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

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.

Figure 1

Classes of the slave-bosonic theories for the Kondo effect in QDs. Their range of quantitative reliability with respect to the Kondo temperature $T_{\text{K}}$ ($T_{\text{K}}/\Gamma \sim \exp [-2\pi ({\mu }_0-{\epsilon }_{\text{d}})/\Gamma ]$, where $\Gamma$ is the total QD-contacts coupling strength, which is twice that of Ref. 6, ${\mu }_0$ the QD chemical potential, ${\epsilon }_{\text{d}}$ the single-particle energy level of the QD) is given by their horizontal location. The vertical location of a class shows its type, numerical, analytical, or both. As to our field theory, what the blue class shows is what one generally expects from nonperturbative field theories in the vicinity of the zero slave-bosonic field configuration.

Figure 2

The equilibrium and nonequilibrium TDOS in the Kondo regime. Here $kT=0.005\Gamma$, ${\mu }_0-{\epsilon }_{\mathrm{d}}=1.876\Gamma$, $W=100\Gamma$. In equilibrium, $V=0$, there is a sharp many-particle peak at the Fermi energy. In nonequilibrium, $V\ne 0$, the peak is reduced and split into two lower peaks (see inset). The total spectral weights of the equilibrium and nonequilibrium situations are almost the same, differing by $1\%$, which is due to the accuracy of the numerical integration involved in the total spectral weight.

Figure 3

The mechanism of the Kondo resonance formation in the nonperturbative Keldysh field theory in the vicinity of the zero slave-bosonic field configuration. Here $V=0$, $kT=0.008\Gamma$, ${\mu }_0-{\epsilon }_{\mathrm{d}}=1.95\Gamma$.

Figure 4

The Kondo peak in the differential conductance. The inset shows the temperature dependence of the differential conductance maximum at $V=0$. The solid line is the result of our nonperturbative Keldysh field theory. The circles show the experimental data of Ref. 3.

Figure 5

Comparison of our field theory with the numerical renormalization group theory. The solid line is obtained from the numerical renormalization group calculations (Refs. 27, 28, 29). The circles show the differential conductance maximum obtained from our field theory for the values of the parameters used in Fig. 4.

Figure 6

Keldysh field-theoretic prediction of the Kondo temperature dependence on the QD single-particle energy level $({\mu }_0-{\epsilon }_{\mathrm{d}})/\Gamma$.

Figure 7

Comparison of the universal temperature dependence of the differential conductance maximum in our field theory and in the numerical renormalization group theory. The solid line is the result of the numerical renormalization group calculations (Refs. 27, 28, 29). The circles show the result obtained from our field theory.