Aharonov-Bohm effect in higher genus materials

Flux periodicity of conducting electrons on a closed surface with genus two $g=2$ (double torus) are investigated theoretically. We examine flux periodicity of the ground-state energy and of the wave functions as a function of applied magnetic field. A fundamental flux period of the ground-state energy is twice a fundamental unit of magnetic flux for uniformly applied magnetic field, which is shown to be valid for a simple ladder geometry and carbon double torus. Flux periodicity of the wave functions in a double torus is complicate as compared with a simple torus ($g=1$), and an adiabatic addition of magnetic fluxes does not provide a good quantum number for the energy eigenstates. The results are extended to higher genus materials and the implications of the results are discussed.