The Lorenz number in CeCoIn${}_5$ inferred from the thermal and charge Hall currents

The thermal Hall conductivity ${\kappa }_{xy}$ and Hall conductivity ${\sigma }_{xy}$ in CeCoIn${}_5$ are used to determine the Lorenz number ${\cal L}_H$atlowtemperature$T$. This enables the separation of the observed thermal conductivity into its electronic and non-electronic parts. We uncover evidence for a charge-neutral, field-dependent thermal conductivity, which we identify with spin excitations. At low $T$, these excitations dominate the scattering of charge carriers. We show that suppression of the spin excitations in high fields leads to a steep enhancement of the electron mean-free-path, which leads to an interesting scaling relation between the magnetoresistance, thermal conductivity and ${\sigma }_{xy}$.