Asymptotic limit of high spatial dimensions and thermodynamic consistence

The question of thermodynamic consistence and $\mathrm{\Phi }$-derivability of the asymptotic limit of high spatial dimensions for quantum itinerant models is addressed. It is shown that although the irreducible $n$-particle Green functions are local, reducible vertex functions retain different momentum dependence. As a consequence, the vertex corrections to conductivity do not generally vanish in the mean-field limit. The mean-field theory is a $\mathrm{\Phi }$-derivable approximation only if regular nonlocal or anomalous local external sources are admitted.