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

TL;DR

This paper investigates how the symmetry parameter $eta$ influences transport statistics in strongly disordered systems, revealing that its effect persists even when disorder is very high, contrary to some theoretical expectations.

Contribution

The study demonstrates through numerical analysis that the distribution forms of transport parameters depend on $eta$ regardless of disorder strength, challenging assumptions about the role of $eta$ in the strongly localized regime.

Findings

Distribution $p()$ scales as $^eta$ near zero

Distribution $p()$ approaches $ ext{exp}(-c^2)$ at large $$

The influence of $eta$ remains significant even 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 $\Delta\to\infty$.