Nonanalytic behavior of the spin susceptibility in clean Fermi systems

The wavevector and temperature dependent static spin susceptibility, $\chi_s(Q,T)$, of clean interacting Fermi systems is considered in dimensions $1\leq d \leq 3$. We show that at zero temperature $\chi_s$ is a nonanalytic function of $|Q|$, with the leading nonanalyticity being $|Q|^{d-1}$ for 1<d<3, and $Q^2\ln|Q|$ for d=3. For the homogeneous spin susceptibility we find a nonanalytic temperature dependence $T^{d-1}$ for 1<d<3. We give qualitative mode-mode coupling arguments to that effect, and corroborate these arguments by a perturbative calculation to second order in the electron-electron interaction amplitude. The implications of this, in particular for itinerant ferromagnetism, are discussed. We also point out the relation between our findings and established perturbative results for 1-d systems, as well as for the temperature dependence of $\chi_s(Q=0)$ in d=3.