Orbital Susceptibility of an Interaction Electron Gas

The wave-vector and frequency-dependent orbital susceptibility (${\chi }^{\mathrm{orb}}$) of an interacting electron gas is expressed in terms of suitable vertex functions. This makes the theory parallel to that of the spin susceptibility (${\chi }^{\mathrm{sp}}$) of this system. The equation for the vertex function is solved in the statically screened exchange approximation by a variational method introduced earlier by one of the authors. The static long-wavelength limit of (${\chi }^{\mathrm{orb}}$) is shown to be related to the difference between the $f$- and $p$-wave decomposition of the effective interaction, whereas ${\chi }^{\mathrm{sp}}$ is related to the difference between the corresponding $p$- and $s$-wave parts in the same limit. The classic result that ${\chi }^{\mathrm{orb}}$ is minus one-third of ${\chi }^{\mathrm{sp}}$ for the noninteracting system is modified when the interactions are included. Explicit results are given for a model Yukawa interaction. From these, it follows that, for very short-range interactions, (${\chi }^{\mathrm{sp}}$) reduces to the Stoner-enhanced form while ${\chi }^{\mathrm{orb}}$ is unaffected. The momentum dependence of the interaction is thus more important for the determination of ${\chi }^{\mathrm{orb}}$ than for ${\chi }^{\mathrm{sp}}$. In the unscreened Coulomb limit as well as for small screening, our results reduce to those obtained earlier by Kanazawa and Matsudaira. Several errors in the existing expressions for ${\chi }^{\mathrm{orb}}(\overrightarrow{\mathrm{q}}, q_0)$ are corrected in this work.