Electron correlations in a multivalley electron gas and Fermion-boson conversion

The correlation energy ${\mathrm{\varepsilon }}_{\mathit{c}}$, the chemical-potential shift due to the correlation, ${\mathrm{\mu }}_{\mathit{c}}$, the compressibility κ, and the renormalization factor at the Fermi surface, ${\mathit{z}}_{\mathit{F}}$, are calculated for the paramagnetic valley-unpolarized state of the multivalley electron gas that may be realized in a doped multivalley semiconductor or multivalley semimetal. The calculations are based on an improved version of the effective-potential expansion method, which is known to give accurate ${\mathrm{\varepsilon }}_{\mathit{c}}$, ${\mathrm{\mu }}_{\mathrm{c}}$, κ, and ${\mathit{z}}_{\mathit{F}}$ for a single-valley electron gas at metallic densities. The valley degeneracy is found to be a good parameter to connect the total energies of the electron gas of both paramagnetic and ferromagnetic states with that of the charged-boson system. Based on this connection, the possibility of a valley-polarized state is discussed.