Stabilization of flux states on two-dimensional lattices

The total energy of two-dimensional electrons in a uniform magnetic field is systematically calculated for the square lattice, the triangular lattice, and the honeycomb lattice for various ratios of transfer integrals. It has many cusps as a function of the magnetic field at which the Fermi energy jumps across a gap. For a fixed electron density, the lowest energy with respect to the magnetic field (including the zero-field case) is realized when the magnetic field gives one flux unit per electron in agreement with the proposal of Hasegawa, Lederer, Rice, and Wiegmann [Phys. Rev. Lett. 63, 907 (1989)]. The density of states is calculated analytically for the square lattice. The anyon lattice gas, which obeys fractional statistics, is discussed. In the mean-field treatment of the flux, the boson gas has the lowest energy.