Strong-coupling theory of superfluid transition temperatures for paramagnon models: Application toHe3

The triplet superconducting transition temperature $T_c$ for $l=1\mathrm{}$ paramagnon-induced pairing is computed within a Matsubara formulation of conventional strong-coupling theory, as a function of the interaction parameter $\overline{I}$. Using the scattering amplitudes of Fermi-liquid theory to fix $\overline{I}$ for ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ at each pressure, we apply our results to ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The computed values of $T_c$ differ by less than a factor of 5 from those measured experimentally but the (slight) pressure dependence and the effective mass ratio $Z=\frac{m^{*}}{m}$ are incorrect. If $Z$ is adjusted to be in better accord with experiment, we then obtain reasonable agreement with the measured magnitude and pressure dependence of $T_c$ over the entire pressure range. General features of the paramagnon model are (i) $T_c\left(\overline{I}\right)$ has a maximum value of ${10}^{-2\mathrm{}}$∼${10}^{-3\mathrm{}}$ of the Fermi temperature $T_F$ at $\overline{I}\simeq 0.995$; it seems doubtful that even under the most ideal conditions the paramagnon mechanism can be used to obtain high-temperature superconductors; (ii) $T_c\left(\overline{I}\right)$ vanishes by $\overline{I}=1\mathrm{}$. (iii) Below $\overline{I}\sim 0.97$, the exponential form $T_c={\overline{\omega }}_c{e}^{-\frac{b}{\lambda }}$ is obtained where $b$ is close to unity, ${\overline{\omega }}_c$ is a constant of order $\frac{T_F}{10}$, and $\lambda$ is the renormalized coupling constant $\frac{{\lambda }^{\Delta }}{Z}$. The previously proposed analogous expression involving the spin-fluctuation frequency ${\omega }_{\mathrm{sf}}$,