive the thermodynamic potential as a function of magnetization and show that it contains, in addition to regular terms, a nonanalytic parallel to M parallel to(3) term, which becomes M-4/T at finite T. We show that regular (M-2,M-4,...) terms originate from fermions with energies of order of the bandwidth, while the nonanalytic term comes from low-energy fermions. We consider the vicinity of a ferromagnetic QCP by generalizing the Eliashberg treatment of the spin-fermion model to finite magnetic field, and show that the parallel to M parallel to(3) term crosses over to a non-Fermi-liquid form parallel to M parallel to(7/2) near a QCP. The prefactor of the parallel to M parallel to(7/2) term is negative, which indicates that the system undergoes a first-order rather than a continuous transition to ferromagnetism. We compare two scenarios of the breakdown of a continuous QCP: a first-order instability and a spiral phase; the latter may arise from the nonanalytic dependence of chi on the momentum. In a model with a long-range interaction in the spin channel, we show that the first-order transition occurs before the spiral instability.