Variational theory of superconductivity and application to the low-density electron gas

The method of effective-potential expansion is employed to describe superconductivity in strongly correlated electron systems. We have obtained a gap equation in which both the pairing potential $V_p$(k,k’) and the single-particle energy ɛ${̃}_{\mathrm{k}}$ are expanded in terms of the effective potential. When the ring approximation or the random-phase approximation is used to evaluate each term in the expansion, $V_p$(k,k’) and ɛ${̃}_{\mathrm{k}}$ have the form $V_p^{\mathrm{KMK}(\mathrm{k}}$,k’)/$z_{\mathrm{k}}$ $z_{\mathrm{k}\mathcal{’}}$, and ${\varepsilon }_{\mathrm{k}}^{\mathrm{*}}$/$z_{\mathrm{k}}$, respectively, where $V_p^{\mathrm{KMK}}$ is the pairing potential derived by Kirzhnits, Maksimov, and Khomskii (KMK), $z_{\mathrm{k}}$ is the renormalization factor, and ${\varepsilon }_{\mathrm{k}}^{\mathrm{*}}$ tends to the bare single-particle energy in the weak-coupling limit. Thus our theory may be viewed as an extension of the KMK theory to the strong-coupling region. The present general theory is applied to the electron gas in order to investigate whether the Coulomb interaction alone can cause superconductivity. We have improved on the two-body approximation to describe the normal state of the electron gas and have determined the effective potential variationally. With the use of the effective potential thus obtained, we have evaluated all terms up to second order for both $V_p$(k,k’) and ɛ${̃}_{\mathrm{k}}$. The pairing potential includes the plasmon-mediated attractive and the paramagnon-mediated repulsive parts as well as many complicated vertex corrections. We have solved the gap equation numerically and obtained the result that superconductivity appears at rather low carrier densities (i.e., $r_s$>3.9).