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

Abstract

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 $\ensuremath{\chi}$ 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 $T$, $\ensuremath{\mid}H\ensuremath{\mid}$ 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 $T$, $\ensuremath{\mid}H\ensuremath{\mid}$ scaling. We derive the thermodynamic potential as a function of magnetization and show that it contains, in addition to regular terms, a nonanalytic ${\ensuremath{\mid}M\ensuremath{\mid}}^{3}$ term, which becomes ${M}^{4}∕T$ at finite $T$. We show that regular $({M}^{2},{M}^{4},\dots{})$ 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 ${\ensuremath{\mid}M\ensuremath{\mid}}^{3}$ term crosses over to a non-Fermi-liquid form ${\ensuremath{\mid}M\ensuremath{\mid}}^{7∕2}$ near a QCP. The prefactor of the ${\ensuremath{\mid}M\ensuremath{\mid}}^{7∕2}$ 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 $\ensuremath{\chi}$ 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.