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arXiv:1203.4456 · Physical Review B

Universal low-temperature crossover in two-channel Kondo models

Andrew K. Mitchell, Eran Sela

Published Jun 18, 2012• 61 citations
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Abstract

An exact expression is derived for the electron Green function in two-channel Kondo models with one and two impurities, describing the crossover from non-Fermi liquid (NFL) behavior at intermediate temperatures to standard Fermi liquid (FL) physics at low temperatures. Symmetry-breaking perturbations generically present in experiment ensure the standard low-energy FL description, but the full crossover is wholly characteristic of the unstable NFL state. Distinctive conductance lineshapes in quantum dot devices should result. We exploit a connection between this crossover and one occurring in a classical boundary Ising model to calculate real-space electron densities at finite temperature. The single universal finite-temperature Green function is then extracted by inverting the integral transformation relating these Friedel oscillations to the $t$ matrix. Excellent agreement is demonstrated between exact results and full numerical renormalization group calculations.

DOI
10.1103/physrevb.85.235127

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An exact expression is derived for the electron Green function in two-channel Kondo models with one and two impurities, describing the crossover from non-Fermi liquid (NFL) behavior at intermediate temperatures to standard Fermi liquid (FL) physics at low temperatures. Symmetry-breaking perturbations generically present in experiment ensure the standard low-energy FL description, but the full crossover is wholly characteristic of the unstable NFL state. Distinctive conductance lineshapes in quantum dot devices should result. We exploit a connection between this crossover and one occurring in a classical boundary Ising model to calculate real-space electron densities at finite temperature. The single universal finite-temperature Green function is then extracted by inverting the integral transformation relating these Friedel oscillations to the ttt matrix. Excellent agreement is demonstrated between exact results and full numerical renormalization group calculations.
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.
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.
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.
We study triangular clusters of three spin-1/2 Kondo or Anderson impurities that are coupled to two conduction leads. In the case of Kondo impurities, the model takes the form of an antiferromagnetic Heisenberg ring with Kondo-type exchange coupling to continuum electrons. We show that this model exhibits many types of the behavior found in various simpler one- and two-impurity models, thereby enabling the study of crossovers between a number of Fermi-liquid (FL) and non-Fermi-liquid (NFL) fixed points. In particular, we explore a direct crossover between the two-impurity Kondo-model NFL fixed point and the two-channel Kondo-model NFL fixed point. We show that the concept of the two-stage Kondo effect applies even in the case when the first-stage Kondo state is of NFL type. In the case of Anderson impurities, we consider the transport properties of three coupled quantum dots. This class of models includes, as limiting cases, the familiar serial double quantum dot and triple quantum dot nanostructures. By extracting the quasiparticle scattering phase shifts, we compute the low-temperature conductance as a function of the interimpurity tunneling coupling. We point out that due to the existence of exponentially low-temperature scales, there is a parameter range where the stable ``zero-temperature'' fixed point is essentially never reached (not even in numerical renormalization group calculations). The zero-temperature conductance is then of no interest and it may only be meaningful to compute the conductance at finite temperature. This illustrates the perils of studying the conductance in the ground state and considering thermal fluctuations only as a small correction.
We study finite-frequency transport properties of the double dot system recently constructed to observe the two-channel Kondo effect [R. M. Potok et al., Nature 446, 167 (2007)]. We derive an analytical expression for the frequency-dependent linear conductance of this device in the Kondo regime. We show how the features characteristic of the two-channel Kondo quantum critical point emerge in this quantity, which we compute using the results of conformal field theory as well as numerical renormalization group methods. We determine the universal crossover functions describing non-Fermi-liquid vs Fermi-liquid crossovers and also investigate the effects of a finite magnetic field.
The long-thought charge Kondo effects have recently been experimentally realized in the quantum Hall regime. This experiment, supported by numerics, exemplifies the realization of two-channel Kondo state, a non-Fermi Liquid, and its crossover to the one-channel counterpart, a Fermi liquid. Scaling up such a platform, we find a hierarchy of non-Fermi Liquids and their tunable crossovers based on a renormalization group analysis. Utilizing results from a conformal field theory, we further examine the universal conductances of this strongly correlated system and their finite temperature scaling, which elucidate the sharp distinctions between charge and spin Kondo physics.