Transport in chaotic quantum dots: Effects of spatial symmetries which interchange the leads

We investigate the effect of spatial symmetries on phase coherent electronic transport through chaotic quantum dots. For systems which have a spatial symmetry that interchanges the source and drain leads, we find in the framework of random matrix theory that the density of the transmission eigenvalues is independent of the number of channels $N$ in the leads. As a consequence, the weak localization correction to the conductance vanishes in these systems, and the shot noise suppression factor $F$ is independent of $N$. We confirm this prediction by means of numerical calculations for stadium billiards with various lead geometries. These calculations also uncover transport signatures of partially preserved symmetries.

Figure 1

Distribution of transmission eigenvalues $\rho \left(\tau \right)$ (normalized to the sample size), for $N=1$, in stadium geometries with lead-transposing left-right symmetry [(a), (b)] and without global symmetry [(c), (d)]. The numerical data (histogram) are compared with the corresponding RMT predictions (solid lines). Due to the horizontal alignment of the entrance lead in (c) a fraction of back-reflected paths have an up-down reflected partner trajectory of equal length (see, e.g., pair of dashed lines). Although for a class of trajectories the corresponding partner is cut short by the exit lead (see, e.g., pair of solid lines) transport is strongly influenced by this partial symmetry. Cavity area (a) $A=(4+\pi )∕2$, (b), (c), (d) $A=4+\pi$, lead widths (a) $d=0.125$, (b), (c), (d) $d=0.25$, and entrance lead tilt angle in (d) $\delta =11.8°$.

Figure 2

Energy-averaged Fano factor in different mode intervals. Upper panel: Numerical data for systems with a lead-transposing symmetry [see Figs. 1a, 1b] compared with the corresponding mode-independent random-matrix prediction ($F=1∕4$, solid line). Lower panel: Analogous plot for systems without global symmetry [see Figs. 1c, 1d]. The deviations to the RMT result [solid line, numerically obtained from Eq. (3) for $N>2$] for the geometry depicted in Fig. 1c can be explained in terms of the partial symmetry of short trajectories in the structure.