We present a unified theory of quantum phase transitions for half-filled quantum dots (QDs) coupled to gapped host bands. We augment the bands by an additional weakly coupled metallic lead, which allows us to analyze the system by using standard numerical renormalization-group techniques. The ground-state properties of the systems without the additional metallic lead are then extrapolated in a controlled way from the broadened subgap spectral functions. We show that a broad class of narrow-gap-semiconductor tunneling densities of states (TDOSs) support the existence of two distinct phases known from their superconducting counterpart: the 0 phase, which is marked by the singlet ground state, and the $\pi$ phase regime with the doublet ground state. To keep a close analogy with the superconducting case, we focus on the influence of particle-hole asymmetry of the TDOS on the subgap spectral features. Nevertheless, we also discuss the possibility of inducing singlet-doublet quantum phase transitions in experimental setups by varying the filling of the QD. In addition, for gapped TDOS functions with smoothed gap edges, we demonstrate that all subgap peaks may leak out of the gap into the continuous part of the spectrum, an effect that has no counterpart in the superconducting Anderson model.

• Received 9 December 2021
• Revised 24 March 2022
• Accepted 11 April 2022

DOI:https://doi.org/10.1103/PhysRevB.105.205412