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We present a general framework how to investigate stability of solutionswithin a single self-consistent renormalization scheme being a parquet-typeextension of the Baym-Kadanoff construction of conserving approximations. Toobtain a consistent description of one- and two-particle quantities, needed forthe stability analysis, we impose equations of motion on the one- as well onthe two-particle Green functions simultaneously and introduce approximations intheir input, the completely irreducible two-particle vertex. Thereby we do notloose singularities caused by multiple two-particle scatterings. We find acomplete set of stability criteria and show that each instability, singularityin a two-particle function, is connected with a symmetry-breaking orderparameter, either of density type or anomalous. We explicitly study the Hubbardmodel at intermediate coupling and demonstrate that approximations with staticvertices get unstable before a long-range order or a metal-insulator transitioncan be reached. We use the parquet approximation and turn it to a workablescheme with dynamical vertex corrections. We derive a qualitatively new theorywith two-particle self-consistence, the complexity of which is comparable withFLEX-type approximations. We show that it is the simplest consistent and stabletheory being able to describe qualitatively correctly quantum critical pointsand the transition from weak to strong coupling in correlated electron systems.Comment: REVTeX, 26 pages, 12 PS figure

Stability of self-consistent solutions for the Hubbard model at intermediate and strong coupling