Backflow correlations in the Hubbard model: An efficient tool for the study of the metal-insulator transition and the large-$U$ limit

We show that backflow correlations in the variational wave function for the Hubbard model greatly improve the previous results given by the Slater-Jastrow state, usually considered in this context. We provide evidence that, within this approach, it is possible to have a satisfactory connection with the strong-coupling regime. Moreover, we show that, for the Hubbard model on the lattice, backflow correlations are essentially short range, inducing an effective attraction between empty (holons) and doubly occupied sites (doublons). In the presence of frustration, we report the evidence that the metal to Mott-insulator transition is marked by a discontinuity of the double occupancy, together with a similar discontinuity of the kinetic term that does not change the number of holons and doublons, while the other kinetic terms are continuous across the transition. Finally, we show the estimation of the charge gap, obtained by particle-hole excitations à la Feynman over the ground-state wave function.

Figure 1

Upper panel: Ground-state energy as a function of the range of backflow parameters $l_{\max }$ on the 2D square lattice for $t^{\prime }/t=0.4$ and $0.75$ at $U/t=8$. Range equal to zero corresponds to no backflow correlations in the wave function. Lower panel: Optimized backflow parameters ${\eta }_l$ up to the fourth distance ($l=4$). Results for $U/t=3$ and $t^{\prime }/t=0.4$ correspond to a metallic state; the other ones correspond to insulating states. Data are shown for a lattice with $L=98$ sites.

Figure 2

Optimized backflow parameters ${\eta }_l$ up to the fourth distance ($l=4$) for the 1D lattice. The point at $U/t_1=5$ lies in the metallic phase, while the other points are located in the insulating region of the phase diagram. Data are shown for a lattice with $L=120$ sites.

Figure 3

The energy (in unit of $4t_1^2/U$), as a function of $t_1/U$, for the 1D Hubbard model with $t_2/t_1=1$. Solid (empty) circles denote the results with (without) backflow correlations. Data are presented for an $L=120$ lattice size and lines are fits of the points at strong coupling. The DMRG energy for the corresponding Heisenberg model with $J_2/J_1=1$ is shown by an arrow.[21]

Figure 4

Upper panel: Fourier transform of the optimized Jastrow factor $v_q$ multiplied by $\left|q\right|{}^2$ as a function of $\left|q\right|$, along the $(1,1)$ direction of the Brillouin zone of the square lattice, at $t^{\prime }/t=0.75$ for $U/t=7$ (triangles), $U/t=8$ (squares), and $U/t=10$ (circles). Solid (empty) symbols refer to the presence (absence) of backflow correlations, while different colors correspond to different lattice sizes: red ($L=98$), blue ($L=162$), and black ($L=242$). Lower panel: The same quantity for a 1D lattice with $t_2/t_1=0.9$. Data are shown for $U/t_1=5.4$ (squares), $U/t_1=5.6$ (circles), $U/t_1=6$ (up-triangles), $U/t_1=8$ (down-triangles), and $U/t_1=10$ (diamonds) on an $L=120$ lattice size. Solid (empty) symbols refer to the presence (absence) of backflow correlations.

Figure 5

Upper row: Hopping terms of a spin up electron from site $i$ to site $j$ that contribute to ${\mathcal{K}}_0$. Lower row: hopping terms of a spin up electron from site $i$ to site $j$ that contribute to ${\mathcal{K}}_{\pm }$. Lattice sites $i$ and $j$ are represented as boxes and spin up/down electrons inside them represent the electronic configurations.

Figure 6

Upper panel: Density of doubly occupied sites $D$ as a function of $U/t$. Lower panel: ${\mathcal{K}}_0$ (circles) and ${\mathcal{K}}_{\pm }$ (squares) as a function of $U/t$. Solid (empty) symbols refer to the presence (absence) of backflow correlations in the wave function. The metal-insulator transition is marked by an arrow. Data refer to a frustrated square lattice with $t^{\prime }/t=0.75$ and $L=162$.

Figure 7

The same as in Fig. 6, but for $t^{\prime }=0$.

Figure 8

The same as in Fig. 6, but for a 1D lattice with $t_2/t_1=0.9$ and $L=120$.

Figure 9

Charge gap $E_q$ for $q\to 0$ for the 1D Hubbard model with $t_2/t_1=0.9$, on $L=120$ and $L=240$ lattices. Results without backflow correlations for the $L=120$ case are shown for comparison. Inset: Charge gap for the 1D Hubbard model with $t_2=0$ on an $L=120$ lattice size. Axis labels are the same as those of the main plot.