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

TL;DR

This paper investigates quantum interference effects in disordered insulators using random matrix theory, revealing how eigenvalue spectra and conductance fluctuations depend on system size, disorder strength, and magnetic field, especially in three-dimensional models.

Contribution

It extends random matrix theory analysis to three-dimensional disordered insulators, elucidating the spectral regimes and magnetic field effects on quantum transmission and localization.

Findings

Eigenvalue spacing remains Wigner-like for certain size-to-localization ratios.

Magnetic field induces orthogonal-unitary crossover affecting conductance.

Large magneto-conductance fluctuations are observed, similar to sample-to-sample variations.

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 ${\bf t}$. We recall how the eigenvalue distribution of ${\bf t.t}^{\dagger}$ 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 ${\bf t.t}^{\dagger}$ 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