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arXiv:cond-mat/9611096 · Physical Review Letters

Unified Description of Fermi and Non-Fermi Liquid Behavior in a Conserving Slave Boson Approximation for Strongly Correlated Impurity Models

Johann Kroha, P. Wölfle, Theo A. Costi

Published Jul 14, 1997• 48 citations
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Abstract

We show that the presence of Fermi or non-Fermi liquid behavior in the $\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(M)$ Anderson impurity models may be read off the infrared threshold exponents governing the spinon and holon dynamics in a slave boson representation of these models. We construct a conserving $T$-matrix approximation which recovers the exact exponents with good numerical accuracy. Our approximation includes both coherent spin flip scattering and charge fluctuation processes. For the single-channel case the tendency to form bound states drastically modifies the low energy behavior. For the multichannel case in the Kondo limit the bound state contributions are unimportant.

DOI
10.1103/physrevlett.79.261

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We show that the presence of Fermi or non-Fermi liquid behavior in the SU(N)\ifmmode×\else\texttimes\fiSU(M)\mathrm{SU}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(M)SU(N)\ifmmode×\else\texttimes\fiSU(M) Anderson impurity models may be read off the infrared threshold exponents governing the spinon and holon dynamics in a slave boson representation of these models. We construct a conserving TTT-matrix approximation which recovers the exact exponents with good numerical accuracy. Our approximation includes both coherent spin flip scattering and charge fluctuation processes. For the single-channel case the tendency to form bound states drastically modifies the low energy behavior. For the multichannel case in the Kondo limit the bound state contributions are unimportant.
We use the numerical renormalization group to calculate the auxiliary spectral functions of the $U=\ensuremath{\infty}$ Anderson impurity model. The slave-boson and pseudo-fermion spectral functions diverge at the threshold with exponents ${\ensuremath{\alpha}}_{b}$ and ${\ensuremath{\alpha}}_{f}$ given by the x-ray photoemission and the x-ray absorption exponents, respectively. In contrast to the noncrossing approximation, the exact exponents obtained here depend on the impurity occupation number. In general, vertex corrections in the convolution formulas for physical Green's functions are singular at the threshold and may not be neglected in the Fermi liquid regime.
We formulate a general approach for studying the low-frequency response of an interacting quantum dot connected to leads in the presence of oscillating gate voltages. The energy dissipated is characterized by the charge relaxation resistance, which under the loose assumption of Fermi liquid behavior at low energy, is shown to depend only on static charge susceptibilities. The predictions of the scattering theory are recovered in the noninteracting limit while the effect of interactions is simply to replace densities of states by charge susceptibilities in formulas. In order to substantiate the Fermi liquid picture in the case of a quantum RCRCRC geometry, we apply a renormalization group analysis and derive the low-energy Hamiltonian for two specific models: the Anderson and the Coulomb blockade models. The Anderson model is shown, using a field theoretical approach based on Barnes slave bosons, to map onto the Kondo model. We recover the well-known expression of the Kondo temperature for the asymmetric Anderson model and compute the charge susceptibility. The Barnes slave bosons are extended to the Coulomb blockade model where the renormalization-group analysis can be carried out perturbatively up to zero energy. All calculations agree with the Fermi liquid nature of the low-energy fixed point and satisfy the Friedel sum rule.
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.
We study a generalized multichannel single-impurity Kondo model, in which the impurity spin is described by a representation of the SU(N)\mathrm{SU}(N)SU(N) group that combines bosonic and fermionic degrees of freedom. The impurity spin states are described by Abrikosov pseudofermions, and we make use of a method initiated by Popov and Fedotov that allows a proper handling of the fermionic constraint. The partition function is derived within a path integral approach. We use renormalization group techniques to calculate the $\ensuremath{\beta}$ scaling function perturbatively in powers of the Kondo coupling constant, which is justified in the weak coupling limit. The truncated expansion is valid in the overscreened (Nozi\`eres-Blandin) regime, for an arbitrary SU(N)\mathrm{SU}(N)SU(N) group and any value of the parameters characterizing the impurity spin representation. The intermediate coupling fixed point is identified. We derive the temperature dependence of various physical quantities at low TTT, controlled by a unique critical exponent, and show that the physics of the system in the overscreened regime governed by the intermediate coupling fixed point is characterized by a non-Fermi liquid behavior. Our results are in accordance with those obtained by other methods, as Bethe ansatz and boundary conformal field theory, in the case of various impurity spin symmetries. We establish in a unified way that the Kondo models in which the impurity spin is described successively by a fundamental, symmetric, antisymmetric, and mixed symmetry representation yield all the same low-energy physics in the overscreened regime. Possible generalizations of the analysis we present to the case of arbitrary impurity spin representations of SU(N)\mathrm{SU}(N)SU(N) are also discussed.
We study how the non-Fermi-liquid nature of the overscreened multichannel Kondo impurity model affects the response to a BCS pairing term that, in the absence of the impurity, opens a gap $\mathrm{\ensuremath{\Delta}}$. We find that the low-energy spectrum in the limit $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$ actually does not correspond to the spectrum strictly at $\mathrm{\ensuremath{\Delta}}=0$. In particular, in the two-channel Kondo model, the $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$ ground state is an orbitally degenerate spin singlet, while it is an orbital singlet with a residual spin degeneracy at $\mathrm{\ensuremath{\Delta}}=0$. In addition, there are fractionalized spin-1/2 subgap excitations whose energy in units of $\mathrm{\ensuremath{\Delta}}$ tends toward a finite and universal value when $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$, as if the universality of the anomalous power-law exponents that characterize the overscreened Kondo effect turned into universal energy ratios when the scale invariance is broken by $\mathrm{\ensuremath{\Delta}}\ensuremath{\ne}0$. This intriguing phenomenon can be explained by the renormalization flow toward the overscreened fixed point and the gap cutting off the orthogonality catastrophe singularities. We also find other non-Fermi-liquid features at finite $\mathrm{\ensuremath{\Delta}}$: the local density of states lacks coherence peaks, the states in the continuum above the gap are unconventional, and the boundary entropy is a nonmonotonic function of temperature. The persistent subgap excitations are characteristic of the non-Fermi-liquid fixed point of the model, and thus depend on the impurity spin and the number of screening channels.
We show how the electron-gas methods of Luttinger, Ward, and Nozi\`eres can be applied to an SU(N)\mathrm{SU}(N)SU(N), multichannel generalization of the infinite-UUU Anderson impurity model within a Schwinger boson treatment. Working to all orders in a 1∕N1∕N1∕N expansion, we show how the Friedel-Langreth relationship and the Yamada-Yosida-Yoshimori and the Shiba-Korringa relations can be derived under the assumption that the spinon and holon fields are gapped. One of the remarkable features of this treatment is that the Landau amplitudes depend on the exchange of low-energy virtual spinons and holons. We end the paper with a discussion on the extension of our approach to the lattice, where the spinon-holon gap is expected to close at a quantum critical point.