Asymptotic limit of high spatial dimensions and thermodynamic consistence of mean-field theories

The question of the thermodynamic consistence of mean-field theories for lattice itinerant models defined from the asymptotic limit of high spatial dimensions is addressed. We show that the local ansatz for the self-energy and the thermodynamic potential generating all physical quantities via functional derivatives does not lead to the complete high-dimensional asymptotics of correlation and vertex functions. This can be achieved only if either regular nonlocal or anomalous local external sources, order parameters, are introduced in the generating functional. The full vertex function then gives rise to new quantum phases and nontrivial corrections to the conductivity.