We present a general framework for investigating the stability of solutions being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we use explicit equations for one- and two-particle Green functions simultaneously. We formulate a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density or anomalous type. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices become unstable with respect to spin flips, before the Mott-Hubbard metal-insulator transition in the paramagnetic phase or the Kondo strong-coupling regime can be reached. Further on, we use the parquet approximation and turn it into a workable scheme with dynamical vertex corrections. We obtain a theory with two-particle self-consistence, the complexity of which is comparable with fluctuation-exchange-type approximations. We show that it is the simplest consistent theory free of spurious phases being able to correctly describe qualitatively, quantum critical points and the transition from weak to strong coupling in correlated electron systems.

• Received 10 February 1999

DOI:https://doi.org/10.1103/PhysRevB.60.11345