Magnetic translation symmetry on the lattice
  Magnetic translation symmetry on a finite periodic square lattice is investigated for an arbitrary uniform magnetic field in arbitrary dimensions. It can be used to classify eigenvectors of the Hamiltonian. The system can be converted to another system of half or lower dimensions. A higher dimensional generalization of Harper equation is obtained for tight-binding systems. 
Ken-ichi Sekiguchi, Tomohiro Okamoto and Takanori Fujiwara