The Flux-Phase of the Half-Filled Band

TL;DR

This paper proves that for a half-filled electron band on a periodic bipartite lattice, the optimal magnetic flux per plaquette is always π, extending previous results to include various interactions, temperatures, and higher dimensions.

Contribution

It generalizes the flux-phase conjecture by removing assumptions of uniform flux, including interactions, and applying to positive temperatures and higher dimensions.

Findings

Optimal flux per plaquette is π for half-filled bands.

Results apply to various lattice types including hexagonal.

The theorem holds at positive temperatures and with interactions.

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).