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arXiv:1108.2434 · Physical Review B

1/(N−1) expansion based on a perturbation theory in U for the Anderson model with N -fold degeneracy

Akira Oguri, Rui Sakano, Takuro Fujii

Published Sep 22, 2011• 10 citations
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Abstract

We study low-energy properties of the $N$-fold degenerate Anderson model. Using a scaling that takes $u=(N\ensuremath{-}1)U$ as an independent variable in place of the Coulomb interaction $U$, the perturbation series in $U$ is reorganized as an expansion in powers of $1/(N\ensuremath{-}1)$. We calculate the renormalized parameters, which characterize the Kondo state, to the next leading order in the $1/(N\ensuremath{-}1)$ expansion at half-filling. The results, especially the Wilson ratio, agree very closely with the exact numerical renormalization group results at $N=4$. This ensures the applicability of our approach to $N>4$, and we present highly reliable results for nonequilibrium Kondo transport through a quantum dot.

DOI
10.1103/physrevb.84.113301

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We apply recently developed $1/(N\ensuremath{-}1)$ expansion to a particle-hole asymmetric SU(NNN) Anderson model with finite Coulomb interaction UUU. To leading order in $1/(N\ensuremath{-}1)$ it describes the Hartree-Fock random-phase approximation (HF-RPA), and the higher-order corrections describe systematically the fluctuations beyond the HF-RPA. We show that the next-leading-order results of the renormalized parameters for the local Fermi-liquid state agree closely with the numerical renormalization group results at N=4N=4N=4. This ensures the reliability of the next-leading-order results for $N\ensuremath{\geqslant}4$, and we examine the NNN dependence of the local Fermi-liquid parameters. Our expansion scheme uses the standard Feynman diagrams and has wide potential applications.
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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 VVV. 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/dVdI/dVdI/dV exactly up to terms of order (eV)2(eV)^2(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)f_{\text{eff}}(ω)feff​(ω), which at eV=0eV=0eV=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)f_{\text{eff}}(ω)feff​(ω). The residual interaction between the quasiparticles U~\widetilde{U}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~\widetilde{U}U. We also discuss the application of the RPT to a high-bias region beyond the Fermi-liquid regime.
Using different techniques, and Fermi-liquid relationships, we calculate the variation with the applied magnetic field (up to second order) of the zero-temperature equilibrium conductance through a quantum dot described by the impurity Anderson model. We focus on the strong-coupling limit $U\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$, where UUU is the Coulomb repulsion and $\mathrm{\ensuremath{\Delta}}$ is half the resonant-level width, and consider several values of the dot level energy Ed{E}_{d}Ed​, ranging from the Kondo regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$ to the intermediate-valence regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy. We have mainly used the density-matrix renormalization group (DMRG) and the numerical renormalization group (NRG) combined with renormalized perturbation theory (RPT). Results for the dot occupancy and magnetic susceptibility from the DMRG and NRG+RPT\mathrm{NRG}+\mathrm{RPT}NRG+RPT are compared with the corresponding Bethe ansatz results for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, showing an excellent agreement once Ed{E}_{d}Ed​ is renormalized by a constant Haldane shift. For $U<3\mathrm{\ensuremath{\Delta}}$ a simple perturbative approach in UUU agrees very well with the other methods. The conductance decreases with the applied magnetic field for dot occupancies ${n}_{d}\ensuremath{\sim}1$ and increases for ${n}_{d}\ensuremath{\sim}0.5$ or ${n}_{d}\ensuremath{\sim}1.5$ regardless of the value of UUU. We also relate the energy scale for the magnetic-field dependence of the conductance with the width of the low-energy peak in the spectral density of the dot.