We use the poor man's scaling approach to study the phase boundaries of a pair of quantum impurity models featuring a power-law density of states ρ(ε) ∝ |ε| r , either vanishing (for r > 0) or diverging (for r < 0) at the Fermi energy ε = 0, that gives rise to quantum phase transitions between local-moment and Kondo-screened phases. For the Anderson model with a pseudogap (i.e., r > 0), we find the phase boundary for (a) 0 < r < 1/2, a range over which the model exhibits interacting quantum critical points both at and away from particle-hole (p-h) symmetry, and (b) r > 1, where the phases are separated by first-order quantum phase transitions that are accessible only for broken p-h symmetry. For the p-h-symmetric Kondo model with easy-axis or easy-plane anisotropy of the impurity-band spin exchange, the phase boundary and scaling trajectories are obtained for both r > 0 and r < 0. Throughout the regime of weak-to-moderate impurity-band coupling in which poor man's scaling is expected to be valid, the approach predicts phase boundaries in excellent qualitative and good quantitative agreement with the nonperturbative numerical renormalization group, while also establishing the functional relations between model parameters along these boundaries.

An experimental investigation of the transition from Fowler–Nordheim (FN) field emission to space-charge-limited (SCL) flows in a nanogap is presented. Electrodes with gap size D∼30–70 nm corresponding to D/λo up to a maximum of ∼2×103, where λo is the de Broglie wavelength of the space-charge-electrons, are experimented. The transition from the FN field emission to the classical SCL flow is a function of the applied bias and lies in the range 5–15 V. The equilibrium transmitted current density for the 50 nm sample indicates a transition from the FN to the quantum SCL flow at ∼0.4 V with D/λo of ∼35 and then gradually to the classical SCL behavior as the voltage is increased beyond ∼9 V. The experiments indicate no sharp demarcation between the different regimes.

The Anderson impurity model with a density of states ρ(ε) ∝ |ε| r containing a power-law pseudogap centered on the Fermi energy (ε = 0) features for 0 < r < 1 a Kondo-destruction quantum critical point (QCP) separating Kondo-screened and local-moment phases. The observation of mixed valency in quantum critical β-YbAlB4 has prompted study of this model away from particle-hole symmetry. The critical spin response associated with all Kondo destruction QCPs has been shown to be accompanied, for r = 0.6 and noninteger occupation of the impurity site, by a divergence of the local charge susceptibility on both sides of the QCP. In this work, we use the numerical renormalization-group method to characterize the Kondo-destruction charge response using five critical exponents, which are found to assume nontrivial values only for 0.55 r < 1. For 0 < r 0.55, by contrast, the local charge susceptibility shows no divergence at the QCP, but rather exhibits nonanalytic corrections to a regular leading behavior. Both the charge critical exponents and the previously obtained spin critical exponents satisfy a set of scaling relations derived from an ansatz for the free energy near the QCP. These critical exponents can all be expressed in terms of just two underlying exponents: the correlation-length exponent ν(r) and the gap exponent ∆(r). The ansatz predicts a divergent local charge susceptibility for ν < 2, which coincides closely with the observed range 0.55 r < 1. Many of these results are argued to generalize to interacting QCPs that have been found in other quantum impurity models.

We study the impurity entanglement entropy Se in quantum impurity models that feature a Kondo-destruction quantum critical point (QCP) arising from a pseudogap in the conduction-band density of states or from coupling to a bosonic bath. On the local-moment (Kondo-destroyed) side of the QCP, the entanglement entropy contains a critical component that can be related to the order parameter characterizing the quantum phase transition. In Kondo models describing a spinSimp, Se assumes its maximal value of ln(2Simp + 1) at the QCP and throughout the Kondo phase, independent of features such as particle-hole symmetry and under-or over-screening. In Anderson models, Se is nonuniversal at the QCP, and at particle-hole symmetry, rises monotonically on passage from the local-moment phase to the Kondo phase; breaking this symmetry can lead to a cusp peak in Se due to a divergent charge susceptibility at the QCP. Implications of these results for quantum critical systems and quantum dots are discussed.