Interaction Correction to the Conductance of a Ballistic Conductor

In disordered metals, electron-electron interactions are the origin of a small correction to the conductivity, the “Altshuler-Aronov correction.” Here we investigate the Altshuler-Aronov correction $\delta G_{\mathrm{AA}}$ of a conductor in which the electron motion is ballistic and chaotic. We consider the case of a double quantum dot, which is the simplest example of a ballistic conductor in which $\delta G_{\mathrm{AA}}$ is nonzero. The fact that the electron motion is ballistic leads to an exponential suppression of $\delta G_{\mathrm{AA}}$ if the Ehrenfest time is larger than the mean dwell time ${\tau }_D$ or the inverse temperature $\hslash /T$.

Figure 1

Schematic drawing of a double quantum dot (left), with a generic set of four trajectories that contributes to the interaction correction to the conductance (right). The wiggly line represents the interaction propagator.

Figure 2

Five configurations of classical trajectories contributing to $\delta G_{\mathrm{AA}}$. Solid (dashed) trajectories represent retarded (advanced) Green functions. The dashed trajectory connecting $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$ is $\beta$. The remaining three trajectories connecting $\mathbf{r}$ and ${\mathbf{r}}_1$, ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, and ${\mathbf{r}}_2$ and ${\mathbf{r}}^{\prime }$ are ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$, respectively. The wiggly lines represent the interaction propagator ${\mathcal{L}}^A$. The interaction correction $\delta G_{\mathrm{AA}}$ has its origin at Friedel oscillations of the density matrix $\rho ({\mathbf{r}}_1,{\mathbf{r}}_2)$ at the positions ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ indicated in the figure. There are five more trajectory configurations that contribute to $\delta G_{\mathrm{AA}}$ which can be obtained by interchanging the roles of advanced and retarded Green functions.