Functional renormalization group approach to the Anderson impurity model

We develop a functional renormalization group approach which describes the low-energy single-particle properties of the Anderson impurity model up to intermediate on-site interactions $U \lesssim 15 \Delta$, where $\Delta$ is the hybridization in the wide-band limit. Our method is based on a generalization of a method proposed by Sch\"{u}tz, Bartosch and Kopietz [Phys. Rev. B 72, 035107 (2005)], using two independent Hubbard-Stratonovich fields associated with transverse and longitudinal spin fluctuations. Although we do not reproduce the exponentially small Kondo scale in the limit $U \to \infty$, the spin fluctuations included in our approach remove the unphysical Stoner instability predicted by mean-field theory for $U > \pi \Delta$. We discuss different decoupling schemes and show that a decoupling which manifestly respects the spin-rotational invariance of the problem gives rise to the lowest quasiparticle weight. To obtain a closed flow equation for the fermionic self-energy we also propose a new truncation scheme of the functional renormalization group flow equations using Dyson-Schwinger equations to express bosonic vertex functions in terms of fermionic ones.