Fig. 3.4 Diagrammatic representation of the Luttinger-Ward functional generating the CTMA for U ! 1. The first diagram constitutes the NCA. The two-loop diagram is excluded, since it is not a skeleton. Solid, dashed, and wiggly lines represent conduction electron, renormalized pseudofermion and auxiliary boson propagators, respectively. The terms with the conduction electron lines running clockwise generate the conduction electron-pseudofermion ladder vertex T(cf) with bosons as rungs (spin fluctuations), while the terms with the conduction electron lines running counter-clockwise generate the conduction electron-empty orbital ladder vertex T(ce) with pseudofermions as rungs (charge fluctuations)

Anderson lattice model [85,86] as well as to phase transitions in dilute, magnetic semiconductors at not too low temperature [87]. The NCA has also been generalized to the case of multiple local orbitals, as in rare earth and transition metal ions, where the NCA correctly produces a distinct Kondo resonance for each crystal-field or spin-orbit split local orbital, each with a characteristic, logarithmic temperature dependence [88,89].

However, in NCA the infrared exponents of the auxiliary particle propagators come out independent of n, aNCA = 1/(N C 1), aNCA = N/(N C 1) [70,71,81], with N the spin degeneracy, in contrast to the Fermi liquid values, (3.62)-(3.64). As a consequence, the NCA fails to describe Fermi liquid behavior at temperatures T ^ 7K, with spurious infrared singularities appearing in physical quantities at energies or temperatures T ^ TK [70,71,79-81]. Since the NCA becomes formally exact for SU(N) symmetric models in the limit N !i, with deviations appearing in O^) [79, 80, 90], this low-T failure is less pronounced for N 1. Note, however, that the deviation of the NCA infrared exponents af and ae is of order 1/N, not 1/N2 as one may have expected. In a magnetic field the NCA also fails even in the high-temperature regime, T > TK, producing a spurious resonance in the impurity spectrum at ! = 0 in addition to the two Zeeman-split Kondo peaks. The low-T failure of the NCA can be traced back to its insufficient inclusion of coherent multiple spin-flip processes which are responsible for the formation of the Kondo singlet state. The origin of the failure in a magnetic field, on the other hand, lies in the fact that NCA neglects the exchange diagram to the conduction electron-impurity spin vertex at second order in the spin coupling J [68, 91]. As a consequence, logarithmic contributions in the potential scattering channel do not cancel even in leading logarithmic order, producing a spurious resonance which does not Zeeman-split in a magnetic field.

3.5.4.2 Anderson Impurity Model with Finite U : SUNCA

At finite Coulomb interaction U, the spin exchange interaction J acquires contributions from both, virtual excitations to the empty and to the doubly occupied impurity states via a Schrieffer-Wolff transformation [92],

Neglecting either one of these contributions would lead to an exponentially wrong Kondo scale TK, because of the exponential dependence of TK on J .A simple generalization of NCA to this case, i.e., adding the second order self-consistent perturbation theory for the two processes, fails to capture the simultaneous contribution of both channels in each order of bare perturbation theory. For a correct treatment of both terms, there must be included, for each diagram with an empty boson line Ge, the corresponding diagram with Ge replaced by a doubly occupied boson line Gd (which amounts to the exchange diagram of the former), and vice versa, on the level of bare perturbation theory [93,94]. The corresponding vertex corrections have first been evaluated in leading self-consistent order by Sakai et al. [95] and by Pruschke and Grewe [96]. The first conserving approximation for finite U, fully symmetric with respect to the empty and double occupied fluctuation channels, was formulated and evaluated by Haule et al. [93] and termed as the symmetrized finite-U NCA (SUNCA). On the level of renormalized perturbation theory (generating functionals), it means that for each dressed b-line there must be included a ladder vertex function with a-lines as rungs, and vice versa. The SUNCA is tractable with relatively moderate numerical effort, since it can be formulated in terms of no higher than three-point vertex functions. The results of a fully self-consistent evaluation of the impurity electron spectral function within SUNCA are shown in Fig. 3.51 in comparison with NRG results. It is seen that the correct Kondo scale (width of the Kondo peak) is reproduced. However, like the NCA, the SUNCA solution still develops a spurious low-T singularity.

There is evidence that both failures of NCA, at low temperature and in a magnetic field, can be cured by a systematic resummation of coherent spin-flip terms to infinite order, which will be discussed in the next section.

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