Nagaoka Ferromagnetism Versus Long-Range Hopping in the One-Dimensional Hubbard-Model

The infinite-U one-dimensional Hubbard model is 'solvable' in the limit that the nearest-neighbour hopping is dominant and infinitesimal longer-range hopping lifts the one-dimensional spin degeneracy. In this limit the Hubbard model maps onto an effective spin interaction: cyclic or ring exchange with a variety of lengths and strengths. For bipartite geometries we find a region of stable ferromagnetism, which originates from the mechanism intrinsic to Nagaoka's ferromagnetism. We look at connectivities that limit exactly to the square-lattice connectivity as the range of the infinitesimal hopping diverges. When we extend this hopping range, we find that the stable region of ferromagnetism shrinks, eventually vanishing as the infinitesimal hopping range diverges and we limit to the square-lattice geometry. Since the calculation involves a non-trivial density of particles in the thermodynamic limit, it suggests how the proposed region of ferronmagnetism may become lost for the case of the two-dimensional square lattice, as numerical simulations predict.