Numerical renormalization-group study of the Bose-Fermi Kondo model

We extend the numerical renormalization-group method to Bose-Fermi Kondo models (BFKMs), describing a local moment coupled to a conduction band and a dissipative bosonic bath. We apply the method to the Ising-symmetry BFKM with a bosonic bath spectral function $\eta(\omega)\propto \omega^s$, of interest in connection with heavy-fermion criticality. For $0<s<1$, an interacting critical point, characterized by hyperscaling of exponents and $\omega/T$-scaling, describes a quantum phase transition between Kondo-screened and localized phases. Connection is made to other results for the BFKM and the spin-boson model.Comment: 4 pages, 4 figure

Single-particle dynamics of the Anderson model: a two-self-energy description within the numerical renormalization group approach

Single-particle dynamics of the Anderson impurity model are studied using both the numerical renormalization group (NRG) method and the local moment approach (LMA). It is shown that a 'two-self-energy' description of dynamics inherent to the LMA, as well as a conventional 'single-self-energy' description, arise within NRG; each yielding correctly the same local single-particle spectrum. Explicit NRG results are obtained for the broken symmetry spectral constituents arising in a two-self-energy description, and the total spectrum. These are also compared to analytical results obtained from the LMA as implemented in practice. Very good agreement between the two is found, essentially on all relevant energy scales from the high-energy Hubbard satellites to the low-energy Kondo resonance.Comment: 12 pages, 6 figure

Local quantum phase transition in the pseudogap Anderson model: scales, scaling and quantum critical dynamics

The pseudogap Anderson impurity model provides a paradigm for understanding local quantum phase transitions, in this case between generalised fermi liquid and degenerate local moment phases. Here we develop a non-perturbative local moment approach to the generic asymmetric model, encompassing all energy scales and interaction strengths and leading thereby to a rich description of the problem. We investigate in particular underlying phase boundaries, the critical behaviour of relevant low-energy scales, and single-particle dynamics embodied in the local spectrum. Particular attention is given to the resultant universal scaling behaviour of dynamics close to the transition in both the GFL and LM phases, the scale-free physics characteristic of the quantum critical point itself, and the relation between the two.Comment: 39 pages, 19 figure

A Local Moment Approach to magnetic impurities in gapless Fermi systems

A local moment approach is developed for the single-particle excitations of a symmetric Anderson impurity model (AIM), with a soft-gap hybridization vanishing at the Fermi level with a power law r > 0. Local moments are introduced explicitly from the outset, and a two-self-energy description is employed in which the single-particle excitations are coupled dynamically to low-energy transverse spin fluctuations. The resultant theory is applicable on all energy scales, and captures both the spin-fluctuation regime of strong coupling (large-U), as well as the weak coupling regime. While the primary emphasis is on single particle dynamics, the quantum phase transition between strong coupling (SC) and (LM) phases can also be addressed directly; for the spin-fluctuation regime in particular a number of asymptotically exact results are thereby obtained. Results for both single-particle spectra and SC/LM phase boundaries are found to agree well with recent numerical renormalization group (NRG) studies. A number of further testable predictions are made; in particular, for r < 1/2, spectra characteristic of the SC state are predicted to exhibit an r-dependent universal scaling form as the SC/LM phase boundary is approached and the Kondo scale vanishes. Results for the `normal' r = 0 AIM are moreover recovered smoothly from the limit r -> 0, where the resultant description of single-particle dynamics includes recovery of Doniach-Sunjic tails in the Kondo resonance, as well as characteristic low-energy Fermi liquid behaviour.Comment: 52 pages, 19 figures, submitted to Journal of Physics: Condensed Matte

Anderson impurities in gapless hosts: comparison of renormalization group and local moment approaches

The symmetric Anderson impurity model, with a soft-gap hybridization vanishing at the Fermi level with power law r > 0, is studied via the numerical renormalization group (NRG). Detailed comparison is 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. 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 are investigated in some detail, for the SC phase in particular. Here, the modified spectral functions 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 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, and 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.Comment: 26 pages, 14 figures, submitted for publicatio

Criticality of the Mean-Field Spin-Boson Model: Boson State Truncation and Its Scaling Analysis

The spin-boson model has nontrivial quantum phase transitions at zero temperature induced by the spin-boson coupling. The bosonic numerical renormalization group (BNRG) study of the critical exponents $\beta$ and $\delta$ of this model is hampered by the effects of boson Hilbert space truncation. Here we analyze the mean-field spin boson model to figure out the scaling behavior of magnetization under the cutoff of boson states $N_{b}$. We find that the truncation is a strong relevant operator with respect to the Gaussian fixed point in $0<s<1/2$ and incurs the deviation of the exponents from the classical values. The magnetization at zero bias near the critical point is described by a generalized homogeneous function (GHF) of two variables $\tau=\alpha-\alpha_{c}$ and $x=1/N_{b}$. The universal function has a double-power form and the powers are obtained analytically as well as numerically. Similarly, $m(\alpha=\alpha_{c})$ is found to be a GHF of $\epsilon$ and $x$. In the regime $s>1/2$, the truncation produces no effect. Implications of these findings to the BNRG study are discussed.Comment: 9 pages, 7 figure

Spectral scaling and quantum critical behaviour in the pseudogap Anderson model

The pseudogap Anderson impurity model provides a classic example of an essentially local quantum phase transition. Here we study its single-particle dynamics in the vicinity of the symmetric quantum critical point (QCP) separating generalized Fermi liquid and local moment phases, via the local moment approach. Both phases are shown to be characterized by a low-energy scale that vanishes at the QCP; and the universal scaling spectra, on all energy scales, are obtained analytically. The spectrum precisely at the QCP is also obtained; its form showing clearly the non-Fermi liquid, interacting nature of the fixed point.Comment: 7 pages, 2 figure

Dynamics of capacitively coupled double quantum dots

We consider a double dot system of equivalent, capacitively coupled semiconducting quantum dots, each coupled to its own lead, in a regime where there are two electrons on the double dot. Employing the numerical renormalization group, we focus here on single-particle dynamics and the zero-bias conductance, considering in particular the rich range of behaviour arising as the interdot coupling is progressively increased through the strong coupling (SC) phase, from the spin-Kondo regime, across the SU(4) point to the charge-Kondo regime; and then towards and through the quantum phase transition to a charge-ordered (CO) phase. We first consider the two-self-energy description required to describe the broken symmetry CO phase, and implications thereof for the non-Fermi liquid nature of this phase. Numerical results for single-particle dynamics on all frequency scales are then considered, with particular emphasis on universality and scaling of low-energy dynamics throughout the SC phase. The role of symmetry breaking perturbations is also briefly discussed.Comment: 14 pages, 6 figure

Numerical Renormalization Group for Impurity Quantum Phase Transitions: Structure of Critical Fixed Points

The numerical renormalization group method is used to investigate zero temperature phase transitions in quantum impurity systems, in particular in the particle-hole symmetric soft-gap Anderson model. The model displays two stable phases whose fixed points can be built up of non-interacting single-particle states. In contrast, the quantum phase transitions turn out to be described by interacting fixed points, and their excitations cannot be described in terms of free particles. We show that the structure of the many-body spectrum of these critical fixed points can be understood using renormalized perturbation theory close to certain values of the bath exponents which play the role of critical dimensions. Contact is made with perturbative renormalization group calculations for the soft-gap Anderson and Kondo models. A complete description of the quantum critical many-particle spectra is achieved using suitable marginal operators; technically this can be understood as epsilon-expansion for full many-body spectra.Comment: 14 pages, 12 figure

Dynamics and transport properties of heavy fermions: theory

The paramagnetic phase of heavy fermion systems is investigated, using a non-perturbative local moment approach to the asymmetric periodic Anderson model within the framework of dynamical mean field theory. The natural focus is on the strong coupling Kondo-lattice regime wherein single-particle spectra, scattering rates, dc transport and optics are found to exhibit w/w_L,T/w_L scaling in terms of a single underlying low-energy coherence scale w_L. Dynamics/transport on all relevant (w,T)-scales are encompassed, from the low-energy behaviour characteristic of the lattice coherent Fermi liquid, through incoherent effective single-impurity physics likewise found to arise in the universal scaling regime, to non-universal high-energy scales; and which description in turn enables viable quantitative comparison to experiment.Comment: 27 pages, 12 figure