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arXiv:1210.3227 · Journal of the Korean Physical Society

1/(N − 1) expansion approach to full-counting statistics for the SU(N) Anderson model

Akira Oguri, Rui Sakano

Published Aug 1, 2013• 1 citation
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Abstract

We apply a recently developed 1/( N − 1) expansion to the full-counting statistics for the N -fold degenerate Anderson impurity model in the Kondo regime. This approach is based on the perturbation theory in the Coulomb interaction U and is different from the conventional large- N theories, such as the usual 1/ N expansion and non-crossing approximation. We have confirmed that the calculations carried out up to order 1/( N − 1) 2 agree closely with those of the numerical renormalization group at N = 4, where the degeneracy is still not so large. This ensures the applicability of our approach for N ≥ 4. We present the results of the cumulants of the probability distribution function for a nonequilibrium current through a quantum dot in the particle-hole symmetric case.

DOI
10.3938/jkps.63.423

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We apply a recently developed 1/( N − 1) expansion to the full-counting statistics for the N -fold degenerate Anderson impurity model in the Kondo regime. This approach is based on the perturbation theory in the Coulomb interaction U and is different from the conventional large- N theories, such as the usual 1/ N expansion and non-crossing approximation. We have confirmed that the calculations carried out up to order 1/( N − 1) 2 agree closely with those of the numerical renormalization group at N = 4, where the degeneracy is still not so large. This ensures the applicability of our approach for N ≥ 4. We present the results of the cumulants of the probability distribution function for a nonequilibrium current through a quantum dot in the particle-hole symmetric case.
We study low-energy properties of the NNN-fold degenerate Anderson model. Using a scaling that takes $u=(N\ensuremath{-}1)U$ as an independent variable in place of the Coulomb interaction UUU, the perturbation series in UUU 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=4N=4N=4. This ensures the applicability of our approach to N>4N>4N>4, and we present highly reliable results for nonequilibrium Kondo transport through a quantum dot.
We analyze the full counting statistics of the multiorbital Kondo effect in a quantum dot with SU(NNN) symmetry in the framework of renormalized perturbation theory. The current probability distribution function is calculated for an arbitrary dot-site Coulomb repulsion UUU in the particle-hole-symmetric case. The resulting cumulant up to the leading nonlinear term of applied bias voltages indicates two types of electron transfer, respectively carrying charge eee and 2e2e2e, with different NNN dependences. The cross correlation between different orbital currents shows exponential enhancement with respect to UUU, which directly addresses the formation of the orbital-singlet state.
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.
We study nonequilibrium current fluctuations through a quantum dot, which includes a ferromagnetic Hund's rule coupling J, in the low-energy Fermi liquid regime using the renormalized perturbation theory. The resulting cumulant for the current distribution in the particle-hole symmetric case shows that spin-triplet and spin-singlet pairs of quasiparticles are formed in the current due to the Hund's rule coupling, and these pairs enhance the current fluctuations. In the fully screened higher-spin Kondo limit, the Fano factor takes a value F(b)=(9M+6)/(5M+4) determined by the orbital degeneracy M. We also investigate the crossover between the small and large J limits in the two-orbital case M=2, using the numerical renormalization group approach.
We analyze the full counting statistics (FCS) of the charge transport through the Anderson impurity model (AIM) and similar systems with a single conducting channel. The object of principal interest is the generating function for the cumulants of charge current distribution. We derive an exact analytic formula relating the FCS generating function to the self energy of the system in the presence of the measuring field. We first check that our approach reproduces correctly known results in simple limits, such as the FCS of the resonant level system (AIM without Coulomb interaction). We then proceed to study the FCS for the AIM both perturbatively in the Coulomb interaction and in the Kondo regime at the Toulouse point (we also study a related model of a spinless single-site quantum dot coupled to two half-infinite metallic leads in the Luttinger liquid phase at a special interaction strength). At zero temperature the FCS turns out to be binomial for small voltages. For the generic case of arbitrary energy scales the FCS is shown to be captured very well by generalizations of the Levitov-Lesovik type formula. Surprisingly, the FCS for the AIM indicates a presence of coherent electron pair tunnelling in addition to conventional single-particle processes. By means of perturbative expansions around the Toulouse point we succeeded in showing the universality of the binomial FCS at zero temperature in linear response. Based on our general formula for the FCS we then argue for a more general binomial theorem stating that the linear response zero-temperature FCS for any interacting single-channel setup is always binomial.
We study the full counting statistics (FCS) of electron tunneling through a multiterminal quantum dot in the Kondo regime within the slave-boson mean-field theory. By employing the method of Gogolin and Komnik for calculating the FCS generating function based on the nonequilibrium Green's function [Phys. Rev. B 73, 195301 (2006)], we obtain the counting field $\ensuremath{\lambda}$-dependent self-consistent equations for the mean values of the slave-boson fields and the explicit expression for the derivative of the adiabatic potential of the system with respect to the counting fields. Performing a perturbative expansion to the first order of $\ensuremath{\lambda}$, we find an extra contribution to the shot noise due to the bias-induced Bose field fluctuation, and then confirm that the nonequilibrium particle number fluctuation plays an important role in the current noise of the Kondo dot: enhancement of the current autocorrelation and a positive current cross-correlation.