Approximate solution of variational wave functions for strongly correlated systems: Description of bound excitons in metals and insulators

An approximate solution scheme, similar to the Gutzwiller approximation, is presented for the Baeriswyl and the Baeriswyl-Gutzwiller variational wavefunctions. The phase diagram of the one-dimensional Hubbard model as a function of interaction strength and particle density is determined. For the Baeriswyl wavefunction a metal-insulator transition is found at half-filling, where the metallic phase ($U<U_c$) corresponds to the Hartree-Fock solution, the insulating phase is one with finite double occupations arising from bound excitons. This transition can be viewed as the "inverse" of the Brinkman-Rice transition. Close to but away from half filling, the $U>U_c$ phase displays a finite Fermi step, as well as double occupations originating from bound excitons. As the filling is changed away from half-filling bound excitons are supressed. For the Baeriswyl-Gutzwiller wavefunction at half-filling a metal-insulator transition between the correlated metallic and excitonic insulating state is found. Away from half-filling bound excitons are suppressed quicker than for the Baeriswyl wavefunction.