A local moment approach is developed for single-particle excitations of a symmetric Anderson impurity model (AIM) with a soft-gap hybridization vanishing at the Fermi level: I | | r , with r >0. Local moments are introduced explicitly from the outset, and a two-self-energy description is employed in which 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 where it is perturbatively exact for those r -domains in which perturbation theory in U is non-singular. While the primary emphasis is on single-particle dynamics, the quantum phase transition between strong-coupling (SC) and local moment (LM) phases can also be addressed directly; for the spin-fluctuation regime in particular a number of asymptotically exact results are thereby obtained, notably for the behaviour of the critical U c ( r ) separating SC/LM states and the Kondo scale m ( r ) characteristic of the SC phase. Results for both single-particle spectra and SC/LM phase boundaries are found to agree well with recent numerical renormalization group (NRG) studies; and a number of further testable predictions are made. Single-particle spectra are examined systematically for both SC and LM states; in particular, for all 0 r <½, 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 wings of the Kondo resonance, as well as characteristic low-energy Fermi liquid behaviour and the exponential diminution with U of the Kondo scale itself. The normal AIM is found to represent a particular case of more generic behaviour characteristic of the r >0 SC phase which, in agreement with conclusions drawn from recent NRG work, may be viewed as a non-trivial but natural generalization of Fermi liquid physics.
A local moment approach is developed for single-particle excitations of a symmetric Anderson impurity model (AIM) with a soft-gap hybridization vanishing at the Fermi level: I | | r , with r >0. Local moments are introduced explicitly from the outset, and a two-self-energy description is employed in which 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 where it is perturbatively exact for those r -domains in which perturbation theory in U is non-singular. While the primary emphasis is on single-particle dynamics, the quantum phase transition between strong-coupling (SC) and local moment (LM) phases can also be addressed directly; for the spin-fluctuation regime in particular a number of asymptotically exact results are thereby obtained, notably for the behaviour of the critical U c ( r ) separating SC/LM states and the Kondo scale m ( r ) characteristic of the SC phase. Results for both single-particle spectra and SC/LM phase boundaries are found to agree well with recent numerical renormalization group (NRG) studies; and a number of further testable predictions are made. Single-particle spectra are examined systematically for both SC and LM states; in particular, for all 0 r <½, 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 wings of the Kondo resonance, as well as characteristic low-energy Fermi liquid behaviour and the exponential diminution with U of the Kondo scale itself. The normal AIM is found to represent a particular case of more generic behaviour characteristic of the r >0 SC phase which, in agreement with conclusions drawn from recent NRG work, may be viewed as a non-trivial but natural generalization of Fermi liquid physics.
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.
The local moment approach is extended to the orbitally degenerate (SU(2N)) Anderson impurity model (AIM). Single-particle dynamics are obtained over the full range of energy scales, focusing on particle-hole symmetry in the strongly-correlated regime where the onsite Coulomb interaction leads to many-body Kondo physics with entangled spin and orbital degrees of freedom. The approach captures many-body broadening of the Hubbard satellites and recovers the correct exponential vanishing of the Kondo scale for all N, and its universal scaling spectra are found to be in very good agreement with numerical renormalization group (NRG) results. In particular the high-frequency logarithmic decays of the scaling spectra, obtained here in closed form for arbitrary N, coincide essentially perfectly with available numerics from the NRG. A particular case of an anisotropic Coulomb interaction, in which the model represents a system of N 'capacitively coupled' SU(2) AIMs, is also discussed. Here the model is generally characterized by two low-energy scales, the crossover between which is seen directly in its dynamics.
We study correlated quantum impurity models that undergo a local quantum phase transition (QPT) from a strong coupling, Fermi liquid phase to a non-Fermi liquid phase with a globally doubly degenerate ground state. Our aim is to establish what can be shown exactly about such ``local moment'' (LM) phases, of which the permanent (zero-field) local magnetization is a hallmark, and an order parameter for the QPT. A description of the zero-field LM phase is shown to require two distinct self-energies, which reflect the broken symmetry nature of the phase and together determine the single self-energy of standard field theory. Distinct Friedel sum rules for each phase are obtained, via a Luttinger theorem embodied in the vanishing of appropriate Luttinger integrals. By contrast, the standard Luttinger integral is nonzero in the LM phase but found to have universal magnitude. A range of spin susceptibilites are also considered, including that corresponding to the local order parameter, whose exact form is shown to be RPA-like, and to diverge as the QPT is approached. Particular attention is given to the pseudogap Anderson model, including the basic physical picture of the transition, the low-energy behavior of single-particle dynamics, the quantum critical point itself, and the rather subtle effect of an applied local field. A two-level impurity model that undergoes a QPT (``singlet-triplet'') to an underscreened LM phase is also considered, for which we derive on general grounds some key results for the zero-bias conductance in both phases.
The pseudogap Anderson impurity model provides a paradigm for understanding local quantum phase transitions, in this case between generalised Fermi liquid (GFL) 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 D (ω). 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.
Single-particle dynamics of the Anderson impurity model in the presence of a magnetic field H are considered, using a recently developed local moment approach that encompasses all energy scales, field and interaction strengths. For strong coupling in particular, the Kondo scaling regime is recovered. Here the frequency (ω/ω K ) and field ( H /ω K ) dependence of the resultant universal scaling spectrum is obtained in large part analytically, and the field-induced destruction of the Kondo resonance investigated. The scaling spectrum is found to exhibit the slow logarithmic tails recently shown to dominate the zero-field scaling spectrum. At the opposite extreme of the Fermi level, it gives asymptotically exact agreement with results for statics known from the Bethe ansatz. Good agreement is also found with the frequency and field-dependence of recent numerical renormalization group calculations. Differential conductance experiments on quantum dots in the presence of a magnetic field are likewise considered, and appear to be well accounted for by the theory. Some new exact results for the problem are also established.
We generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozi\`eres that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives with respect to the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe ansatz and numerical renormalization group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature, and small bias voltage, and compute the coefficients of the $\ensuremath{\sim}{B}^{2}, \ensuremath{\sim}{T}^{2}$, and $\ensuremath{\sim}{V}^{2}$ terms exactly in terms of the Fermi-liquid parameters. The coefficients of T2,V2, and B2 are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.