Quantum transmission in disordered insulators: random matrix theory and transverse localization

Abstract

We consider quantum interferences of classically allowed or forbidden electronic trajectories in disordered dielectrics. Without assuming a directed path approximation, we represent a strongly disordered elastic scatterer by its transmission matrix $\mathbf{t}$. We recall how the eigenvalue distribution of ${t.t}^{†}$ can be obtained from a certain ansatz leading to a Coulomb gas analogy at a temperature ${\beta }^{-1}$ which depends on the system symmetries. We recall the consequences of this random matrix theory for quasi--$1d$ insulators and we extend our study to microscopic three dimensional models in the presence of transverse localization. For cubes of size $L$, we find two regimes for the spectra of ${t.t}^{†}$ as a function of the localization length $\xi$. For $L/\xi \approx 1-5$, the eigenvalue spacing distribution remains close to the Wigner surmise (eigenvalue repulsion). The usual orthogonal--unitary cross--over is observed for {\it large} magnetic field change $\Delta B\approx {\Phi }_0/{\xi }^2$ where ${\Phi }_0$ denotes the flux quantum. This field reduces the conductance fluctuations and the average log--conductance (increase of $\xi$) and induces on a given sample large magneto--conductance fluctuations of typical magnitude similar to the sample to sample fluctuations (ergodic behaviour). When $\xi$ is of the order of the

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