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The Holstein-Hubbard Model at Half-filling : A Static Auxiliary Field Study

We study the Holstein-Hubbard model at half filling to explore the ordered phases such as the charge density wave and antiferromagnet. The Coulomb interaction is rewritten in terms of auxiliary fields. By treating the auxiliary fields and phonons as classical, we obtain real space features of the system and transition between the phases from weak to strong coupling. When both interactions are weak, mutual competition between them leads to a metallic phase in an otherwise insulator dominated phase diagram. Spatial correlations induced by thermal fluctuations lead to pseudogap features at intermediate range of coupling.