Abstract. A 1D Dirac tight-binding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrödinger 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.
We study spectral properties of some discrete Dirac operators with nonzero potential only at some sparse and suitably randomly distributed positions. As observed in the corresponding Schrödinger operators, we determine the Hausdorff dimension of its spectral measure and identify a sharp spectral transition from point to singular continuous. C
An 1D tight-binding version of the Dirac equation is considered; after checking that it recovers the usual discrete Schrödinger equation in the nonrelativistic limit, it is found that for two-valued Bernoulli potentials the zero mass case presents absence of dynamical localization for specific values of the energy, albeit it has no continuous spectrum. For the other energy values (again excluding some very specific ones) the Bernoulli Dirac system is localized, independently of the mass.
We establish dynamical localization for random Dirac operators on the d-dimensional lattice, with d ∈ {1, 2, 3}, in the three usual regimes: large disorder, band edge and 1D. These operators are discrete versions of the continuous Dirac operators and consist in the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by different scalar potentials, which are sequences of independent and identically distributed random variables according to an absolutely continuous probability measure with bounded density and of compact support. We prove the exponential decay of fractional moments of the Green function for such models in each of the above regimes, i.e., (j) throughout the spectrum at larger disorder, (jj) for energies near the band edges at arbitrary disorder and (jjj) in dimension one, for all energies in the spectrum and arbitrary disorder. Dynamical localization in theses regimes follows from the fractional moments method. The result in the one-dimensional regime contrast with one that was previously obtained for 1D Dirac model with Bernoulli potential.
Quantum dynamical lower bounds for continuous and discrete one-dimensional Dirac operators are established in terms of transfer matrices. Then such results are applied to various models, including the Bernoulli-Dirac one and, in contrast to the discrete case, critical energies are also found for the continuous Dirac case with positive mass.
Consider the family of Schrödinger operators (and also its Dirac version) on 2 (Z) or 2 (N)where S is a transformation on (compact metric) Ω, F is a real Lipschitz function and W is a (sufficiently fast) power-decaying perturbation. Under certain conditions it is shown that H W ω,S presents quasi-ballistic dynamics for ω in a dense G δ set. Applications include potentials generated by rotations of the torus with analytic condition on F , doubling map, Axiom A dynamical systems and the Anderson model. If W is a rank one perturbation, examples of H W ω,S with quasi-ballistic dynamics and point spectrum are also presented.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.