Gutzwiller scheme for electrons and phonons: The half-filled Hubbard-Holstein model

We analyze the ground-state properties of strongly correlated electrons coupled with phonons by means of a generalized Gutzwiller wave function, which includes phononic degrees of freedom. We study in detail the paramagnetic half-filled Hubbard-Holstein model, where the electron-electron interaction can lead to a Mott transition and the electron-phonon coupling to a bipolaronic transition. We critically discuss the quality of the proposed wave function in describing the various transitions and crossovers that occur as a function of the parameters. Previous variational attempts are recovered as particular choices of the wave function, while keeping all the variational freedom allows to access regions of the phase diagram, otherwise inaccessible within previous variational approaches.

Figure 1

Phase diagram of the half-filled Hubbard-Holstein model in the $U-\lambda$ plane as obtained in GPW for three different values of $\gamma$. The solid lines represent the metal-insulators transitions and the dashed lines are the bipolaron formation lines obtained by imposing condition Eq. (29) as discussed in Sec. 4 (the inset shows the narrow region where a bipolaronic metal is found for $\gamma =0.2$). The thin dotted lines are the antiadiabatic $\left(\gamma \to \infty \right)$ transition lines symmetric to the $U=\lambda D$ line, also displayed. The vertical arrows in the middle panel mark the boundary of the region in which the transition to the BPI is of first order.

Figure 2

(Color online) Typical evolution of $Z$ as a function of $\lambda$ with and without Hubbard repulsion. Parameters are $\gamma =4,0.6,0.2$ from top to bottom and $U=0$ and $U=0.8U_c$ in the left and right panels, respectively. The open circles are the phonon contribution to the hopping renormalization coming from ${\left|⟨{\varphi }_1|{\varphi }_2⟩\right|}^2$.

Figure 3

GPW results (top panels) for $P\left(x\right)$ as a function of $e\text{-ph}$ coupling constant at $\gamma =0.6$ in the presence of Hubbard repulsion (right panel, $U/U_c=0.8$ and $\lambda$ ranging from 2.8 to 3.12) and with $U=0$ (left panel, $\lambda$ ranging from 1 to 1.3). The bottom panels show the lattice distribution function as obtained with VSQ for the same parameters.

Figure 4

(Color online) Comparison of the variational on-site energies obtained for the paramagnetic solution in the absence of Hubbard repulsion (upper panel) and close to the critical value $U_c=8\left|{\epsilon }_0\right|$ for the purely electronic Mott transition (lower panel) for $\gamma =0.2$.

Figure 5

(Color online) Comparison of the projected phonon wave functions for $\gamma =0.2$ as obtained with GPW and with the variational Ansätze discussed in the text. The first two panels from left are $U=0,\lambda =1$ and rightmost panel is $U=0.8U_c,\lambda =2.8$). The thin dotted line is the corresponding atomic solution.

Figure 6

(Color online) Comparison of the effective displacement and width of projected phonon wave functions in the metallic solution before the BPI is reached as obtained with GPW and with the variational Ansätze discussed in the text (parameters are $\gamma =0.6$ and $U=0.8U_c$). The thin dotted line is the atomic solution.