Fractional flux periodicity in tori composed of square lattice

This paper describes a study of the fractional flux periodicity of the ground state in planar systems on a square lattice whose boundary is compacted into a torus. It is found that, in the long length and large diameter limit of the torus, the groundstate energy and persistent currents exhibit a fractional period of the fundamental unit of magnetic flux depending on the twist around the torus axis. We discuss the possible relationship between the twist and genus of the torus and comment on the fractional flux periodicity in a cylinder.