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arXiv:1104.2757 · Physical Review B

Slave-boson Keldysh field theory for the Kondo effect in quantum dots

Sergey Smirnov, Milena Grifoni

Published Sep 7, 2011• 20 citations
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Abstract

We present a nonequilibrium nonperturbative field theory for the Kondo effect in strongly interacting quantum dots at finite temperatures. Unifying the slave-boson representation with the Keldysh field integral, an effective Keldysh action is derived and explored in the vicinity of the zero slave-bosonic field configuration. The theory properly reflects the essential features of the Kondo physics and at the same time significantly simplifies a field-theoretic treatment of the phenomenon, avoiding complicated saddle-point analysis or $1/N$ expansions, used so far. Importantly, our theory admits a closed analytical solution which explains the mechanism of the Kondo effect in terms of an interplay between the real and the imaginary parts of the slave-bosonic self-energy. It thus provides a convenient nonperturbative building block, playing the role of a ``free propagator,'' for more advanced theories. We finally demonstrate that already this simplest possible field theory is able to correctly reproduce experimental data on the Kondo peak observed in the differential conductance, correctly predicts the Kondo temperature, and, within its applicability range, has the same universal temperature dependence of the conductance as the one obtained in numerical renormalization group calculations.

DOI
10.1103/physrevb.84.125303

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We present a nonequilibrium nonperturbative field theory for the Kondo effect in strongly interacting quantum dots at finite temperatures. Unifying the slave-boson representation with the Keldysh field integral, an effective Keldysh action is derived and explored in the vicinity of the zero slave-bosonic field configuration. The theory properly reflects the essential features of the Kondo physics and at the same time significantly simplifies a field-theoretic treatment of the phenomenon, avoiding complicated saddle-point analysis or 1/N1/N1/N expansions, used so far. Importantly, our theory admits a closed analytical solution which explains the mechanism of the Kondo effect in terms of an interplay between the real and the imaginary parts of the slave-bosonic self-energy. It thus provides a convenient nonperturbative building block, playing the role of a ``free propagator,'' for more advanced theories. We finally demonstrate that already this simplest possible field theory is able to correctly reproduce experimental data on the Kondo peak observed in the differential conductance, correctly predicts the Kondo temperature, and, within its applicability range, has the same universal temperature dependence of the conductance as the one obtained in numerical renormalization group calculations.
Kondo physics in nonequilibrium interacting nanoscale devices is an attractive fundamental many-particle phenomenon with a rich potential for applications. Due to enormous complexity its clear and flexible theory is still highly desirable. We develop a physically transparent analytical theory capable of correctly describing the Kondo effect in strongly interacting systems at temperatures close to and above the Kondo temperature. We derive a nonequilibrium Keldysh field theory valid for a system with any finite electron-electron interaction which is much stronger than the coupling of the system to contacts. Finite electron-electron interactions are treated involving as many slave-boson degrees of freedom as one needs for a concrete many-body system. In a small vicinity of the zero slave-bosonic field configuration weak slave-bosonic oscillations, induced by the dot-contacts tunneling, are described by an effective Keldysh action quadratic in the slave-bosonic fields. For clarity the theory is presented for the single impurity Anderson model but the construction of the Keldysh field integral is universal and applicable to systems with more complex many-body spectra.
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 electric current in the nonequilibrium Kondo model at zero magnetic field, using real-time perturbation theory in the Schwinger-Keldysh formulation. We show that the perturbative coefficients to all orders have a finite limit at large switch-on time $({t}_{0}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty})$, and we give a prescription for general operators to give finite coefficients in this limit. We explain how this is related to the fact that the leads play the role of thermal baths and allow relaxation to occur and the steady state to form. This proves perturbatively that a steady state is reached in the Schwinger-Keldysh formulation, and specifies which operators correspond to quantities that have a well-defined value in the steady state. Then, we show that the steady state can be described by a special type of density matrix (related to Hershfield's conjecture for the particular example of the nonequilibrium Kondo model.) In the second part of the paper, we perform a renormalization-group analysis of the perturbative series. We give a general argument that strongly suggests that the perturbative series of any average in the steady state satisfies the equilibrium Callan-Symanzik equations, and we show in detail how it works to one-loop order for the electric current operator inside any average. We finally compute to two-loops order the average of the electric current in the steady state, and perform a renormalization-group improvement. From this, we give a universal prescription, valid in the perturbative regime, for comparing the effect of the electric current to that of the temperature on the ``Kondo cloud.''
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.
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.
A general theoretical formulation for the effect of a strong on-site Coulomb interaction on the time-dependent electron transport through a quantum dot under the influence of arbitrary time-varying bias voltages and/or external fields is presented, based on slave bosons and the Keldysh nonequilibrium Green's function (GF) techniques. To avoid the difficulties of computing double-time GFs, we generalize the propagation scheme recently developed by Croy and Saalmann to combine the auxiliary-mode expansion with the celebrated Lacroix's decoupling approximation in dealing with the second-order correlated GFs and then establish a closed set of coupled equations of motion, called second-order quantum rate equations (SOQREs), for an exact description of transient dynamics of electron correlated tunneling. We verify that the stationary solution of our SOQREs is able to correctly describe the Kondo effect on a qualitative level. Moreover, a comparison with other methods, such as the second-order von Neumann approach and Hubbard-I approximation, is performed. As illustrations, we investigate the transient current behaviors in response to a step voltage pulse and a harmonic driving voltage, and linear admittance as well, in the cotunneling regime.