Scaling law of the thermal conductivity in nodal superconductors

The scaling law of the thermal conductivity in d-wave superconductors in the vortex state has been considered by Simon-Lee$^{1}$ and K\"{u}bert-Hirschfeld$^2$. Here we present simple expressions of $\kappa_{xx}$ and $\kappa_{xy}$ for a class of nodal superconductors in the clean limit (i.e. $\Gamma \ll v\sqrt{eH}$, $T \ll \Delta$ where $\Gamma$ is the quasi-particle scattering rate, $v$ the Fermi velocity of quasiparticle and $T$ the temperature). When the above condition is satisfied, $\kappa_{xx}$ and $\kappa_{xy}$ are simple function of $v\sqrt {eH} /T$. The present result is compared with the recent thermal conductivity data of the heavy fermion superconductor CePt$_3$Si by Izawa et al. \par\noindent [1] S.H. Simon and P.A. Lee, PRL 78, 1548 (1997). \par\noindent [2] C. K\"{u}bert and P.J. Hirschfeld, PRL 80, 4963 (1998).