Spectral and Localization Properties for the One-Dimensional Bernoulli Discrete Dirac Operator

TL;DR

This paper investigates the spectral and dynamical properties of a one-dimensional Bernoulli Dirac operator, revealing localization phenomena and delocalization in specific cases, and compares behaviors across different potentials and masses.

Contribution

It establishes the spectral nature of the Bernoulli Dirac operator and compares dynamical behaviors for various potentials and mass parameters, including the nonrelativistic limit.

Findings

Spectrum is pure point for typical Bernoulli potentials.

Zero mass case exhibits dynamical delocalization at specific energies.

Massive case shows dynamical localization except at some energies.

Abstract

A 1D Dirac tight-binding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schr?odinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases.