Wiedemann-Franz law and non-vanishing temperature scale across the field-tuned quantum critical point of YbRh$_2$Si$_2$

The in-plane thermal conductivity $\kappa$ and electrical resistivity $\rho$ of the heavy-fermion metal YbRh$_2$Si$_2$ were measured down to 50 mK for magnetic fields $H$ parallel and perpendicular to the tetragonal $c$ axis, through the field-tuned quantum critical point, $H_c$, at which antiferromagnetic order ends. The thermal and electrical resistivities, $w \equiv L_0 T/\kappa$ and $\rho$, show a linear temperature dependence below 1~K, typical of the non-Fermi liquid behaviour found near antiferromagnetic quantum critical points, but this dependence does not persist down to $T=0$. Below a characteristic temperature $T^\star \simeq 0.35$~K, which depends weakly on $H$, $w(T)$ and $\rho(T)$ both deviate downward and converge as $T \to 0$. We propose that $T^\star$ marks the onset of short-range magnetic correlations, persisting beyond $H_c$. By comparing samples of different purity, we conclude that the Wiedemann-Franz law holds in YbRh$_2$Si$_2$, even at $H_c$, implying that no fundamental breakdown of quasiparticle behaviour occurs in this material. The overall phenomenology of heat and charge transport in YbRh$_2$Si$_2$ is similar to that observed in the heavy-fermion metal CeCoIn$_5$, near its own field-tuned quantum critical point.