Universal temperature corrections to Fermi liquid theory in an interacting electron system

Abstract

We calculate analytically the effective mass and the quasiparticle renormalization factor in an electron liquid with long-range Coulomb interactions between electrons in two and three dimensions in the leading order density expansion. We concentrate on the temperature dependence of the effective mass in the limit $T∕{T}_{\mathrm{F}}⪡{r}_{\mathrm{s}}⪡1$ and show that the leading temperature correction is linear in two dimensions and proportional to ${T\phantom{\rule{0.1em}{0ex}}}^{2}\phantom{\rule{0.3em}{0ex}}\mathrm{ln}(1∕T)$ in three dimensions (positive in both cases). We explicitly calculate the coefficients, which are shown to be universal density independent parameters of the order of unity (in the high-density limit). The singular temperature corrections are due to the singularity in the dynamic dielectric function at $\ensuremath{\omega}\ensuremath{\sim}{v}_{\mathrm{F}}q$ and $q⪡2{p}_{\mathrm{F}}$. In two dimensions, we predict a nonmonotonic effective mass temperature dependence and find that the maximum occurs at a temperature ${T}^{*}\ensuremath{\sim}{T}_{\mathrm{F}}{r}_{\mathrm{s}}\phantom{\rule{0.3em}{0ex}}{\mathrm{ln}}^{\ensuremath{-}1}(1∕{r}_{\mathrm{s}})$. We also study the quasiparticle renormalization factor in both three and two dimensions.