Nonanalytic behavior of the spin susceptibility in clean Fermi systems

The wave vector and temperature-dependent static spin susceptibility, ${\mathrm{\chi }}_{\mathrm{s}}$(Q,T), of clean interacting Fermi systems is considered in dimensions 1⩽d⩽3. We show that at zero temperature ${\mathrm{\chi }}_{\mathrm{s}}$ is a nonanalytic function of |Q|, with the leading nonanalyticity being |Q ${\mathrm{|}}^{\mathrm{d}\mathrm{-}1}$ for 1<d<3, and ${\mathbf{Q}}^2$ln |Q| for d=3. For the homogeneous spin susceptibility we find a nonanalytic temperature dependence ${\mathrm{T}}^{\mathrm{d}\mathrm{-}1}$ 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 one-dimensional systems, as well as for the temperature dependence of ${\mathrm{\chi }}_{\mathrm{s}}$(Q=0) in d=3.