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High transition temperature superconductors in cuprates exhibit the charge-density-wave fluctuations and the ferromagnetic time-reversal-symmetry-breaking fluctuation in the polar Kerr rotation experiments. We demonstrate that they share the same root of origin, and the underlying mechanism also leads to the pseudogap formation. The pseudogap formation, the charge-density-wave fluctuation, and the time-reversal-symmetry-breaking fluctuation are the consequent phenomena of the correlation. They are the basic notions in strongly correlated electron systems. Keywords: Pseudogap, Fluctuations, Charge density wave, Time-reversal-symmetry breaking, Strongly correlated electrons

The effective lattice models in strongly correlated electron systems are 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.