Qualitative breakdown of the noncrossing 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 noncrossing (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 $S_{\mathrm{imp}}(T=0)\to ln\sqrt{2},$ 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 $S_{\mathrm{imp}}(T=0)\to 0$. Our results indicate that the very well known NCA should be used with caution close to the symmetric point of the Anderson model.

Figure 1

Impurity contribution to the total entropy as a function of temperature for several values of the degeneracy $N$ and $E_f=-4$. The temperatures are scaled by the corresponding Kondo ones, $T_K/\mathrm{\Delta }\approx 0.007,0.016,$ and 0.1 for $N=2,4,$ and 6, respectively. The horizontal dashed line stands for a guide indicating the value of $g=1+N$. The inset shows the low-temperature behavior.

Figure 2

Upper panel: NCA impurity contribution to the total entropy as a function of temperature, for several values of the Coulomb repulsion and $N=2$ and $E_f=-4$. The temperatures are scaled by the corresponding Kondo ones, $T_K/\mathrm{\Delta }\approx 0.007,0.0065,$ and 0.004 for $U=\infty ,20,$ and 8, respectively. Lower panel: Same results as in the upper panel for $U=8$ and $E_f=-4$ and for the impurity entropy of the 2CH, $U=\infty$ model at the same energy level $E_f=-4$ (red dashed line).

Figure 3

Upper panel: Comparison of the impurity entropy for the symmetric $N=2$ AIM calculated within OCA and the corresponding one for the overscreened model using Eq. (10) at the same energy level $E_f=-4$. Lower panel: Impurity contribution to the total entropy as a function of temperature, within the OCA approximation, for several values of the degeneracy $N$ and for $U=8$, $E_f=-4$. The temperatures are scaled by the corresponding Kondo ones: $T_K/\mathrm{\Delta }\approx 0.06,0.185,$ and 0.6 for $N=2,4,$ and 6, respectively. The inset shows the low-temperature behavior.