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arXiv:cond-mat/9704024 · Physical review. B, Condensed matter

Low-energy fixed points of the σ-τ and the O(3) symmetric Anderson models

R. Bulla, Alex C. Hewson, Guang-Ming Zhang

Published Nov 1, 1997• 17 citations
arXivHTMLPDF

Abstract

We study the single-channel (compactified) models, the \ensuremath{\sigma}-\ensuremath{\tau} model, and the O(3) symmetric Anderson model, which were introduced by Coleman et al., and Coleman and Schofield, as a simplified way to understand the low-energy behavior of the isotropic and anisotropic two-channel Kondo systems. These models display both Fermi-liquid and marginal-Fermi-liquid behavior and an understanding of the nature of their low-energy fixed points may give some general insights into the low-energy behavior of other strongly correlated systems. We calculate the excitation spectrum at the non-Fermi-liquid fixed point of the \ensuremath{\sigma}-\ensuremath{\tau} model using conformal field theory, and show that the results are in agreement with those obtained in recent numerical renormalization group (NRG) calculations. For the O(3) Anderson model we find further logarithmic corrections in the weak-coupling perturbation expansion to those obtained in earlier calculations, such that the renormalized interaction term now becomes marginally stable rather than marginally unstable. We derive a Ward identity and a renormalized form of the perturbation theory that encompasses both the weak- and strong-coupling regimes and show that the $\ensuremath{\chi}/\ensuremath{\gamma}$ ratio is $8/3$ (\ensuremath{\chi} is the total susceptibility, spin plus isospin), independent of the interaction $U$ and in agreement with the NRG calculations.

DOI
10.1103/physrevb.56.11721

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We study the single-channel (compactified) models, the \ensuremath{\sigma}-\ensuremath{\tau} model, and the O(3) symmetric Anderson model, which were introduced by Coleman et al., and Coleman and Schofield, as a simplified way to understand the low-energy behavior of the isotropic and anisotropic two-channel Kondo systems. These models display both Fermi-liquid and marginal-Fermi-liquid behavior and an understanding of the nature of their low-energy fixed points may give some general insights into the low-energy behavior of other strongly correlated systems. We calculate the excitation spectrum at the non-Fermi-liquid fixed point of the \ensuremath{\sigma}-\ensuremath{\tau} model using conformal field theory, and show that the results are in agreement with those obtained in recent numerical renormalization group (NRG) calculations. For the O(3) Anderson model we find further logarithmic corrections in the weak-coupling perturbation expansion to those obtained in earlier calculations, such that the renormalized interaction term now becomes marginally stable rather than marginally unstable. We derive a Ward identity and a renormalized form of the perturbation theory that encompasses both the weak- and strong-coupling regimes and show that the \ensuremathχ/\ensuremathγ\ensuremath{\chi}/\ensuremath{\gamma}\ensuremathχ/\ensuremathγ ratio is 8/38/38/3 (\ensuremath{\chi} is the total susceptibility, spin plus isospin), independent of the interaction UUU and in agreement with the NRG calculations.
The O(3) symmetric Anderson model is an example of a system which has a stable low energy marginal Fermi liquid fixed point for a certain choice of parameters. It is also exactly equivalent, in the large U limit, to a localized model which describes the spin degrees of freedom of the linear dispersion two channel Kondo model. We first use an argument based on conformal field theory to establish this precise equivalence with the two channel model. We then use the numerical renormalization group (NRG) approach to calculate both one-electron and two-electron response functions for a range of values of the interaction strength U. We compare the behaviours about the marginal Fermi liquid and Fermi liquid fixed points and interpret the results in terms of a renormalized Majorana fermion picture of the elementary excitations. In the marginal Fermi liquid case the spectral densities of all the Majorana fermion modes display a |omega| dependence on the lowest energy scale, and in addition the zero Majorana mode has a delta function contribution. The weight of this delta function is studied as a function of the interaction U and is found to decrease exponentially with U for large U. Using the equivalence with the two channel Kondo model in the large U limit, we deduce the dynamical spin susceptibility of the two channel Kondo model over the full frequency range. We use renormalized perturbation theory to interpret the results and to calculate the coefficient of the ln omega divergence found in the low frequency behaviour of the T=0 dynamic susceptibility.
We discuss the two-channel Kondo problem with a pseudogap density of states $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{r}$ of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic r=0r=0r=0 case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For r>rmaxr>{r}_{\mathrm{max}}r>rmax​, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions r=0r=0r=0 and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.
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{T}^{2}, {V}^{2}T2,V2, and B2{B}^{2}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.
We study an exchange interaction in which conduction electrons with pseudospin Sc{\mathrm{S}}_{\mathrm{c}}Sc​=3/2 interact with the impurity spin SI{\mathrm{S}}_{\mathrm{I}}SI​=1/2. Due to the overscreening of the impurity spin by higher conduction electron spin, a nontrivial intermediate-coupling-strength fixed point is realized. Using the numerical renormalization group (NRG), we show that the low-energy spectra are described by a non-Fermi-liquid excitation spectrum. A conformal field theory analysis is compared with NRG results and excellent agreement is obtained. Using the double-fusion rule to generate the operator spectrum with the conformal theory, we find that the specific heat coefficient and magnetic susceptibility will diverge as ${\mathrm{T}}^{\mathrm{\ensuremath{-}}2\mathrm{/}3}$, that the scaling dimension of an applied magnetic field is 5/6, and that exchange anisotropy is always relevant. We discuss the possible relevance of our work to two-level-system Kondo materials and dilute cerium alloys, and we point out a paradox in understanding the Bethe-ansatz solutions to the multichannel Kondo model.
We reformulate the two-channel Kondo model to explicitly remove the unscattered charge degrees of freedom. This procedure permits us to move the non-Fermi-liquid fixed point to infinite coupling where we can apply a perturbative strong-coupling expansion. The fixed-point Hamiltonian involves a three-body Majorana zero mode whose scattering effects give rise to marginal self-energies. The compactified model is the N=3 member of a family of O(N) Kondo models that can be solved by semiclassical methods in the large N limit. For odd N, fermionic ``kink'' fluctuations about the N=\ensuremath{\infty} mean-field theory generate a fermionic N-body bound state which asymptotically decouples at low energies. For N=3, our semiclassical methods fully recover the non-Fermi-liquid physics of the original two-channel model. Using the same methods, we find that the corresponding O(3) Kondo lattice model develops a spin-gap and a gapless band of coherently propagating three-body bound states. Its strong-coupling limit offers a rather interesting realization of a marginal Fermi liquid.
Numerical renormalization group simulations have shown that the underscreened spin-1 Kondo impurity model with power-law bath density of states (DOS) $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{r}$ possesses various intermediate-coupling fixed points, including a stable non-Fermi-liquid phase. In this paper we discuss the corresponding universal low-energy theories, obtain thermodynamic quantities and critical exponents by renormalization group analysis together with suitable ϵϵϵ expansions, and compare our results with numerical data. Whereas the particle-hole symmetric critical point can be controlled at weak coupling using a simple generalization of the spin-1∕21∕21∕2 model, we show that the stable non-Fermi-liquid fixed point must be accessed near strong coupling via a mapping onto an effective ferromagnetic Seff=1∕2{S}_{\mathrm{eff}}=1∕2Seff​=1∕2 model with singular bath DOS with exponent ${r}_{\mathrm{eff}}=\ensuremath{-}r<0$. In addition, we consider the particle-hole asymmetric critical fixed point, for which we propose a universal field theory involving the crossing between doublet and triplet levels.