Lieb, E. H.
THE FLUX PHASE OF THE HALF-FILLED BAND
(42K, Plain TeX)
ABSTRACT. 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 {\it 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 $D \geq 2$ dimensions if there is
periodicity in $D-1$ directions (e.g., the cubic lattice has the lowest
energy if there is flux $\pi$ in each square face). \smallskip}