Abstract
The effective lattice models in strongly correlated electron systems are \emph{derived} in particular for the cuprate superconductors, that incorporate the quantum fluctuations of the spin Berry's phase and the antiferromagnetic fluctuation. Consistent with the field-theoretical approach, the density modulation, the weak ferromagnetism, and the superconductivity are reproduced. We discussed the pros and cons of the effective-model approach and demonstrate that both positive and negative Hubbard model are the effective models subject to the occurrence of the quantum fluctuations of the correlation degrees of freedom.
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