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A Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems

We perform a detailed numerical study of the conductance $G$ through one-dimensional (1D) tight-binding wires with on-site disorder. The random configurations of the on-site energies $ε$ of the tight-binding Hamiltonian are characterized by long-tailed distributions: For large $ε$, $P(ε)\sim 1/ε^{1+α}$ with $α\in(0,2)$. Our model serves as a generalization of 1D Lloyd's model, which corresponds to $α=1$. First, we verify that the ensemble average $\left\langle -\ln G\right\rangle$ is proportional to the length of the wire $L$ for all values of $α$, providing the localization length $ξ$ from $\left\langle-\ln G\right\rangle=2L/ξ$. Then, we show that the probability distribution function $P(G)$ is fully determined by the exponent $α$ and $\left\langle-\ln G\right\rangle$. In contrast to 1D wires with standard white-noise disorder, our wire model exhibits bimodal distributions of the conductance with peaks at $G=0$ and $1$. In addition, we show that $P(\ln G)$ is proportional to $G^β$, for $G\to 0$, with $β\leα/2$, in agreement to previous studies.