The 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 π 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 ≥ 2 dimensions if there is periodicity in D − 1 directions (e.g., the cubic lattice has the lowest energy if there is flux π in each square face). PACS: 05.30.Fk, 75.10.Lp For Physical Review Letters, 73, 2158 (1994) The flux phase conjecture states that the ground state (g.s.) energy minimizing magnetic flux through a square, planar lattice on which free electrons hop is π per plaquette when the electron filling factor is 1/2 [1-4]. (Zeeman terms are excluded.) This conjecture, along with extensions to positive temperature, higher dimensional geometries and allowance for some electron-electron interactions, will be proved here. If the sites of the lattice are interpreted as atoms in a solid then flux π would correspond to magnetic fields available only on neutron stars. The significance of the flux phase is thus not primarily as a literal interpretation in terms of physical magnetic fields. One interesting interpretation concerns mean field calculations connected with superconductivity. The main interest, however, in the author’s view, is that it shows that diamagnetism (which states that the optimal flux is zero — and which is correct when the electron density is very small) can be reversed when the density is high. Indeed, it can be maximally