On the role of the symmetry parameter $\beta$ in the strongly localized regime

Abstract

The generalization of the Dorokhov-Mello-Pereyra-Kumar equation for the description of transport in strongly disordered systems replaces the symmetry parameter $\beta$ by a new parameter $\gamma$, which decreases to zero when the disorder strength increases. We show numerically that although the value of $\gamma$ strongly influences the statistical properties of transport parameters $\Delta$ and of the energy level statistics, the form of their distributions always depends on the symmetry parameter $\beta$ even in the limit of strong disorder. In particular, the probability distribution is $p(\Delta)\sim \Delta^\beta$ when $\Delta\to 0$ and $p(\Delta) \sim \exp (-c\Delta^2)$ in the limit $\Delta\to\infty$.