Divergence of the Classical trajectories and Weak Localization

We study the weak localization correction (WLC) to transport coefficients of a system of electrons in a static long-range potential (e.g. an antidot array or ballistic cavity). We found that the weak localization correction to the current response is delayed by the large time $t_E = \lambda^{-1} |\ln \hbar|$, where $\lambda$ is the Lyapunov exponent. In the semiclassical regime $t_E$ is much larger than the transport lifetime. Thus, the fundamental characteristic of the classical chaotic motion, Lyapunov exponent, can be found by measuring the frequency or temperature dependence of WLC.