Hubbard model with magnetic field in an infinite number of spatial dimensions

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Abstract
This paper presents an investigation of the Hubbard model with magnetic field in the limit of infinite spatial dimension. The Hubbard model is mapped onto a local atomic Hamiltonian generalized by an auxiliary Kadanoff-Baym field. This mapping has already been applied in order to solve rigorously the spinless Falicov-Kimball model, a special case of the Hubbard model. The treatment of the full model cannot be performed rigorously. Therefore the rigorous solution of an imaginary-time discretized effective Hamiltonian is introduced. This effective Hamiltonian reproduces the Lie-Trotter formula for the corresponding partition function. This leads to a rigorous lower bound on the grand-canonical free energy. The treatment allows for spatially homogeneous and antiferromagnetically ordered states where the direction of antiferromagnetism is chosen perpendicular to the external field. The asymptotic behavior of the time discretization is studied. Results are provided for the half-filled case where the magnetic configurations that are included here yield the lowest free energies. In principle, the chemical potential is not restricted to this case.

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