Continuum tight binding models and quantum transport in the presence of dynamical disorder

Continuum limits of various tight binding linear chain lattices used in the author's recent study of quantum transport in the presence of dynamical disorder, are analyzed from the point of view of their energy level spectra when disorder is absent. These spectra show a linear dispersion similar to that in the Luttinger model, and describe the energy levels of the corresponding discrete systems in the range of midband wavevectors. Next, the more conventional longwavelength continuum limit, which describes the energy levels of the actual discrete systems near the bottom of the bands is discussed. On the basis of these properties it is argued that the applicability of the continuum models to the study of dynamical properties is restricted to low frequencies in the range of low lying excitations, near the midband Fermi level in half-filled band situations in the case of the midband models, and near the bottom of a nearly empty band for the longwavelength models. Finally, it is shown that in the presence of dynamic disorder the longwavelength continuum limit of a single-band tight-binding model leads to nondiffusive motion, with a mean squared displacement <x ²(t)>t ³, fort.