Double expansion with respect to $U$ and $1/(N-1)$ for an SU($N$) impurity Anderson model

We apply a new large-$N$ scheme for an SU($N$) impurity Anderson model [1,2] to the Green's function for finite frequency $\omega$ and finite Coulomb interaction $U$. This approach is essentially different from the conventional large-$N$ theories, such as the non-crossing approximation and its extensions which are based on a perturbation expansion in the hybridization strength $V$. Our expansion scheme, which uses $1/(N-1)$ and the scaled interaction $u \equiv (N-1)U$ as a set of two independent variables, gives the Hartree-Fock (HF) results at zeroth order. Then, to leading order in $1/(N-1)$ it describes the Hartree-Fock random phase approximation (HF-RPA). The higher-order corrections systematically describe the fluctuations beyond the HF-RPA. It was shown that the renormalized local-Fermi-liquid parameters, calculated up to order $1/(N-1)^2$, agree closely with the exact NRG results at $N=4$ where the degeneracy is still not so large [1,2]. We discuss the $\omega$ dependence of the Green's function to clarify both the low- and high-energy features. \\[4pt] [1] A.O., R.\ Sakano, and T.\ Fujii, PRB {\bf 84}, 113301 (2011).\\[0pt] [2] A.O., PRB {\bf 85}, 155404 (2012).