Cantor Spectra for Double Exchange Model

We numerically study energy spectra and localization properties of the double exchange model at irrational filling factor. To obtain variational ground state, we use a mumerical technique in momentum space by ``embedded’’ boundary condition which has no finite size effect a priori. Although the Hamiltonian has translation invariance, the ground state spontaneously exhibits a self-similarity. Scaling and multi-fractal analysis for the wave functions are performed and the scaling indices $\alpha$’s are obtained. The energy spectrum is found to be a singular continuous, so-called the Cantor set with zero Lebesque measure.