We study the statistics of the conductance $g$ through one-dimensional disordered systems where electron wave functions decay spatially as $\left|\psi \right|\sim \exp (-\lambda r^{\alpha })$ for $0<\alpha <1$, $\lambda$ being a constant. In contrast to the conventional Anderson localization where $\left|\psi \right|\sim \exp (-\lambda r)$ and the conductance statistics is determined by a single parameter, the mean free path, here we show that when the wave function is anomalously localized ($\alpha <1$), the full statistics of the conductance is determined by the average $\langle \ln g\rangle$ and the power $\alpha$. Our theoretical predictions are verified numerically by using a random hopping tight-binding model at zero energy, where due to the presence of chiral symmetry in the lattice there exists an anomalous localization; this case corresponds to the particular value $\alpha =1/2$. To test our theory for other values of $\alpha$, we introduce a statistical model for random hopping in the tight-binding Hamiltonian.

• Received 2 February 2012

DOI:https://doi.org/10.1103/PhysRevB.85.235450