the minimum position.

Results for the Superconducting Ground State
============================================

Case of Simple 2D Hubbard Model
-------------------------------

We studied the cases of the $d$-, extended $s$- ($s^*$-) and $s$-wave gap functions as follows: $$\begin{aligned}
\begin{array}{ccccc}
d & \Delta_{\mib k} &=& \Delta (\cos k_x - \cos k_y ),
\label{eq:dw}
      \\
s^* & \Delta_{\mib k} &=& \Delta (\cos k_x + \cos k_y ),
     \\
s & \Delta_{\mib k} &=& \Delta
\end{array}\end{aligned}$$ The sizes of the lattice we treated are $6 \times 6$ and $10 \times 10$ having electron density close to unity with slight hole doping into the half-filled state.

Results indicating the occurrence of the $d$-wave superconductivity were obtained even for the case of $N_e = 32$ on the $6 \times 6$ lattice with the periodic and the antiperiodic boundary conditions (b.c.’s) for the $x$- and the $y$-direction, respectively, with $U = 8$ and $t' = 0$. This set of b.c.’s was chosen so that $\Delta_{\mib k}$ does not vanish for any ${\mib k}$-points possibly occupied by electrons; otherwise zero division occurs in eq. (\[eq:ajl\]). $E_g/N_s$ is minimized at $\Delta \sim 0.10$. Here $g = 0.3038$ and $\mu = - 0.48$. The energy gain due to the SC gap formation, i.e., 