FLUX PHASE OF THE HALF-FILLED BAND

The conjecture is verified that the optimum, energy minimizing, magnet ic flux for a half-filled band of electrons hopping on a planar, bipar tite graph is pi per square plaquette. We require only that the graph has periodicity in one direction and the result includes the hexagonal lattice (with flux 0 per hexagon) as a special case. The theorem goes beyond previous conjectures in several ways: (1) It does not assume, a priori, that all plaquettes have the same flux (as in Hofstadter's m odel). (2) A Hubbard-type on-site interaction of any sign, as well as certain longer range interactions, can be included. (3) The conclusion holds for positive temperature as well as the ground state. (4) The r esults hold in D greater than or equal to 2 dimensions if there is per iodicity in D - 1 directions (e.g., the cubic lattice has the lowest e nergy if there is flux pi in each square face).