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Qualitative breakdown of the non-crossing approximation for the symmetric one-channel Anderson impurity model at all temperatures

The Anderson impurity model is studied by means of the self-consistent hybridization expansions in its non-crossing (NCA) and one-crossing (OCA) approximations. We have found that for the one-channel spin-$1/2$ particle-hole symmetric Anderson model, the NCA results are qualitatively wrong for any temperature, even when the approximation gives the exact threshold exponents of the ionic states. Actually, the NCA solution describes an overscreened Kondo effect, because it is the same as for the two-channel infinite-$U$ single level Anderson model. We explicitly show that the NCA is unable to distinguish between these two very different physical systems, independently of temperature. Using the impurity entropy as an example, we show that the low temperature values of the NCA entropy for the symmetric case yield the limit $S_{imp}(T=0)\rightarrow \ln\sqrt{2},$ which corresponds to the zero temperature entropy of the overscreened Kondo model. Similar pathologies are predicted for any other thermodynamic property. On the other hand, we have found that the OCA approach lifts the artificial mapping between the models and restores correct properties of the ground-state, for instance, a vanishing entropy at low enough temperatures $S_{imp}(T=0)\rightarrow0$. Our results indicate that the very well known NCA should be used with caution close to the symmetric point of the Anderson model.