  We investigate interacting disordered zigzag nanoribbons at low doping, using
the Hubbard model to treat electron interactions within the density matrix
renormalization group and Hartree-Fock method. Extra electrons that are
inserted into an interacting disordered zigzag nanoribbon divide into anyons.
Furthermore, the fractional charges form a new disordered anyon phase with a
highly distorted edge spin density wave, containing numerous localized magnetic
moments residing on the zigzag edges, thereby displaying spin-charge separation
and a strong non-local correlation between the opposite zigzag edges. We make
the following new predictions, which can be experimentally tested: (1) In the
low doping case and weak disorder regime, the soft gap in the tunneling density
of states is replaced by a sharp peak at the midgap energy with two
accompanying peaks. The $e^-/2$ fractional charges that reside on the boundary
of the zigzag edges are responsible for these peaks. (2) We find that the
midgap peak disappears as the doping concentration increases. The presence of
$e-/2$ fractional charges will be strongly supported by the detection of these
peaks. Doped zigzag ribbons may also exhibit unusual transport, magnetic, and
inter-edge tunneling properties.
