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arXiv:1310.8324

Reply to the reply to "Comment on Universal out-of-equilibrium transport in Kondo-correlated quantum dots"

Armando A. Aligia

Published Oct 30, 2013• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1310.8324

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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.

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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.

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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.
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.
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.
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.
Comment [arXiv:cond-mat.stat.mech., 1602.02087v1 (2016)] has raised questions claiming that the nonlocal model Hamiltonian presented in [Phys. Rev. E 92, 012123 (2015)] is equivalent to the standard (short-ranged) Φ^4 theory. These claims are based on a low momentum expansion of the interaction vertex that cannot be applied to the vertex factors containing both low and high momenta inside the loop-integrals. Elaborating upon the important steps of the momentum shell decimation scheme, employed in the renormalization-group calculation, we explicitly show the interplay of internal (high) and external (low) momenta determining the loop integrals for self-energy and vertex functions giving rise to corrections (to the bare parameters) different from those of the standard (short-ranged) Φ^4 theory. Employing explicit mathematical arguments, we show that this difference persists when the range of interaction is assumed to be long (short) ranged with respect to the lattice constant (correlation-length), yielding the critical exponents as given in the original paper.
This thesis comprises two parts centered around the functional renormalization-group framework: in the first part, I study the role of symmetries and conservation laws in approximate solutions, while in the second part I analyze Friedel oscillations and transport in Luttinger liquids with impurities. The functional renormalization group (fRG) has been developed as a new computational tool in the theory of interacting Fermi systems. The effective behavior of a given microscopic model is calculated by solving coupled differential flow equations for the Green functions with an energy scale as the flow parameter. The symmetries of the microscopic model imply Ward identities between Green and response functions. It is shown that solutions of truncated flow-equation hierarchies satisfy Ward identities if the cutoff bare action is gauge invariant. However, truncations are generally not self-consistent approximations in the sense of Baym and Kadanoff. The fRG is then applied to study Luttinger liquids. By computing the full spatial effective potential of a single impurity, long-range Friedel oscillations are observed in the density profile with the expected power laws for systems with up to 10^7 lattice sites. For a double barrier enclosing a dot region we find temperature regimes in which the conductance follows power laws with universal exponents, as well as non-universal crossover regimes in intermediate parameter regions.