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We present a unified theory of quantum phase transitions for half-filledquantum dots (QDs) coupled to gapped host bands. We augment the bands byadditional weakly coupled metallic lead which allows us to analyze the systemby using standard numerical renormalization group techniques. The ground stateproperties of the systems without the additional metallic lead are thenextrapolated in a controlled way from the broadened subgap spectral functions.We show that a broad class of narrow-gap-semiconductor tunneling densities ofstates (TDOSs) support the existence of two distinct phases known from theirsuperconducting counterpart. Namely, $0$ phase which is marked by the singletground state and the $\pi$ phase regime with the doublet ground state. To keepa close analogy with the superconducting case, we focus on the influence ofparticle-hole asymmetry of the TDOS of the subgap spectral features.Nevertheless, we also discus the possibility of inducing singlet-doubletquantum phase transitions in experimental setups by varying the filling of theQD. In addition, for gapped TDOS functions with smoothed gap edges, wedemonstrate that all subgap peaks may leak out of the gap into the continuouspart of the spectrum, an effect which has no counterpart in the superconductingAnderson model.

Unified theory of quantum phase transitions in quantum dots with gapped host bands