Fig. 2: Density-dependence of the form of the effective energy-fluctuation mediated electron-electron attraction \({V}_{En}^{Pair}({{{{{{{\bf{x}}}}}}}},\tau )\), Eq. (11).
From: Superconductivity from energy fluctuations in dilute quantum critical polar metals
The colored lines show the three energy scales relevant for pairing: ωT(ne) (Eq. (6)), related to the polar order correlation length ξ = cs/ωT(ne), the characteristic momentum-exchange scale 2cskF and the Fermi energy EF, the characteristic energy exchange scale. In each region (separated by gray dashed lines) the dominant scale determines the effective form of the interaction. At low densities \({n}_{e}\ll {n}_{1}\approx \frac{1}{3{\pi }^{2}}{(\frac{{\omega }_{T0}}{2{c}_{s}})}^{3}\) the interaction can be approximated with a local one. On increasing the density \(({n}_{1}\ll {n}_{e}\ll {n}_{2}=3{\pi }^{2}{(\frac{4{m}^{*}{c}_{s}}{\hslash })}^{3})\), the momentum dependence of the interaction becomes prominent first, such that the interaction can be approximated with an instanteneous, but non-local one (12). At the highest densities, strong retardation appears on the scale of order ℏ/EF.
