Semiclassical Theory of Chaotic Conductors

We calculate the Landauer conductance through chaotic ballistic devices in the semiclassical limit, to all orders in the inverse number of scattering channels without and with a magnetic field. Families of pairs of entrance-to-exit trajectories contribute, similarly to the pairs of periodic orbits making up the small-time expansion of the spectral form factor of chaotic dynamics. As a clue to the exact result we find that close self-encounters slightly hinder the escape of trajectories into leads. Our result explains why the energy-averaged conductance of individual chaotic cavities, with disorder or “clean,” agrees with predictions of random-matrix theory.

Figure 1

Pairs of trajectories $\gamma ,{\gamma }^{\prime }$ contributing to the conductance; close self-encounters highlighted by thick lines; trajectory pieces from leads to and in between encounter stretches, termed loops, depicted as thin lines: (a) a Richter-Sieber pair differing in a 2-encounter, (b) a pair of trajectories differing in one 2-encounter and one 3-encounter. In (a), $\gamma$ and ${\gamma }^{\prime }$ differ in the sense of traversal of one loop ($\nu =1$), and the relative orientations of stretches in the 2-encounter are such that $\mu =0$. For (b) we have $\nu =3$; $\mu =-1$ for the 2-encounter and $\mu =1$ for the 3-encounter.