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We disclose a serious deficiency of the Baym-Kadanoff construction ofthermodynamically consistent conserving approximations. There are two verticesin this scheme: dynamical and conserving. The divergence of each indicates aphase instability. We show that each leads to incomplete and qualitativelydifferent behavior at different critical points. The diagrammaticallycontrolled dynamical vertex from the Schwinger-Dyson equation does not obey theWard identity and cannot be continued beyond its singularity. The standardlyused dynamical vertex alone cannot, hence, conclusively decide about thestability of the high-temperature phase. On the other hand, the divergence inthe conserving vertex, obeying the conservation laws, does not invoke criticalbehavior of the spectral function and the specific heat. Moreover, the criticalbehavior of the conserving vertex may become spurious in low-dimensionalsystems. Consequently, the description of the critical behavior of correlatedelectrons becomes consistent and reliable only if the fluctuations of the orderparameter in the conserving vertex lead to a divergence coinciding with that ofthe dynamical one.

Failure of the Baym-Kadanoff construction to match consistently quantum dynamics with thermodynamic critical behavior