Nonanalytic magnetic response of Fermi and non-Fermi liquids

We study the nonanalytic behavior of the static spin susceptibility of two-dimensional fermions as a function of temperature and magnetic field. For a generic Fermi liquid, ${\chi }_s(T,H)=\mathrm{const}+c_1\max \{T,{\mu }_B\mid H\mid \}$, where $c_1$ is shown to be expressed via complicated combinations of the Landau parameters, rather than via the backscattering amplitude, contrary to the case of the specific heat. Near a ferromagnetic quantum critical point, the field dependence acquires a universal form ${\chi }_s^{-1}\left(H\right)=\mathrm{const}-c_2{\mid H\mid }^{3∕2}$, with $c_2>0$. This behavior implies a first-order transition into a ferromagnetic state. We establish a criterion for such a transition to win over the transition into an incommensurate phase.

Figure 1

Schematic phase diagram near a ferromagnetic QCP for $\rho <1$ (see text). $x$ is a control parameter—e.g., pressure. The second-order phase transition becomes first-order below a tricritical point because of the nonanalytic ${\eta }^{7∕2}$ term in Eq. (8). At small $T$, the transition line has an S-shaped form because of the negative $T$ dependence of ${\chi }_s^{-1}\left(T\right)$.