Fractional flux periodicity in tori made of square lattice
  This paper describes a study on fractional flux periodicity of the ground state in planar systems made from a square lattice whose boundary is compacted into a torus. It is pointed out that, in the long length and large diameter limit of a torus, the ground-state energy and persistent currents show 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 a torus, and comment on fractional flux periodicity in a cylinder. 
K. Sasaki, Y. Kawazoe, R. Saito