coefficients, $ \Delta _{ {\bf k} }$ is the DDW order parameter, and $ c _{ {\bf k} \text{ } \sigma }$ and $ c _{ {\bf k}+{\bf Q} \text{ } \sigma }$ are to annihilate electrons at $({\bf k} , \sigma )$ and $({\bf k} +{\bf Q}, \sigma )$, respectively. At half-filling, the zero-temperature chemical potential may lie in the gap between the two DDW energy bands, under which the gapless quasielectrons appear at the nodal points [@Gerami; @Chakravarty]. The lower band is filled with the AF quasielectrons while the upper one is empty. Each filled eigenket $\varphi_{f}$ can be denoted by the two-component wavefunction $$\begin{aligned}
| \varphi _{ f } \rangle =  \left[
\begin{array}{c} 
\varphi  ^{\prime} _{ f }  \\   \varphi ^{\prime \prime} _{ f } 
\end{array}
\right]\end{aligned}$$ with the plane waves $\varphi  ^{\prime} _{ f } = a _{f} ^{\prime} exp (i {\bf k}_{f} \cdot {\bf r})$ and $\varphi ^{\prime \prime} _{ f }  = a _{f} ^{ \prime \prime}  exp (i ({\bf k}_{f} + {\bf Q} ) \cdot {\bf r})$. Here ${\bf k}_{f} \in $ RBZ, and the coefficients $ a _{f} ^{\prime}$ and $ a _{f} ^{ \prime \prime}$ satisfying $( a _{f} ^{ \prime 