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 how bias and tunneling asymmetries affect nonlinear current through a quantum dot with
N discrete levels in the Fermi liquid regime, using an exact low-energy expansion of the current derived up to terms of order
V3 with respect to the bias voltage. The expansion coefficients are described in terms of the phase shift, the linear susceptibilities, and the three-body correlation functions, defined with respect to the equilibrium ground state of the Anderson impurity model. In particular, the three-body correlations play an essential role in the order
V3 term, and their coupling to the nonlinear current depends crucially on the bias and tunnel asymmetries. The number of independent components of the three-body correlation functions increases with
N the internal degrees of the quantum dots, and it gives a variety in the low-energy transport. We calculate the correlation functions over a wide range of electron fillings of the Anderson impurity model with the
SU(N) internal symmetry, using the numerical renormalization group. We find that the order
V3 nonlinear current through the
SU(N) Kondo state, which occurs at electron fillings of 1 and $N\ensuremath{-}1$ for strong Coulomb interactions, significantly varies with the three-body contributions as tunnel asymmetries increase. Furthermore, in the valence fluctuation regime toward the empty or fully occupied impurity state, a sharp peak emerges in the coefficient of
V3 current in the case at which bias and tunneling asymmetries cooperatively enhance the charge transfer from one of the electrodes.
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.
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.
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.