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arXiv:1111.4076 · Physical Review Letters

Universal Out-of-Equilibrium Transport in Kondo-Correlated Quantum Dots: Renormalized Dual Fermions on the Keldysh Contour

Enrique Muñoz, C. J. Bolech, Stefan Kirchner

Published Jan 2, 2013• 59 citations
arXivHTMLPDF

Abstract

The nonlinear conductance of semiconductor heterostructures and single molecule devices exhibiting Kondo physics has recently attracted attention. We address the observed sample dependence of the measured steady state transport coefficients by considering additional electronic contributions in the effective low-energy model underlying these experiments that are absent in particle-hole symmetric setups. A novel version of the superperturbation theory of Hafermann et al. in terms of dual fermions is developed, which correctly captures the low-temperature behavior. We compare our results with the measured transport coefficients.

DOI
10.1103/physrevlett.110.016601

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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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The nonlinear conductance of semiconductor heterostructures and single molecule devices exhibiting Kondo physics has recently attracted attention. We address the observed sample dependence of the measured steady state transport coefficients by considering additional electronic contributions in the effective low-energy model underlying these experiments that are absent in particle-hole symmetric setups. A novel version of the superperturbation theory of Hafermann et al. in terms of dual fermions is developed, which correctly captures the low-temperature behavior. We compare our results with the measured transport coefficients.
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.
Correlations are a central ingredient of modern condensed matter physics. With the advent of mesoscopic physics, it became possible to study correlated systems in non-equilibrium situations, which can be reached by e.g. driving a constant current through a quantum dot. A basic question is how correlations on the one hand and a non-equilibrium current on the other hand influence each other. Here this interplay is studied for the model system of a quantum dot in the Kondo regime using the real time renormalization group (RTRG) method. This method is a functional renormalization group (RG) method allowing a non-perturbative description of correlated systems in non-equilibrium. For the Kondo quantum dot the method is generalized to the case of double vertices. The RTRG method not only provides equations for the calculation of the renormalized coupling between the quantum dot spin and the reservoir electrons, but also equations for the calculation of the non-equilibrium decoherence rate and the non-equilibrium current.The RG equations are derived to second order. This approximation is expected to be valid as long as the quantum dot does not enter the strong coupling regime, i.e. the effective coupling strength between the quantum dot spin and the electrons does not become too large. It is found that this is the case if the bias voltage applied to the dot or the temperature is much larger than the Kondo temperature.It is shown that the RG equations lead to coupling functions depending on the energy of the incident and the scattered electron. This dependence is simplified to the dependence on the average energy of the incident and the scattered electron in the weak coupling regime.A closer inspection of the RG equations for the couplings shows that they not only contain terms leading to a logarithmic increase of the coupling strength but also subleading terms which are unimportant in the weak coupling regime. The RG equations are solved analytically for the case where the subleading terms are neglected. It is found that the coupling functions develop resonances at energies associated with the chemical potentials of the reservoirs. The height of these resonances is determined by the decoherence rate. However, for bias voltages much larger than the Kondo temperature these resonances do not affect the conductance for a two terminal setup, so the conductance does not provide a measure for the decoherence rate in this case. It is found that the conductance for the two terminal setup is a universal function of voltage/Kondo temperature and temperature/Kondo temperature in the weak coupling regime. This function is calculated analytically in the limits of vanishing voltage or temperature. Furthermore, it is found that for zero temperature the decoherence rate is proportional to the non-equilibrium current flowing through the system. This supports the physical picture that a non-equilibrium current is associated with a decoherence rate, which cuts the logarithmic scaling of the coupling strength.Finally, the RG equations are solved numerically for zero temperature including the subleading corrections found before. As expected, the subleading corrections have no impact on the decoherence rate and the conductance in the weak coupling regime. On the other hand, it is found that they lead to strong deviations from the results of the analytic leading logarithmic approximation if the bias voltage approaches the Kondo temperature. Generally, it is found that the subleading corrections lead to a saturation of the decoherence rate and the conductance to some finite value. However, for the numerical solution a numerical regularization parameter has to be introduced. It turns out that the saturation value found depends on its value, so no statements on the precise value of the decoherence rate and the current can be made in this regime.
We present a theory for the Kondo spin-12\frac{1}{2}21​ effect in strongly correlated quantum dots. The theory is applicable at any temperature and voltage. It is based on a quadratic Keldysh effective action parametrized by a universal function. We provide a general analytical form for the tunneling density of states through this universal function for which we propose a simple microscopic model. We apply our theory to the highly asymmetric Anderson model with $U=\ensuremath{\infty}$ and describe its strong-coupling limit, weak-coupling limit, and crossover region within a single analytical expression. We compare our results with a numerical renormalization group in equilibrium and with a real-time renormalization group out of equilibrium and show that the universal shapes of the linear and differential conductance obtained in our theory and in these theories are very close to each other in a wide range of temperatures and voltages. In particular, as in the real-time renormalization group, we predict that at the Kondo voltage the differential conductance is equal to 2/32/32/3 of its maximum.
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.
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.