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 χ 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, H 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, H scaling. We derive the thermodynamic potential as a function of magnetization and show that it contains, in addition to regular terms, a nonanalytic M3 term, which becomes M4 /T at finite T. We show that regular (M2, M4,...) 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 M3 term crosses over to a non-Fermi-liquid form M7/2 near a QCP. The prefactor of the M7/2 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 χ 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.