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arXiv:cond-mat/0506235 · Physical Review B

Universal aspects of nonequilibrium currents in a quantum dot

Benjamin Doyon, Natan Andrei

Published Jun 22, 2006• 106 citations
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Abstract

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.''

DOI
10.1103/physrevb.73.245326

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Kondo effect out of equilibrium

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•Jan 1, 2006•0 citations
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 (t0\ensuremath→\ensuremath−\ensuremath∞)({t}_{0}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty})(t0​\ensuremath→\ensuremath−\ensuremath∞), 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.''
We present here the details of a method [A. B. Culver and N. Andrei, Many-body wavefunctions for quantum impurities out of equilibrium, Phys. Rev. B 103, L201103 (2021)] for calculating the time-dependent many-body wavefunction that follows a local quench. We apply the method to the voltage-driven nonequilibrium Kondo model to find the exact time-evolving wavefunction following a quench where the dot is suddenly attached to the leads at t=0t=0t=0. The method, which does not use Bethe ansatz, also works in other quantum impurity models and may be of wider applicability. We show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction of the Kondo model is a current-carrying nonequilibrium steady state that satisfies the Lippmann-Schwinger equation. We show that the electric current in the time-evolving wavefunction is given by a series expression that can be expanded either in weak coupling or in strong coupling, converging to all orders in the steady-state limit in either case. The series agrees to leading order with known results in the well-studied regime of weak antiferromagnetic coupling and also reveals a universal regime of strong ferromagnetic coupling with Kondo temperature ${T}_{K}^{(F)}=D{e}^{\ensuremath{-}\frac{3{\ensuremath{\pi}}^{2}}{8}\ensuremath{\rho}|J|}$ ($J<0, \ensuremath{\rho}|J|\ensuremath{\rightarrow}\ensuremath{\infty}$). In this regime, the differential conductance dI/dVdI/dVdI/dV reaches the unitarity limit 2e2/h2{e}^{2}/h2e2/h asymptotically at large voltage or temperature.
We present a method for calculating the time-dependent many-body wavefunction that follows a local quench. We apply the method to the voltage-driven nonequilibrium Kondo model to find the exact time-evolving wavefunction following a quench where the dot is suddenly attached to the leads at t=0t=0t=0. The method, which does not use Bethe ansatz, also works in other quantum impurity models (we include results for the interacting resonant level and the Anderson impurity model) and may be of wider applicability. In the particular case of the Kondo model, we show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction is a current-carrying nonequilibrium steady state that satisfies the Lippmann-Schwinger equation. We show that the electric current in the time-evolving wavefunction is given by a series expression that can be expanded either in weak coupling or in strong coupling, converging to all orders in the steady-state limit in either case. The series agrees to leading order with known results in the well-studied regime of weak antiferromagnetic coupling and also reveals another universal regime of strong ferromagnetic coupling, with Kondo temperature ${T}_{K}^{(F)}=D{e}^{\ensuremath{-}\frac{3{\ensuremath{\pi}}^{2}}{8}\ensuremath{\rho}|J|}$ ($J<0, \ensuremath{\rho}|J|\ensuremath{\rightarrow}\ensuremath{\infty}$). In this regime, the differential conductance dI/dVdI/dVdI/dV reaches the unitarity limit 2e2/h2{e}^{2}/h2e2/h asymptotically at large voltage or temperature.
In the previous paper, we found a series expression for the average electric current following a quench in the nonequilibrium Kondo model driven by a bias voltage. Here, we evaluate the steady state current in the regimes of strong and weak coupling. We obtain the standard leading order results in the usual weak antiferromagnetic regime, and we also find a new universal regime of strong ferromagnetic coupling with Kondo temperature TK=De3π28ρJT_K = D e^{\frac{3π^2}{8} ρJ}TK​=De83π2​ρJ. In this regime, the differential conductance dI/dVdI/dVdI/dV reaches the unitarity limit 2e2/h2e^2/h2e2/h asymptotically at large voltage or temperature.
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.
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.
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.