Ferromagnetism in Correlated Electron Systems: A New Class of Rigorous Criteria
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Even after several decades of theoretical work the conditions for the occurence of ferromagnetism in itinerant electron systems, e. g. in the transition metals, are still not well understood. In particular, the simplest lattice model for interacting electrons, the Hubbard model (Gutzwiller, 1963; Hubbard, 1963; Kanamori, 1963), which was originally introduced to clarify precisely this problem, did not provide the hoped-for answer. We now know that the single-band Hubbard model is a generic model for the description of a correlation-induced metal-insulator transition, as well as for the formation of antiferromagnetic order, but not for ferromagnetism. Apparently the on-site interaction, which is totally independent of any lattice properties, does not easily provide a mechanism for the generation of ferromagnetism. In the Hubbard model the lattice structure enters only via the kinetic energy due to nearest-neighbor hopping. It is therefore perhaps not surprising that the rigorous proofs of the stability of ferromagnetism in this model by Nagaoka (1966), Lieb (1989), Mielke (1991, 1992) and Tasaki (1992) apply under conditions which are more specific with regard to the lattice structure than the values of the interaction. Indeed, ferromagnetism was proved to be stable either at U = ∞ (in the case of a single hole moving on certain lattices with loops (Nagaoka, 1966)), or else for all U > 0 (namely in the case of asymmetric bipartite lattices in arbitrary dimensions d > 1 at half filling (Lieb, 1989), or for special (“decorated”) lattices where the single-electron ground state has bulk degeneracy, at sufficiently large filling (Mielke, 1991, 1992; Tasaki, 1992)). For details we refer to the recent reviews by Lieb (1993) and Mielke and Tasaki (1993). Investigations of the stability of the Nagaoka state have recently led to increasingly refined bounds for the critical hole density (Hanisch and Müller-Hartmann, 1993). Nevertheless there still does not exist a rigorous proof of the stability of ferromagnetism in the Hubbard model for conventional lattices (e. g. hypercubic, bcc, fcc) and thermodynamically relevant band fillings n ≾ 1.
KeywordsHubbard Model Near Neighbor Ferromagnetic State Near Neighbor Interaction Half Filling
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