ELECTRON HOPPING IN 3-DIMENSIONAL FLUX STATES

We point out that the eigenvalue problem of an electron hopping on a lattice with arbitrarily oriented three-dimensional flux states can be reduced to a one-dimensional hopping in a suitably chosen gauge. The energy spectra and the density of states are calculated for various flux states. In general, overlapping energy bands are found. In special configurations the bands touch each other (zero gap). Where the bands touch the density of states vanishes. The energy spectrum is calculated for flux states of orientation (2-pi-p/q,2-pi-p/q,2-pi-p/q) and (pi,pi,2-pi-p/q) for all rational values with q < 38 and < 30, respectively. The spectra have some traces of the properties of the two-dimensional model. When the center of the energy band is plotted as a function of p/q a "butterfly" pattern reminiscent of Hofstadter's butterfly emerges. In many cases, the total energy of electrons can be lowered by applying an appropriate uniform magnetic field.