Extended Gutzwiller Approximation for Inhomogeneous System

The generalization of the Gutzwiller approximation to inhomogeneous systems is considered, with extra spin-and-site-dependent fugacity factors included. It is found that the inclusion of fugacity factors reconciles the seemingly contradictory choices of Gutzwiller factors used in the literature. Moreover, from the derivation of the Gutzwiller factors, it is shown that the Gutzwiller approximation breaks the rotational symmetry of the trial wavefunctions, and that different components of the spin-spin interaction need to be renormalized differently under the approximation. Various schemes to restore the rotational symmetry are discussed and are compared with results from variational Monte-Carlo calculations for the two-dimensional square-lattice antiferromagnet. Results along different paths within the full parameter space, which corresponds to different choices of fugacity factors in the literature, are also compared.