Non-interacting single-impurity Anderson model: Solution without using the equation-of-motion method

Abstract

Ground-state properties of the non-interacting symmetric single-impurity Anderson model (SIAM) are derived from the corresponding eigenenergy equation. Explicit formulae are given for the ground-state energy, the hybridization, and the momentum distribution that are essential quantities for variational approaches to the interacting model. Various spectral functions, e.g., the total density of states, the phase shift function, and the impurity spectral function, are shown to agree with those obtained from the equation-of-motion method (see supplementary material). For a constant hybridization strength and a semi-elliptic host density of states it is seen that the impurity spectral function builds up weight at the band edges. Single-particle problems can be solved exactly, in principle. For beginners in many-particle physics, however, it is helpful to know how this is done in practice. The present work shows how to derive explicitly ground-state and spectral properties for the non-interacting symmetric single-impurity Anderson model (SIAM). The formulae are also useful for researchers employing variational approaches to the interacting SIAM.