Flux Phase of the Half-Filled Band

The conjecture is verified that the optimum, energy minimizing, magnetic flux for a half-filled band of electrons hopping on a planar, bipartite 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 model). (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 results hold in $D\ge 2$ dimensions if there is periodicity in $D-1\mathrm{}$ directions (e.g., the cubic lattice has the lowest energy if there is flux $\pi$ in each square face).