We study the higher-order Fermi-liquid relations of Kondo systems for arbitrary impurity-electron fillings, extending the many-body quantum theoretical approach of Yamada and Yosida. It includes, partly, a microscopic clarification of the related achievements based on Nozi\`eres' phenomenological description: Filippone, Moca, von Delft, and Mora [Phys. Rev. B 95, 165404 (2017)]. In our formulation, the Fermi-liquid parameters such as the quasiparticle energy, damping, and transport coefficients are related to each other through the total vertex ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}};{\ensuremath{\sigma}}^{\ensuremath{'}}\ensuremath{\sigma}}(\ensuremath{\omega},{\ensuremath{\omega}}^{\ensuremath{'}};{\ensuremath{\omega}}^{\ensuremath{'}},\ensuremath{\omega})$, which may be regarded as a generalized Landau quasiparticle interaction. We obtain exactly this function up to linear order with respect to the frequencies $\ensuremath{\omega}$ and ${\ensuremath{\omega}}^{\ensuremath{'}}$ using the antisymmetry and analytic properties. The coefficients acquire additional contributions of three-body fluctuations away from half filling through the nonlinear susceptibilities. We also apply the formulation to nonequilibrium transport through a quantum dot, and clarify how the zero-bias peak evolves in a magnetic field.
We study the higher-order Fermi-liquid relations of Kondo systems for arbitrary impurity-electron fillings, extending the many-body quantum theoretical approach of Yamada and Yosida. It includes, partly, a microscopic clarification of the related achievements based on Nozi`eres' phenomenological description: Filippone, Moca, von Delft, and Mora [Phys. Rev. B 95, 165404 (2017)]. In our formulation, the Fermi-liquid parameters such as the quasiparticle energy, damping, and transport coefficients are related to each other through the total vertex \ensuremathΓ\ensuremathσ\ensuremathσ\ensuremath′;\ensuremathσ\ensuremath′\ensuremathσ(\ensuremathω,\ensuremathω\ensuremath′;\ensuremathω\ensuremath′,\ensuremathω), which may be regarded as a generalized Landau quasiparticle interaction. We obtain exactly this function up to linear order with respect to the frequencies \ensuremathω and \ensuremathω\ensuremath′ using the antisymmetry and analytic properties. The coefficients acquire additional contributions of three-body fluctuations away from half filling through the nonlinear susceptibilities. We also apply the formulation to nonequilibrium transport through a quantum dot, and clarify how the zero-bias peak evolves in a magnetic field.
We present a comprehensive Fermi liquid description for thermoelectric transport and current noise, applicable to multilevel quantum dots (QD) and magnetic alloys (MA) without electron-hole or time-reversal symmetry. Our formulation for the low-energy transport is based on an Anderson model with N discrete impurity levels, and is asymptotically exact at low energies, up to the next-leading order terms in power expansions with respect to temperature T and bias voltage eV. The expansion coefficients can be expressed in terms of the Fermi liquid parameters, which include the three-body correlation functions defined with respect to the equilibrium ground state in addition to the linear susceptibilities and the occupation number Nd of impurity electrons. We apply this formulation to the SU(N) symmetric QD and MA, and calculate the correlation functions for N=4 and 6, using numerical renormalization group approach. The three-body correlations are shown to be determined by a single parameter over a wide range of electron fillings $1\ensuremath{\lesssim}{N}_{d}^{}\ensuremath{\lesssim}N\ensuremath{-}1$ for strong Coulomb interactions U, and they also exhibit the plateau structures due to the SU(N) Kondo effects at integer values of Nd. We find that the Lorenz number $L=\ensuremath{\kappa}/(T\ensuremath{\sigma})$ for QD and MA, defined as the ratio of the thermal conductivity $\ensuremath{\kappa}$ to the electrical conductivity $\ensuremath{\sigma}$, deviates from the universal Wiedemann-Franz value ${\ensuremath{\pi}}^{2}/(3{e}^{2})$ as the temperature increases from T=0, showing the T2 dependence, the coefficient for which depends on the three-body correlations away from half filling. Furthermore, we find that the current noise for the SU(4) quantum dots and that for SU(6) show a pronounced difference at the quarter Nd/N=1/4 and 3/4 fillings. In particular, the linear noise for N=4 exhibits a flat peak while the peak for N=6 shows a round shape, reflecting the fact that, at these filling points, the SU(N) Kondo effects occur for $N\ensuremath{\equiv}0$ (mod 4), whereas the intermediate-valence fluctuations occur for $N\ensuremath{\equiv}2$ (mod 4). We also demonstrate the role of three-body correlations on the nonlinear current noise and the other transport coefficients.
We study the three-body Fermi liquid effects in the SU(N) Anderson impurity model within the strong interaction limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$, where the occupation number Nd of the impurity levels varies over the range of 0<Nd<1. The three-body correlations arise among the impurity electrons when the electron-hole symmetry, the time-reversal symmetry, or both are broken by external fields or potentials. They play a key role in low-energy transport, particularly in the next-to-leading-order terms of power expansion with respect to temperature T and bias voltage eV. Using the numerical renormalization group approach, we calculate the differential conductance and nonlinear current noise through quantum dots, as well as the thermal conductivity of both quantum dots and magnetic alloys. Specifically, we focus on the SU(2) and SU(4) cases, demonstrating how the three-body correlations evolve in the strong interaction limit across the 1/N-filling Kondo regime at ${N}_{d}^{}\ensuremath{\simeq}1.0$ and the valence fluctuation regime where the occupation number Nd varies rapidly with the position ${\ensuremath{\epsilon}}_{d}^{}$ of the impurity levels. We also show that the three-body correlations strongly couple with asymmetries in the tunnel couplings between quantum dots and reservoirs. For N=4, this coupling significantly affects the order (eV)3 nonlinear current through the quarter-filling Kondo state, where the electron-hole symmetry is broken. In particular, in the $U\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, the expansion coefficients exhibit clear plateau structures associated with the Kondo effect at ${N}_{d}^{}\ensuremath{\simeq}1.0$. Our formulation and analysis pave the way for deducing the three-body correlations from the low-energy experiments on the next-to-leading-order transport.
This paper is a corrected version of Phys. Rev. B 95, 165404 (2017), which we have retracted because it contained a trivial but fatal sign error that lead to incorrect conclusions. We extend a recently developed Fermi liquid (FL) theory for the asymmetric single-impurity Anderson model [C. Mora et al., Phys. Rev. B 92, 075120 (2015)] to the case of an arbitrary local magnetic field. To describe the system's low-lying quasiparticle excitations for arbitrary values of the bare Hamiltonian's model parameters, we construct an effective low-energy FL Hamiltonian whose FL parameters are expressed in terms of the local level's spin-dependent ground-state occupations and their derivatives with respect to level energy and local magnetic field. These quantities are calculable with excellent accuracy from the Bethe ansatz solution of the Anderson model. Applying this effective model to a quantum dot in a nonequilibrium setting, we obtain exact results for the curvature of the spectral function, cA, describing its leading $\ensuremath{\sim}{\ensuremath{\varepsilon}}^{2}$ term, and the transport coefficients cV and cT, describing the leading $\ensuremath{\sim}{V}^{2}$ and $\ensuremath{\sim}{T}^{2}$ terms in the nonlinear differential conductance. A sign change in cA or cV is indicative of a change from a local maximum to a local minimum in the spectral function or nonlinear conductance, respectively, as is expected to occur when an increasing magnetic field causes the Kondo resonance to split into two subpeaks. We find that the fields BA, BT, and BV at which cA, cT, and cV change sign, respectively, are all of order TK, as expected, with BA=BT=BV=0.75073TK in the Kondo limit.
We extend the Fermi liquid theory of Nozi\`eres by introducing the next-to-leading order corrections to the Fermi liquid fixed point. For a general SU(N) Kondo impurity away from half-filling, this extension is necessary to compute observables (resistivity, current or noise) at low energy. Three additional contributions are identified and their coupling constants are related using an original (and more complete) formulation of the Kondo resonance floating. In the conformal field theory language, a single cubic operator is proposed that produces the same three contributions with the same coupling constants. Comparison with an exact free-energy expansion further relates the leading and next-to-leading order corrections so that a single energy scale, the Kondo temperature, eventually governs the low-energy regime. We compare our results at large N with the approach of Read and Newns and find analytical agreement.
In this report, we describe a recent development in a Fermi liquid theory for the Kondo effect in quantum dots under a finite bias voltage V. Applying the microscopic theory of Yamada and Yosida to a nonequilibrium steady state, we derive the Ward identities for the Keldysh Green's function, and determine the low-energy behavior of the differential conductance dI/dV exactly up to terms of order (eV)2 for the symmetric Anderson model. These results are deduced from the fact that the Green's function at the impurity site is a functional of a nonequilibrium distribution feff(ω), which at eV=0 coincides with the Fermi function. Furthermore, we provide an alternative description of the low-energy properties using a renormalized perturbation theory (RPT). In the nonequilibrium state the unperturbed part of the RPT is determined by the renormalized free quasiparticles, the distribution function of which is given by feff(ω). The residual interaction between the quasiparticles U, which is defined by the full vertex part at zero frequencies, is taken into account by an expansion in the power series of U. We also discuss the application of the RPT to a high-bias region beyond the Fermi-liquid regime.
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, and 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.