Ferromagnetism in Correlated Electron Systems: Generalization of Nagaoka's Theorem

TL;DR

This paper extends Nagaoka's theorem to include various Coulomb interactions in the Hubbard model, demonstrating how exchange interactions stabilize ferromagnetism at finite Hubbard U, especially on non-bipartite lattices.

Contribution

It generalizes Nagaoka's theorem by incorporating additional Coulomb interactions and identifies conditions under which ferromagnetism remains stable at finite U.

Findings

Exchange interaction F stabilizes ferromagnetism at finite U.

For non-bipartite lattices, t ≤ 0 is required for stability.

When X = t, ferromagnetism is stable even if F=0, given certain lattice conditions.

Abstract

Nagaoka's theorem on ferromagnetism in the Hubbard model with one electron less than half filling is generalized to the case where all possible nearest-neighbor Coulomb interactions (the density-density interaction $V$, bond-charge interaction $X$, exchange interaction $F$, and hopping of double occupancies $F'$) are included. It is shown that for ferromagnetic exchange coupling ($F>0$) ground states with maximum spin are stable already at finite Hubbard interaction $U>U_c$. For non-bipartite lattices this requires a hopping amplitude $t\leq0$. For vanishing $F$ one obtains $U_c\to\infty$ as in Nagaoka's theorem. This shows that the exchange interaction $F$ is important for stabilizing ferromagnetism at finite $U$. Only in the special case $X=t$ the ferromagnetic state is stable even for $F=0$, provided the lattice allows the hole to move around loops.