Anderson impurity model in a semiconductor

We consider an Anderson impurity model in which the locally correlated orbital is coupled to a host with a gapped density of states. Single-particle dynamics are studied within a perturbative framework that includes both explicit second-order perturbation theory and self-consistent perturbation theory to all orders in the interaction. Away from particle-hole symmetry, the system is shown to be a generalized Fermi liquid (GFL) in the sense of being perturbatively connectable to the noninteracting limit, and the exact Friedel sum rule for the GFL phase is obtained. We show by contrast that the particle-hole-symmetric point of the model is not perturbatively connected to the noninteracting limit and as such is a non-Fermi liquid for all nonzero gaps. Our conclusions are in agreement with numerical renormalization group studies of the problem.

Figure 1

Real part of the hybridization function ${\Delta }_R\left(\omega \right)$ vs $\omega ∕\delta$ (solid line). The dashed line illustrates the solution of Eq. (13) as described in the text.

Figure 2

Self-energy diagrams up to second order in the interaction $U$. The solid lines represent the noninteracting propagator, and the wavy lines represent $U$.

Figure 3

Skeleton expansion of the self-energy. The exact $\Sigma \left(\omega \right)$ is given by the sum of all skeleton self-energy diagrams constructed from the exact $G\left(\omega \right)$.

Figure 4

Sketch showing how poles in $\Sigma \left(\omega \right)$ lead (via the Dyson equation) to poles in $G\left(\omega \right)$. (a) Solid lines represent the schematic form of ${\Sigma }^I\left(\omega \right)+{\Delta }_I\left(\omega \right)$ with poles in the gap, dashed lines represent the corresponding ${\Sigma }^R\left(\omega \right)+{\Delta }_R\left(\omega \right)$ from Hilbert transformation, and the dotted lines show the constructions used for determing the poles in $G\left(\omega \right)$ from Dyson’s equation [Eq. (5)]. (b) The pole structure of $G\left(\omega \right)$ obtained from case (i) of (a). (c) The pole structure of $G\left(\omega \right)$ obtained from case (ii) of (a).