Spin-channel Keldysh field theory for weakly interacting quantum dots

We develop a low-energy nonequilibrium field theory for weakly interacting quantum dots. The theory is based on the Keldysh field integral in the spin channel of the quantum dot described by the single-impurity Anderson Hamiltonian. The effective Keldysh action is a functional of the Hubbard-Stratonovich magnetization field decoupling the quantum dot spin channel. We expand this action up to the second order with respect to the magnetization field, which allows one to describe nonequilibrium interacting quantum dots at low temperatures and weak electron-electron interactions, up to the contacts-dot coupling energy. Besides its simplicity, an additional advantage of the theory is that it correctly describes the unitary limit, giving the correct result for the conductance maximum. Thus our theory establishes an alternative simple method relevant for investigation of weakly interacting nonequilibrium nanodevices.

Figure 1

Temperature dependence of the differential conductance maximum at the symmetric point obtained from the present spin-channel Keldysh field-integral theory for $U=0.9\Gamma$, ${\epsilon }_{\mathrm{d}}=U/2$. Here $k{T}_0$ is the zero-temperature QD TDOS half-width at half maximum. The red circles show the universal temperature dependence of the differential conductance maximum obtained in the numerical renormalization group theory (Refs. 35 and 36). In this case $k{T}_0$ is the Kondo temperature, which is approximately equal to the zero-temperature QD TDOS half-width at half maximum.

Figure 2

Equilibrium and nonequilibrium QD TDOS (17) at zero temperature, $U=0.8\Gamma$, ${\epsilon }_{\mathrm{d}}=U/2$. The effect of a finite voltage is to decrease and broaden the QD TDOS.

Figure 3

Equilibrium QD TDOS (17) at zero and finite temperatures, $U=0.8\Gamma$, ${\epsilon }_{\mathrm{d}}=U/2$. As one can see, the effect of a finite voltage in Fig. 2 is similar to the effect of a finite temperature in this figure.

Figure 4

Zero-temperature QD differential conductance as a function of the applied voltage. Both interacting and noninteracting cases are shown. Due to the electron-electron interaction, the QD is in the resonant many-particle state where its differential conductance is enhanced at low voltages in comparison with the noninteracting counterpart. The maximum is equal to the correct value $2e^2/h$.