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arXiv:1308.0638

Reply to 'Comment on Universal out-of-equilibrium transport in Kondo-correlated quantum dots'

Enrique Muñoz, C. J. Bolech, Stefan Kirchner

Published Aug 2, 2013• 0 citations
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Abstract

A recent comment on our work (Phys. Rev. Lett., vol. 110, 016601 (2013)) by A.A.Aligia claims that we "made mistakes in the evaluation of the lesser quantities". It is further claimed that the distribution function of the single-particle selfenergy of the interacting region in the Fermi liquid regime, e.g. at small bias voltage, low temperature, and small frequency, is continuous. These claims are based on a comparison of the particle-hole symmetric case with results obtained from the approach of A.A.Aligia. We disagree with these claims and show that the discrepancies that the comment alludes to originate from a violation of Ward identities by the method employed in the comment. A comparison of our approach with the numerical renormalization group shows perfect agreement for the symmetric case.

DOI
10.48550/arxiv.1308.0638

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I reply to the new arguments of Muñoz et al. (arXiv:1308.0638) regarding our controversy on non-equilibrium Green functions of the impurity Anderson model, and to new unfounded criticism to my paper Ref. 1 (arXiv:1110.0816). In particular, I show that the lesser self energy given in Ref. 1 does satisfy a Ward identity. References are updated.

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A recent comment on our work (Phys. Rev. Lett., vol. 110, 016601 (2013)) by A.A.Aligia claims that we "made mistakes in the evaluation of the lesser quantities". It is further claimed that the distribution function of the single-particle selfenergy of the interacting region in the Fermi liquid regime, e.g. at small bias voltage, low temperature, and small frequency, is continuous. These claims are based on a comparison of the particle-hole symmetric case with results obtained from the approach of A.A.Aligia. We disagree with these claims and show that the discrepancies that the comment alludes to originate from a violation of Ward identities by the method employed in the comment. A comparison of our approach with the numerical renormalization group shows perfect agreement for the symmetric case.
It is shown that the lesser and greater self energies and Green functions calculated in a recent work on universal noequilibrium transport in interacting quantum dots [E. Muñoz, C. J. Bolech, and S. Kirchner, Phys. Rev. Lett. 110, 016601 (2013)] are incorrect. The authors start from renormalized second-order perturbation theory in the Coulomb repulsion in the particle-hole symmetric impurity Anderson model. The reported lesser self-energy is compared with the correct one, and with an analytic result valid for small frequency and applied voltage. As a consequence of the mistakes, the conservation of the current is not established and the results are unreliable.
The Comment of A. A. Aligia claims that the superperturbation theory (SPT) approach [E. Mu\~noz, C. J. Bolech, and S. Kirchner, Phys. Rev. Lett. 110, 016601 (2013)] formulated using dual fermions [A. N. Rubtsov, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. B 77, 033101 (2008)] and used by us to compare with numerical renormalization group (NRG) results for the conductance [L. Merker, S. Kirchner, E. Mu\~noz, and T. A. Costi, Phys. Rev. B 87, 165132 (2013)], fails to correctly extend the results of the symmetric Anderson impurity model (SIAM) for general values of the local level Ed{E}_{d}Ed​ in the Kondo regime. We answer this criticism. We also compare new NRG results for cB{c}_{B}cB​, with cB{c}_{B}cB​ calculated directly from the low-field conductance, with new higher-order SPT calculations for this quantity, finding excellent agreement for all Ed{E}_{d}Ed​ and for $U/\ensuremath{\pi}\ensuremath{\Delta}$ extending into the strong coupling regime.
In a recent work [Merker, Kirchner, Mu\~noz, and Costi, Phys. Rev. B 87, 165132 (2013)], the authors compared the results of numerical renormalization group (NRG) and a perturbative approach for the dependence on temperature TTT and magnetic field BBB of the conductance through a quantum dot described by the impurity Anderson model, for small TTT and BBB. We show that the equation used to extract the dependence on BBB from NRG results is incorrect out of the particle-hole symmetric case. As a consequence, in the Kondo regime, the correct NRG results have a weaker dependence on BBB and the disagreement between both approaches increases.
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.
We study the nonequilibrium transport properties of fully (exactly) screened Kondo quantum dots subject to a finite bias voltage or a finite temperature. First, we calculate the Fermi-liquid coefficients of the conductance for models with arbitrary spin, i.e., its leading behavior for small bias voltages or temperatures. Second, we determine the low-temperature behavior of the static susceptibility from the exactly known Bethe ansatz results for the magnetization. Third, we study the crossover from strong to weak coupling in the spin-1/2 and the spin-1 models coupled to one or two screening channels, respectively. Using a real-time renormalization group method we calculate the static and dynamical spin-spin correlation functions for the spin-1/2 model as well as the linear and differential conductance and the static susceptibility for the spin-1 model. We define various Kondo scales and discuss their relations. We assess the validity of the renormalization-group treatment by comparing with known results for the temperature dependence of the linear conductance and static susceptibility as well as the Fermi-liquid behavior at low energies.