Two theorems on the Hubbard model

TLDR

In the Hubbard model the attractive case has a unique S = 0 ground state for even electron filling, while the repulsive bipartite half‑filled case has a unique ground state with spin S = (|B|–|A|)/2. The second theorem confirms a long‑standing conjecture for the balanced bipartite case and provides the first provable example of itinerant‑electron ferromagnetism when |B|≠|A|, holding in all dimensions without requiring a periodic lattice.

Abstract

In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.