\sqrt{ w_{\mib k}^2 + M^2 } )/2]^{1/2}, \\
v_{\mib k} &=& [(1 + w_{\mib k}/  
        \sqrt{ w_{\mib k}^2 + M^2 } )/2]^{1/2}, 
\label{eq:vu} \\
w_{\mib k} &=& (\varepsilon_{\mib k} - \varepsilon_{{\mib k} + {\mib Q}} )/2,\\
\varepsilon_{\mib k} &=& \xi_{\mib k} + \mu,\end{aligned}$$ where $P_{\mbox{\scriptsize G}}$ is the Gutzwiller projection operator defined by eq. (5). Summation over ${\mib k}$ and ${\mib k}'$ in eq. (20) is performed over the filled ${\mib k}$-points, as in the calculation of the normal state variational M. C. energy. ${\mib Q}$ is the SDW wave vector equal to $(\pi, \pi)$. $M$ is the SDW potential amplitude.

Phase Boundary between SC and SDW States
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As shown by the open diamonds in Fig. \[fig:gain-r2\], the energy gain per site in the SDW state rises very sharply from $\rho \sim 0.84$ in the case of $t' = 0$. Already at $\rho = 0.84$ it takes $0.0023$ and is slightly larger than that in the SC state. However at $\rho = 0.80$ there is no more stable SDW state. By extrapolating the sharply rising curve of the SDW energy gain per site as a function of $\rho$, the boundary between the SDW and the SC states is given 