Energy spectrum for a fractal lattice in a magnetic field

To simulate a kind of magnetic field in a fractal environment we study the tight-binding Schr\"odinger equation on a Sierpinski gasket. The magnetic field is represented by the introduction of a phase onto each hopping matrix element. The energy levels can then be determined by either direct diagonalization or recursive methods. The introduction of a phase breaks all the degeneracies which exist in and dominate the zero-field solution. The spectrum in the field may be viewed as considerably broader than the spectrum with no field. A novel feature of the recursion relations is that it leads to a power-law behavior of the escape rate. Green's-function arguments suggest that a majority of the eigenstates are truly extended despite the finite order of ramification of the fractal lattice.