$1/(N-1)$ expansion, NRG, NCA, and exact $T\to \infty$ limit for the Green's function of an SU($N$) Anderson impurity

We study the Green's function of the $N$-orbital Anderson impurity in a wide range of the Coulomb interaction $U$, frequency $\omega$, and temperature $T$, carrying out the calculations with the $1/(N-1)$ expansion [1], the numerical renormalization group (NRG), the non-crossing approximation (NCA), and the exact expression that is available at high temperatures or the high-bias limit of a nonequilibrium steady state [2]. Comparisons of these approaches are made, specifically, for the $N=4$ particle-hole symmetric case. The $1/(N-1)$ expansion is a new large $N$ approach based on the perturbation theory in $U$ [1], and is complementary to the NCA which uses the power series expansion in the hybridization matrix element $V$. The calculations with this approach are carried out up to order $1/(N-1)^2$, which reasonably capture the fluctuations beyond the random phase approximation (RPA) especially at low energies. We also discuss the $N$ dependence for $N > 4$. [1] A.O., R.\ Sakano, and T.\ Fujii, PRB {\bf 84}, 113301 (2011).\\[4pt] [2] A.O. and R.\ Sakano, PRB {\bf 88}, 155424 (2013).