Transport in unconventional superconductors: Application to liquid ${}^3\mathrm{He}$ in aerogel

We consider quite generally the transport of energy and momentum in unconventional superconductors and Fermi superfluids to which both impurity scattering (treated within the $t$-matrix approximation) and inelastic scattering contributes. A new interpolation scheme for the temperature dependence of the transport parameters is presented which preserves all analytical results available for $T\to 0$ and $T\to T_c$ and allows for a particularly transparent physical representation of the results. The two scattering processes are combined using Matthiessen’s rule coupling. This procedure is applied for the first time to ${}^3\mathrm{He}\text{−}\mathrm{B}$ in aerogel. Here, at the lowest temperatures, a universal ratio of the thermal conductivity and the shear viscosity is found in the unitary limit, which is akin to the Wiedemann-Franz law.

Figure 1

(Color online) Pair-breaking parameter $C_{\eta }$ for ${}^3\mathrm{He}\text{−}\mathrm{B}$ for various ${\Gamma }_N∕\Delta$ vs scattering phase shift $c=\cot {\delta }_0$.

Figure 2

(Color online) Total reduced shear viscosity at a pressure of $10\text{bar}$ in the unitary limit vs reduced temperature for various values of the parameter $a_c={\overline{\tau }}_N^i\left(T_c\right)∕{\tau }_N^e$. Note that $a_c=0$ corresponds to the clean limit.

Figure 3

(Color online) Total reduced thermal conductivity, divided by $T∕T_c$, at a pressure of $10\text{bar}$ in the unitary limit vs reduced temperature for various values of the parameter $a_c={\overline{\tau }}_N^i\left(T_c\right)∕{\tau }_N^e$. Again, $a_c=0$ corresponds to the clean limit.

Figure 4

(Color online) Total reduced shear viscosity in the Born limit vs reduced temperature for various values of the parameter $a_c={\overline{\tau }}_N^i\left(T_c\right)∕{\tau }_N^e$.

Figure 5

(Color online) Total reduced thermal conductivity in the Born limit vs reduced temperature for various values of the parameter $a_c={\overline{\tau }}_N^i\left(T_c\right)∕{\tau }_N^e$.