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arXiv:0712.0760 · Physical Review B

Electron transport in the four-lead two-impurity Kondo model: Nonequilibrium perturbation theory with almost degenerate levels

V. Koerting, Jens Paaske, P. Wölfle

Published Apr 16, 2008• 10 citations
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Abstract

The eigenstates of an isolated nanostructure may get mixed by the coupling to external leads. This effect is the stronger, the smaller the level splitting on the dot and the larger the broadening induced by the coupling to the leads. We describe how to calculate the nondiagonal density matrix of the nanostructure efficiently in the cotunneling regime. As an example, we consider a system of two quantum dots in the Kondo regime, the two spins coupled by an antiferromagnetic exchange interaction and each dot tunnel coupled to two leads. Calculating the nonequilibrium density matrix and the corresponding current, we demonstrate the importance of the off-diagonal terms in the presence of an applied magnetic field and a finite bias voltage.

DOI
10.1103/physrevb.77.165122

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The eigenstates of an isolated nanostructure may get mixed by the coupling to external leads. This effect is the stronger, the smaller the level splitting on the dot and the larger the broadening induced by the coupling to the leads. We describe how to calculate the nondiagonal density matrix of the nanostructure efficiently in the cotunneling regime. As an example, we consider a system of two quantum dots in the Kondo regime, the two spins coupled by an antiferromagnetic exchange interaction and each dot tunnel coupled to two leads. Calculating the nonequilibrium density matrix and the corresponding current, we demonstrate the importance of the off-diagonal terms in the presence of an applied magnetic field and a finite bias voltage.
We present improvements of a recently introduced numerical method [E. Arrigoni et al., Phys. Rev. Lett. 110, 086403 (2013)] to compute steady-state properties of strongly correlated electronic systems out of equilibrium. The method can be considered as a nonequilibrium generalization of exact diagonalization based dynamical mean-field theory (DMFT). The key modification for the nonequilibrium situation consists in addressing the DMFT impurity problem within an auxiliary system consisting of the correlated impurity, Nb{N}_{b}Nb​ uncorrelated bath sites, and two Markovian environments (sink and reservoir). Algorithmic improvements in the impurity solver allow to treat efficiently larger values of Nb{N}_{b}Nb​ than previously in DMFT. This increases the accuracy of the results and is crucial for a correct description of the physical behavior of the system in the relevant parameter range including a semiquantitative description of the Kondo regime. To illustrate the approach, we consider a monoatomic layer of correlated orbitals, described by the single-band Hubbard model, attached to two metallic leads. The nonequilibrium situation is driven by a bias voltage applied to the leads. For this system, we investigate the spectral function and the steady-state current-voltage characteristics in the weakly as well as in the strongly interacting limit. In particular, we investigate the nonequilibrium behavior of quasiparticle excitations within the Mott gap of the correlated layer. We find for low-bias voltage Kondo-type behavior in the vicinity of the insulating phase. In particular, we observe a splitting of the Kondo resonance as a function of the bias voltage.
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 apply the adaptive time-dependent density-matrix renormalization-group method (tDMRG) to the study of transport properties of quantum-dot systems connected to metallic leads. Finite-size effects make the usual tDMRG description of the Kondo regime a numerically demanding task. We show that such effects can be attenuated by describing the leads by ``Wilson chains,'' in which the hopping matrix elements decay exponentially away from the impurity $({t}_{n}\ensuremath{\propto}{\ensuremath{\Lambda}}^{\ensuremath{-}n/2})$. For a given system size and in the linear-response regime, results for $\ensuremath{\Lambda}>1$ show several improvements over the undamped $\ensuremath{\Lambda}=1$ case: perfect conductance is obtained deeper in the strongly interacting regime and current plateaus remain well defined for longer time scales. Similar improvements were obtained in the finite-bias regime up to bias voltages of the order of the Kondo temperature. These results show that with the proposed modification, the tDMRG characterization of Kondo correlations in the transport properties can be substantially improved, while it turns out to be sufficient to work with much smaller system sizes. We discuss the numerical cost of this approach with respect to the necessary system sizes and the entanglement growth during the time evolution.
By using the full density matrix approach to spectral functions within the numerical renormalization group method, we present a detailed study of the magnetic field induced splittings in the spin-resolved and the total spectral densities of a Kondo correlated quantum dot described by the single level Anderson impurity model. The universal scaling of the splittings with magnetic field is examined by varying the Kondo scale either by a change of local level position at a fixed tunnel coupling or by a change of the tunnel coupling at a fixed level position. We find that the Kondo-peak splitting $\ensuremath{\Delta}/{T}_{K}$ in the spin-resolved spectral function always scales perfectly for magnetic fields B<8TKB<8{T}_{K}B<8TK​ in either of the two TK{T}_{K}TK​-adjusted paths. Scaling is destroyed for fields B>10TKB>10{T}_{K}B>10TK​. On the other hand, the Kondo peak splitting $\ensuremath{\delta}/{T}_{K}$ in the total spectral function does slightly deviate from the conventional scaling theory in whole magnetic field window along the coupling-varying path. Furthermore, we show the scaling analysis suitable for all field windows within the Kondo regime and two specific fitting scaling curves are given from which certain detailed features at low field are derived. In addition, the scaling dimensionless quantity $\ensuremath{\Delta}/2B$ and $\ensuremath{\delta}/2B$ are also studied and they can reach and exceed 1 in the large magnetic field region, in agreement with a recent experiment [T. M. Liu et al., Phys. Rev. Lett. 103, 026803 (2009)].
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We consider the transport properties in an applied magnetic field of the spin S=1/2S=1/2S=1/2 Kondo model with ferromagnetic exchange coupling to electronic reservoirs, a description relevant for the strong-coupling limit of underscreened spin S=1S=1S=1 Kondo impurities. Because the ferromagnetic Kondo interaction is marginally irrelevant, perturbative methods should prove accurate down to low energies. For the purpose of this study, we use a combination of Majorana diagrammatic theory with density matrix numerical renormalization group simulations. In the standard case of antiferromagnetic Kondo exchange, we first show that our simple analytical technique recovers results for the T\mathcal{T}T-matrix and spin relaxation at weak coupling (above the Kondo temperature), which were previously obtained using functional renormalization tools. Considering then the ferromagnetic case, we demonstrate how the low-energy Kondo anomaly splits for arbitrary small values of the Zeeman energy, in contrast to fully screened Kondo impurities near the strong-coupling Fermi liquid fixed point, and in agreement with recent experimental findings for spin S=1S=1S=1 molecular quantum dots.