Fermi Surface Effect on Lorenz Number of Correlated Metal

We investigate an effect that an ideal Lorenz number $L_{\rm i}$ of correlated metal shows peculiar Fermi surface dependence, which is caused by the onset of a particular channel of Umklapp scattering. We evaluate $L_{\rm i}$ for some simple models and transition metals, and note that $L_{\rm i}$ for Na$_x$CoO$_2$ decreases sensitively as $x$ approaches an Umklapp threshold around $x_c \simeq 0.6$.