Abstract
We present a general theory of coexisting charge-density-wave (CDW) and superconductivity $d$-wave gaps for the two-dimensional (2D) Hubbard model. This motivates the description of the normal state of the underdoped cuprates by the previous fluctuation-exchange equations with a phenomenological CDW $d$-wave gap. The resulting neutron-scattering intensity, spin-lattice relaxation rate ${1/T}_{1}$, magnetic susceptibility, resistivity, and photoemission intensity are in qualitative agreement with the data on underdoped high-${T}_{c}$ cuprates. The ${T}_{c}$ decreases and the crossover temperature ${T}_{*}$ for ${1/T}_{1}T$ increases with increasing amplitude of the CDW gap.
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