end part of the large error bar attached to the data point. In the above-mentioned figure the energy gain vanishes around $\rho \sim 0.56$ but in Fig. \[fig:gain-r\] of the present work it vanishes around $\rho \sim 0.72$. We believe that the present results are more improved than the results of Giamarchi et al. because of improved accuracy of computation.

Competition with the SDW State
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Magnetic State of the 2D Hubbard Model
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When the density of doped holes is small or zero, the 2D Hubbard model takes an antiferromagnetic state as its ground state. With increase of doped hole density, the magnetic order is destroyed and SC appears. At what hole density does the SC state start? We have investigated the transition between the pure SC and the pure uniform SDW states by computing the energy of the SDW state by means of the variational Monte Carlo method. The trial SDW wave function is written as \[\] $$\begin{aligned}
\Psi_{\mbox{\scriptsize SDW}} &=& P_{\mbox{\scriptsize G}}
                            \psi_{\mbox{\scriptsize SDW}}, \\
\psi_{\mbox{\scriptsize SDW}} &=& 
\prod_{\mib k} (u_{\mib k}c_{{\mib k}\uparrow}^{\dag}
        + v_{\mib k}c_{{\mib k}+{\mib Q}\uparrow}^{\dag})
\prod_{{\mib k}'} (u_{{\mib k}'}c_{{\mib k}'\downarrow}^{\dag}
        - v_{{\mib k}'}c_{{\mib k}'+{\mib Q}\downarrow}^{\dag})
                        |0>, \\
u_{\mib k} &=& [(1 - w_{\mib k}/  
        