Functional renormalization group for nonequilibrium quantum many-body problems

We extend the concept of the functional renormalization for quantum many-body problems to nonequilibrium situations. Using a suitable generating functional based on the Keldysh approach, we derive a system of coupled differential equations for the $m$-particle vertex functions. The approach is completely general and allows calculations for both stationary and time-dependent situations. As a specific example we study the stationary state transport through a quantum dot with local Coulomb correlations at finite bias voltage employing two different truncation schemes for the infinite hierarchy of equations arising in the functional renormalization group scheme.

Figure 1

(Color online) Keldysh contour.

Figure 2

Diagrammatic form of the flow equation for ${\gamma }_1^{\Lambda }$. The slashed line stands for the single scale propagator ${\widehat{\mathcal{S}}}^{\Lambda }$.

Figure 3

Diagrammatic form of the flow equation for ${\gamma }_2^{\Lambda }$. The slashed line stands for the single-scale propagator ${\widehat{\mathcal{S}}}^{\Lambda }$, the unslashed line for ${\widehat{G}}^{\Lambda }$.

Figure 4

Flow of ${\Sigma }^{--,\Lambda }∕\Gamma$ with $\Lambda ∕\Gamma$ for $U∕\Gamma =1$, $V_G∕\Gamma =0.5$, and $V_B=0$. The full curve shows the real part; the dashed shows the imaginary part of ${\Sigma }^{--,\Lambda }$.

Figure 5

(Color online) Flow of ${\Sigma }^{--,\Lambda }∕\Gamma$ with $\Lambda ∕\Gamma$ for $U∕\Gamma =15$, $V_G∕\Gamma =6$, and $V_B=0$. The full and dashed curves show real and imaginary parts obtained from the integration $\Lambda =\infty \to \Lambda =0$, the dashed-dotted and dotted curves real and imaginary parts obtained from an integration $\Lambda =0\to \Lambda =\infty$, using the solution from (45) as initial value for ${\Sigma }^{--,\Lambda }$. The crosses denote the analytical solution (44).

Figure 6

(Color online) Imaginary part of ${\Lambda }_p$ determined from (44) and $V_G+{\Sigma }^{--,{\Lambda }_p}={\Lambda }_p+i\Gamma$ for different values of $U$ and $V_B=0$ as function of $V_G$. For $U∕\Gamma =15$ and 20, there exist an interval $[V_G^a,V_G^b]$ where $\mathrm{Im}{\Lambda }_p>0$, whereas for small $U$ or $V_G\notin [V_G^a,V_G^b]$ we always have $\mathrm{Im}{\Lambda }_p<0$.

Figure 7

(Color online) Flow of ${\Sigma }^{--,\Lambda }∕\Gamma$ with $\Lambda$ for $U∕\Gamma =1$ and $5$ for $V_G∕\Gamma =0.5$ at $V_B∕\Gamma =0$ and $1$ (thick curves: real part; thin curves: imaginary part). The curve for $\mathrm{Im}{\Sigma }^{--,\Lambda }∕\Gamma$ at $U∕\Gamma =1$ and $V_B∕\Gamma =1$ (thin dashed line) lies on top of the corresponding zero bias curve and is thus not visible.

Figure 8

(Color online) Conductance normalized to $G_0=2e^2∕h$ as function of $V_G$ for $U∕\Gamma =2$ and several values of the bias voltage $V_B$.

Figure 9

(Color online) Flow of ${\Sigma }^{--,\Lambda }∕\Gamma$ with $\Lambda$ for $U∕\Gamma =1$ (left panel) and $U∕\Gamma =15$ (right panel) at $V_G=U∕2$ and $V_B=0$. The full curves show the real part; the dashed shows the imaginary part of ${\Sigma }^{--,\Lambda }$. The stars at the vertical axis denote the values as obtained from the imaginary-time fRG.[39]

Figure 10

(Color online) Flow of ${\Sigma }^{--,\Lambda }∕\Gamma$ and ${\Sigma }^{-+,\Lambda }∕\Gamma$ with $\Lambda$ for $U∕\Gamma =1$ (left panel) and $U∕\Gamma =15$ (right panel), $V_G∕\Gamma =U∕2$ and $V_B∕\Gamma =1$. The full curves show the real part; the dashed shows the imaginary part of ${\Sigma }^{--,\Lambda }$; and the dotted-dashed show the imaginary part of ${\Sigma }^{-+,\Lambda }$. The real part for the latter is zero.

Figure 11

(Color online) Current normalized to $J_0=G_0\frac{\Gamma }{e}$ (after reintroducing $e$ and $\hslash$) as function of $V_B$ for $U∕\Gamma =1$, $6$, and $15$ and $V_G=0$. For $U∕\Gamma =15$, we find a region of negative differential conductance in the region $\mid V_B∕\Gamma \mid \approx 0.5$ (cf. Fig. 12).

Figure 12

(Color online) Differential conductance $G$ as function of $V_B$ for $V_G=0$ and various values of $U$. For $U∕\Gamma >5$, a distinct minimum around $V_B∕\Gamma \approx 0.5$, appears.

Figure 13

(Color online) Differential conductance $G$ as function of $V_G$ for different values of $V_B$ and $U∕\Gamma =1$ (upper panel) and $U∕\Gamma =15$ (lower panel). Note the extended plateau at $V_B=0$ for $U∕\Gamma =15$, which is a manifestation of the pinning of spectral weight at the Fermi level.