Spin and Charge Susceptibility of a Ferromagnetic Electron Gas

In the ferromagnetic state of an electron gas where the energy bands are spin split, the charge potential produces a spin polarization besides a charge polarization and a magnetic field produces a charge polarization besides a spin polarization. Thus, in a ferromagnetic electron gas, besides the spin susceptibility to a magnetic field ${\chi }^{\mathrm{mm}}$ and the charge susceptibility to a charge potential ${\chi }^{\mathrm{ee}}$, we introduce two nondiagonal susceptibilities, the spin response to a charge potential ${\chi }^{\mathrm{me}}$ and the charge response to a magnetic field ${\chi }^{\mathrm{em}}$. We present a general formulation for these four susceptibilities and discuss them within the Hartree-Fock or random-phase approximation, taking into account the long-range nature of the Coulomb interaction properly. The importance of the nondiagonal susceptibilities in analyzing observed spin and charge polarization around an impurity is pointed out.