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Abstract:

We develop a local moment approach to static properties of the symmetric Anderson model in the presence of a magnetic field, focussing in particular on the strong-coupling Kondo regime. The approach is innately simple and physically transparent; but is found to give good agreement, for essentially all field strengths, with exact results for the Wilson ratio, impurity magnetization, spin susceptibility and related properties.

Abstract:

The symmetric Anderson impurity model with a hybridization vanishing at the Fermi level. ΔI ∝ |ωVBAR^r, is studied via the numerical renormalization group (NRG) at T = O; 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) ∝ |ωVBAR^r D(ω) are found to contain a generalized Kondo resonance that is ubiquitously pinned at the Fermi level; and which exhibits a characteristics 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.

Abstract:

We consider single-particle spectra of a symmetric narrow-band Anderson impurity model, where the host bandwidth D is small compared to the hybridization strength Δ0. Simple second-order perturbation theory (2PT) in U is found to produce a rich spectral structure that leads to rather good agreement with extant Lanczos results and offers a transparent picture of the underlying physics. It also leads naturally to two distinct regimes of spectral behavior, Δ0Z/D≫1 and ≪1 (with Z the quasiparticle weight), whose existence and essential characteristics are discussed and shown to be independent of 2PT itself. The self-energy ∑i(ω) is also examined beyond the confines of PT. It is argued that on frequency scales of order ω∼Δ0D, the self-energy in strong coupling is given precisely by the 2PT result, and we point out that the resultant poles in ∑i(ω) connect continuously to that characteristic of the atomic limit. This in turn offers a natural rationale for the known inability of the skeleton expansion to capture such behavior, and points to the intrinsic dangers of partial infinite-order summations that are based on PT in U.

Abstract:

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: ΔI α |ω|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 Uc(r) separating SC/LM states and the Kondo scale ωm(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 < 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-Šunjić 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. © 2000 IOP Publishing Ltd.

Abstract:

We consider a symmetric Anderson impurity model with a soft-gap hybridization vanishing at the Fermi level, ΔI ∝ |ω|r with r > 0. Three facets of the problem are examined. First the non-interacting limit, which despite its simplicity contains much physics relevant to the U > 0 case: it exhibits both strong coupling (SC) states (for r < 1) and local moment states (for r > 1), with characteristic signatures in both spectral properties and thermodynamic functions. Second, we establish general conditions upon the interaction self-energy for the occurence of a SC state for U > 0. This leads to a pinning theorem, whereby the modified spectral function A(ω) = |ω|r D(ω) is pinned at the Fermi level ω = 0 for any U where a SC state obtains; it generalizes to arbitrary r the pinning condition upon D(ω = 0) familiar in the normal r = 0 Anderson model. Finally, we consider explicitly spectral functions at the simplest level: second order perturbation theory in U, which we conclude is applicable for r < 1/2 and r > 1 but not for 1/2 < r < 1. Characteristic spectral features observed in numerical renormalization group calculations are thereby recovered, for both SC and LM phases; and for the SC state the modified spectral functions are found to contain a generalized Abrikosov-Suhl resonance exhibiting a characteristic low-energy Kondo scale with increasing interaction strength.