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 wave functions. The phase diagram of the one-dimensional Hubbard model as a function of interaction strength and particle density is determined. For the Baeriswyl wave function a metal-insulator transition is found at half filling, where the metallic phase $\left(U<U_c\right)$ 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 suppressed. For the Baeriswyl-Gutzwiller wave function 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 wave function.

Figure 1

Kinetic and potential energies as a function of the variational parameter at half filling.

Figure 2

Kinetic and potential energies as a function of the variational parameter at quarter filling.

Figure 3

Comparison of the energy of the GA-B scheme to well known results: GA-G (Gutzwiller approximation applied to the GWF), exact Gutzwiller, and exact results. $U_{\text{BG}}^{\ast }$ indicates the transition point for the Baeriswyl-Gutzwiller wave function.

Figure 4

Critical interaction strength as a function of particle density. Closed circles indicate systems in which the large interaction phase contains bound excitons.

Figure 5

Double occupation as a function of interaction strength for various fillings.

Figure 6

Density as a function of $k$ vector at different fillings for various values of the interaction strength.

Figure 7

Double occupation as a function of interaction strength for the Baeriswyl-Gutzwiller wave function for filling $n=0.98$.