Real-time effective-action approach to the Anderson quantum dot

The nonequilibrium time evolution of an Anderson quantum dot is investigated. The quantum dot is coupled between two leads forming a chemical-potential gradient. We use Kadanoff-Baym dynamic equations within a nonperturbative resummation of the $s$-channel bubble chains. The effect of the resummation leads to the introduction of a frequency-dependent four-point vertex. The tunneling to the leads is taken into account exactly. The method allows the determination of the transient as well as stationary transport through the quantum dot, and results are compared with different schemes discussed in the literature (functional renormalization group, iterative real-time summation of the path integral, time-dependent matrix renormalization group, and quantum Monte Carlo methods).

Figure 1

The schematic representation of the energy levels in the Anderson dot system, for the zero magnetic field case. The solid line below the dotted line represents the energy level of the first electron occupying the dot; the solid line above is the level of the second electron occupying the dot.

Figure 2

2PI diagrams of the loop expansion of ${\Gamma }_2[D]$. (a) The two lowest-order diagrams of the loop expansion described in Secs. 3a and 3b, with each vertex coupling up-spins to down-spins. Black dots represent the bare vertex $~U$, solid blue (dashed green) lines represent the up-spin (down-spin) propagator $D_{\sigma \sigma },\sigma =\uparrow ,\downarrow$. (b) Diagram representing the resummation approximation explained in Sec. 3c. The wiggly line is the scalar propagator $G_{11}$. This propagator is represented as a sum of bubble-chain diagrams; see Eq. (35). The same diagram with down-spins also appears in ${\Gamma }_2[D]$.

Figure 3

Time evolution of the two-time correlation function $F_{11}(0,t)$ for an isolated quantum dot: comparison of 2PI and exact results obtained by directly solving the Schrödinger equation. The magnetic field is zero. While the exact evolution shows revivals, the oscillations are damped out within the 2PI approach.

Figure 4

The same time evolution as in Fig. 3: comparison of 2PI and exact results. The difference between the exact and 2PI results, as well as the difference between the Taylor-expanded exact and the exact results, are shown.

Figure 5

The stationary current (55) through the quantum dot at $U=0$ as a function of the bias voltage. The comparison of the 2PI results and the results of Ref. 14, which in this case are both exact, serves as a benchmark test.

Figure 6

Comparison of the transient currents obtained with the 2PI equations and with the tDMRG method, at the symmetric point: $E_0=-U/2$. The parameters are $U=3\Gamma$, $T=0.1\Gamma$. Note that the different initial slopes are due to differently chosen initial conditions. While initial correlations between dot and leads are present in the tDMRG calculation, there are no such correlations in the 2PI approach.

Figure 7

The modification of the asymptotic stationary current due to nonvanishing interactions as functions of the bias voltage. The results obtained with the 2PI method are compared with those of the ISPI calculations given in Ref. 36 and with perturbative results from Ref. 53.

Figure 8

The time-dependent current through the dot is shown for different couplings $U$, bias voltage $V$, and temperature $T$. We compare results obtained within the second-order truncation of the 2PI effective action with those derived after $s$-channel resummation. For $U=2\Gamma$ there is perfect agreement. Note that, for the bigger coupling, the Kondo temperature (6) is $T_K=0.33\Gamma$, such that the system is already in the Kondo regime.

Figure 9

Comparison of the results obtained with different methods at the symmetric point $E_0=-U/2$. The stationary current is shown as a function of the bias voltage for several coupling strengths. The 2PI and ISPI methods use $T=0.1\Gamma$.

Figure 10

The stationary current in the mixed-valence regime for the coupling of $U=2\Gamma$, and $E_0=\Gamma$. See Ref. 34 for a detailed discussion of theother methods.

Figure 11

The stationary current as a function of the bias voltage, for several magnetic field strengths.

Figure 12

The stationary current is plotted as a function of the bias voltage for several temperatures.

Figure 13

The effective coupling for various couplings as a function of the frequency.