Coexistence of $d_{x^2-y^2}$ superconductivity and antiferromagnetism in the two-dimensional $t-J$ model and numerical estimation of Gutzwiller factors

The coexisting phase of the d-wave superconducting and the antiferromagnetic order in the two-dimensional t-$J$ model is reexamined using a variational Monte Carlo method in order to estimate the Gutzwiller factors. A trial wave function is used which is consistent with the mean-field solution. The Gutzwiller factors are determined by comparing the numerically obtained expectation values with the mean-field values. We find the z component of the Gutzwiller factor of exchange interaction is enhanced compared with the $\mathrm{xy}$ components. This effect does not appear in the conventional Gutzwiller approximation and is essential for the stabilization of the antiferromagnetic order in the presence of d-wave superconductivity. We discuss several applications of the present Gutzwiller factors to the problems with antiferromagnetism and d-wave superconductivity.