Role of backflow correlations for the nonmagnetic phase of the $t\text{–}{t}^{\prime }$ Hubbard model

We introduce an efficient way to improve the accuracy of projected wave functions, widely used to study the two-dimensional Hubbard model. Taking the clue from the backflow contribution, whose relevance has been emphasized for various interacting systems on the continuum, we consider many-body correlations to construct a suitable approximation for the ground state at intermediate and strong couplings. In particular, we study the phase diagram of the frustrated $t-t^{\prime }$ Hubbard model on the square lattice and show that, thanks to backflow correlations, an insulating and nonmagnetic phase can be stabilized at strong coupling and sufficiently large frustrating ratio $t^{\prime }/t$.

Figure 1

(Color online) Results for 18 electrons on 18 sites as a function of $U/t$. Upper panels: Accuracy of energy $\Delta E=\left(E_0-E_v\right)$, $E_v$, and $E_0$ being the variational and the exact values, respectively. Lower panels: Overlap between the exact ground state and the variational wave functions. The BCS state with long-range Jastrow factor is denoted by blue triangles, the BCS state with backflow correlations and Jastrow term by red circles. The results for $\Delta E$ considering one Lanczos step upon the BCS state, i.e., $\left(1+\alpha \mathcal{H}\right)|\Psi ⟩$, are also shown (black squares).

Figure 2

(Color online) Variational energies per site (in unit of $J=4t^2/U$) for the BCS state with a Jastrow factor, with and without backflow correlations, and 98 sites. The results for the wave function with antiferromagnetic order and no BCS pairing are also shown. Arrows indicate the variational results obtained by applying the full Gutzwiller projection to the mean-field states for the corresponding Heisenberg models.

Figure 3

(Color online) Lower panel: Phase diagram of the frustrated $t-t^{\prime }$ Hubbard model as obtained by comparing the variational energies of the backflow wave functions. Upper panel: Comparison between the variational (VMC) energies per site (with backflow correlations) and the FN ones for $U/t=16$ and 98 sites.

Figure 4

(Color online) Upper panel: Variational (empty symbols) and FN (full symbols) results for $S\left(\pi ,\pi \right)$ divided by $N$. Both calculations have been done by using the projected BCS wave function; $U/t=16$ and $t^{\prime }/t=0$ (triangles), $U/t=24$ and $t^{\prime }/t=0.7$ (circles), and $U/t=8$ and $t^{\prime }/t=0.75$ (squares). Lines are guides to the eye. Lower panel: Variational results for $N\left(q\right)$ divided by $\left|q\right|$ for 98 (empty symbols) and 162 (full symbols) sites and $t^{\prime }/t=0.75$; from top to bottom: $U/t=4$, 6, 7, 8, and 16.