Fractal character of wave functions in one-dimensional incommensurate systems

The electronic wave functions of simple one-dimensional systems with a modulation potential incommensurate with that of the underlying lattice are determined by a direct diagonalization method. The existence of the metal-insulator transition is also obtained by a renormalization-group method. Numerical evidence for a fractal character of the wave functions is obtained and the fractal dimensionality D is calculated as a function of the strength of the modulation potential $V_0$. At the critical point $V_0$=2t, we find that D=0.80±0.15. The wave functions can also be characterized by the localization length $l_c$ and the amplitude correlation length ξ.