Renormalization-group results of electronic states in a one-dimensional system with incommensurate potentials

The density of states, localized or extended nature of the eigenstates, and the mobility edge in a one-dimensional tight-binding Hamiltonian with nearest-neighbor election hopping integral t and a sinusoidally modulated potential incommensurate with that of the underlying lattice are investigated with the renormalization-group method. A combined analysis on the local density of states and the real part of the local Green’s function reveals complicated structures of the density of states, the eigenfunctions, and the global and local mobility edges. provided that t is larger than a threshold value, which has also been determined. In this region of most interest, we found it necessary to have a chain of at least ${10}^6$ atoms in order to achieve convergent results. The rate of approaching the fixed point changes drastically across the mobility edges around which the Anderson localization transition can be either sharp or gradual.