We use the numerical renormalization group method to study an Anderson impurity in a conduction band with the density of states varying as with r > 0. We find two different fixed points: a local-moment fixed point with the impurity effectively decoupled from the band and a strong-coupling fixed point with a partially screened impurity spin. The specific heat and the spin susceptibility show power-law behaviour with different exponents in the strong-coupling and local-moment regimes. We also calculate the impurity spectral function which diverges (vanishes) with in the strong-coupling (local-moment) regime.
We use the numerical renormalization group method to study an Anderson impurity in a conduction band with the density of states varying as with r > 0. We find two different fixed points: a local-moment fixed point with the impurity effectively decoupled from the band and a strong-coupling fixed point with a partially screened impurity spin. The specific heat and the spin susceptibility show power-law behaviour with different exponents in the strong-coupling and local-moment regimes. We also calculate the impurity spectral function which diverges (vanishes) with in the strong-coupling (local-moment) regime.
We use the numerical renormalization group method to investigate the spectral properties of a single-impurity Anderson model with a gap $\ensuremath{\delta}$ across the Fermi level in the conduction-electron spectrum. For any finite $\ensuremath{\delta}>0$, at half filling the ground state of the system is always a doublet. Away from half filling a quantum phase transition (QPT) occurs as function of the gap value $\ensuremath{\delta}$, and the system evolves from the strong-coupling (SC) Kondo-type state, corresponding to $\ensuremath{\delta}<{\ensuremath{\delta}}_{C}$ toward a localized moment (LM) regime for $\ensuremath{\delta}>{\ensuremath{\delta}}_{C}$. The opening of the gap leads to the formation of one (two) bound states when the system is in the SC (LM) regime. The evolution across the QPT of their positions and the corresponding weights together with the dynamic properties of the model are investigated.
We study a pseudogap Anderson-Holstein model of a magnetic impurity level that hybridizes with a conduction band whose density of states vanishes in power-law fashion at the Fermi energy, and couples, via its charge, to a nondispersive bosonic mode (e.g., an optical phonon). The model, which we treat using poor-man's scaling and the numerical renormalization group, exhibits quantum phase transitions of different types depending on the strength of the impurity-boson coupling. For weak impurity-boson coupling, the suppression of the density of states near the Fermi energy leads to quantum phase transitions between strong-coupling (Kondo) and local-moment phases. For sufficiently strong impurity-boson coupling, however, the bare repulsion between a pair of electrons in the impurity level becomes an effective attraction, leading to quantum phase transitions between strong-coupling (charge Kondo) and local-charge phases. Even though the Hamiltonian exhibits different symmetries in the spin and charge sectors, the thermodynamic properties near the two types of quantum phase transition are closely related under spin-charge interchange. Moreover, the critical responses to a local magnetic field (for small impurity-boson coupling) and to an electric potential (for large impurity-boson coupling) are characterized by the same exponents, whose values place these quantum-critical points in the universality class of the pseudogap Anderson model. One specific case of the pseudogap Anderson-Holstein model may be realized in a double-quantum-dot device, where the quantum phase transitions manifest themselves in the finite-temperature linear electrical conductance.
Spin susceptibility of Anderson impurities is a key quantity in understanding the physics of Kondo screening. Traditional numerical renormalization group (NRG) calculation of the impurity contribution ${\ensuremath{\chi}}_{\mathrm{imp}}$ to susceptibility, defined originally by Wilson in a flat wide band, has been generalized before to structured conduction bands. The results brought about non-Fermi-liquid and diamagnetic Kondo behaviors in ${\ensuremath{\chi}}_{\mathrm{imp}}$, even when the bands are not gapped at the Fermi energy. Here, we use the full density-matrix (FDM) NRG to present high-quality data for the local susceptibility ${\ensuremath{\chi}}_{\mathrm{loc}}$ and to compare them with ${\ensuremath{\chi}}_{\mathrm{imp}}$ obtained by the traditional NRG. Our results indicate that those exotic behaviors observed in ${\ensuremath{\chi}}_{\mathrm{imp}}$ are unphysical. Instead, the low-energy excitations of the impurity in arbitrary bands only without gap at the Fermi energy are still a Fermi liquid and paramagnetic. We also demonstrate that unlike the traditional NRG yielding ${\ensuremath{\chi}}_{\mathrm{loc}}$ less accurate than ${\ensuremath{\chi}}_{\mathrm{imp}}$, the FDM method allows a high-precision dynamical calculation of ${\ensuremath{\chi}}_{\mathrm{loc}}$ at much reduced computational cost, with an accuracy at least one order higher than ${\ensuremath{\chi}}_{\mathrm{imp}}$. Moreover, artifacts in the FDM algorithm to ${\ensuremath{\chi}}_{\mathrm{imp}}$ and origins of the spurious non-Fermi-liquid and diamagnetic features are clarified. Our work provides an efficient high-precision algorithm to calculate the spin susceptibility of impurity for arbitrary structured bands, while negating the applicability of Wilson's definition to such cases.
Quantum impurity models are the prototypical examples of quantum many-body dynamics which manifests in their spectral and transport properties. Single channel Anderson(and Kondo model) leads to the Fermi liquid ground state in the strong coupling regime which corresponds to a stable infrared fixed point at which the quantum impurity gets completely screened by the conduction electrons. Quantum impurity models with non-trivial density of states exhibit quantum phase transition and this quantum criticality lies in the universality class of local quantum critical systems. In this paper, we report first study of the flow equation renormalization of gapped Kondo model which has gapped density of states, the gap being at the Fermi level. Flow equation renormalization group method has proved to be one of the very robust renormalization methods to study Kondo physics both in equilibrium as well as in non-equilibrium. Here we have shown that this method can also be employed to study local quantum criticality. We have calculated the flow equations for the Kondo coupling and solved them for various values of the gap parameter and we find that there is suppression of Kondo divergence as gap is increased which signifies that as gap is increased, renormalization flow goes away from the strong coupling fixed point. We have also calculated the spin susceptibility and we find that as gap is increased, susceptibility goes over to the Curie behaviour and hence confirming the renormalization flow towards the local moment fixed point.
The symmetric Anderson impurity model with a hybridization vanishing at the Fermi level, Δ I ∝|ω| r , is studied via the numerical renormalization group (NRG) at T = 0; and detailed comparison made with predictions arising from the local moment approach (LMA), a recently developed many-body theory which is found to provide a remarkably successful description of the problem. Results for the `normal' ( r = 0) impurity model are obtained as a specific case, and likewise compared. Particular emphasis is given both to single-particle excitation dynamics, and to the transition between the strong-coupling (SC) and local moment (LM) phases of the model. Scaling characteristics and asymptotic behaviour of the SC/LM phase boundaries are considered. Single-particle spectra D (ω) are investigated in some detail, for the SC phase in particular. Here, in accordance with a recently established result, the modified spectral functions ℱ(ω)∝|ω| r D (ω) are found to contain a generalized Kondo resonance that is ubiquitously pinned at the Fermi level; and which exhibits a characteristic low-energy Kondo scale, ω K ( r ), that narrows progressively upon approach to the SC→LM transition, where it vanishes. Universal scaling of the spectra as the transition is approached thus results. The scaling spectrum characteristic of the normal Anderson model is recovered as a particular case, that exemplifies behaviour characteristic of the SC phase generally, and which is captured quantitatively by the LMA. In all cases the r -dependent scaling spectra are found to possess characteristic low-energy asymptotics, but to be dominated by generalized Doniach-Sunjic tails, in agreement with LMA predictions.
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 $\ensuremath{\rho}(\ensuremath{\varepsilon})\ensuremath{\propto}{|\ensuremath{\varepsilon}|}^{r}$, either vanishing (for r>0) or diverging (for r<0) at the Fermi energy $\ensuremath{\varepsilon}=0$, 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.