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arXiv:cond-mat/0203506 · Journal of the Physical Society of Japan

Crossover Temperature from Non-Fermi Liquid to Fermi Liquid Behavior in Two Types of Impurity Kondo Model

Satoshi Yotsuhashi, Hideaki Maebashi

Published Jul 15, 2002• 12 citations
arXivHTMLPDF

Abstract

Numerical renormalization-group results on entropy of the anisotropic two-channel Kondo model with the band-width cutoff ($D$) in the presence of a magnetic field ($h$) are obtained to determine crossover temperature from the non-Fermi liquid to Fermi liquid fixed point. It is found that the crossover temperature ($T_{\rm x}$) is given by $T_{\rm x} \equiv {r} T_{\rm K} \sim D(ΔJ/J_{\rm av})^2 e^{-1/J_{\rm av}}$ when $(h /T_{\rm K})^2 \ll r \ll 1 $, where $T_{\rm K}$, $J_{\rm av}$ and $ΔJ$ are the Kondo temperature, the average and difference of the exchange coupling constants, respectively. This result indicates that non-Fermi liquid behavior can be seen even if $ΔJ > T_{\rm K}$. Robust similarities of the crossover behavior in the region around the non-Fermi liquid critical point to that of the two-impurity Kondo model are also discussed.

DOI
10.1143/jpsj.71.1705

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Numerical renormalization-group results on entropy of the anisotropic two-channel Kondo model with the band-width cutoff (DDD) in the presence of a magnetic field (hhh) are obtained to determine crossover temperature from the non-Fermi liquid to Fermi liquid fixed point. It is found that the crossover temperature (TxT_{\rm x}Tx​) is given by Tx≡rTK∼D(ΔJ/Jav)2e−1/JavT_{\rm x} \equiv {r} T_{\rm K} \sim D(ΔJ/J_{\rm av})^2 e^{-1/J_{\rm av}}Tx​≡rTK​∼D(ΔJ/Jav​)2e−1/Jav​ when (h/TK)2≪r≪1(h /T_{\rm K})^2 \ll r \ll 1 (h/TK​)2≪r≪1, where TKT_{\rm K}TK​, JavJ_{\rm av}Jav​ and ΔJΔJΔJ are the Kondo temperature, the average and difference of the exchange coupling constants, respectively. This result indicates that non-Fermi liquid behavior can be seen even if ΔJ>TKΔJ > T_{\rm K}ΔJ>TK​. Robust similarities of the crossover behavior in the region around the non-Fermi liquid critical point to that of the two-impurity Kondo model are also discussed.
Symmetry-breaking perturbations destabilize the critical points of the two-channel and two-impurity Kondo models, thereby leading to a crossover from non-Fermi liquid behavior to standard Fermi liquid physics. Here we use an analogy between this crossover and one occurring in the boundary Ising model to calculate the full crossover Green function analytically. In remarkable agreement with our numerical renormalization group calculations, the single exact function applies for an arbitrary mixture of the relevant perturbations in each model. This rich behavior resulting from finite channel asymmetry, interlead charge transfer, and/or magnetic field should be observable in quantum dot or tunneling experiments.
An asymptotically exact solution is presented for the two-impurity Kondo model for a finite region of the parameter space surrounding the critical point. This region is located in the most interesting intermediate regime where the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is comparable to the Kondo temperature. After several exact simplifications involving reduction to one dimension and Abelian bosonization, the critical point is explicitly identified, making clear its physical origin. By using controlled low-energy projection, an effective Hamiltonian is derived for the finite region in the phase diagram around the critical point. The completeness of the effective Hamiltonian is rigorously proved from general symmetry considerations. The effective Hamiltonian is solved exactly not only at the critical point but also for the surrounding Fermi-liquid phase. Analytic crossover functions from the critical to Fermi-liquid behavior are derived for the specific heat and staggered susceptibility. It is shown that applying a uniform magnetic field has negligible effect on the physical behavior inside our solution region. A detailed comparison is made with the numerical renormalization-group and conformal-field-theory results. The excellent agreement is exploited to argue for the universality of the critical point, which in turn implies universal behavior everywhere inside our solution region.
Entanglement of spin and orbital Kondo effect is investigated on the basis of a Kondo-type exchange model with twofold orbital degeneracy. By using Wilson's numerical renormalization-group method, we examine dynamical and thermal properties respecting the difference in time-reversal property of multipole operators. In the presence of particle-hole symmetry, the model has a new non-Fermi-liquid fixed point with a fractional entropy. The spectral intensity of the quadrupole susceptibility diverges in the zero-frequency limit, while the dipole susceptibility shows a Fermi-liquid-like behavior. This is understood by mapping to the two-channel Kondo model, in which the dipole moment is mapped onto the operators with the scaling dimension Δm=1Δ_m=1Δm​=1, while the quadrupole moment onto the operators with another scaling dimension Δe=1/2Δ_e=1/2Δe​=1/2. Even for a fairly particle-hole asymmetric case with the Fermi-liquid ground state, the non-Fermi-liquid behavior has significant influences in electric and thermal properties.
The multichannel Kondo model exhibits non-Fermi-liquid behavior in the overscreened case, when the number of channels, k, is greater than twice the size of the impurity spin, s. We show that, for overscreening, exchange anisotropy is irrelevant at the low-temperature fixed point for s=1/2 or s=k/2-1/2, but relevant for all other values of s. However, an external field or channel asymmetry is relevant, producing crossover to Fermi-liquid fixed points with different phase shifts for each spin in the first case and for each channel in the second. These results are elucidated by explicit comparison of analytic finite-size spectra derived from conformal field theory with those obtained from numerical renormalization-group calculations. The relevance of the results (for k=2 and s=1/2) to the quadrupolar Kondo Hamiltonian (which has been proposed as a model for many uranium-based heavy-fermion materials) will be briefly discussed. Also, a larger, a/Isymplectic symmetry Sp(2k) is shown to be present in the k-channel Kondo model.
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.