Compact numerical solutions to the two-dimensional repulsive Hubbard model obtained via nonunitary similarity transformations

Similarity transformation of the Hubbard Hamiltonian using a Gutzwiller correlator leads to a non-Hermitian effective Hamiltonian, which can be expressed exactly in momentum-space representation and contains three-body interactions. We apply this methodology to study the two-dimensional Hubbard model with repulsive interactions near half filling in the intermediate interaction strength regime $\left(U/t=4\right)$. We show that at optimal or near optimal strength of the Gutzwiller correlator, the similarity-transformed Hamiltonian has extremely compact right eigenvectors, which can be sampled to high accuracy using the full configuration interaction quantum Monte Carlo (FCIQMC) method and its initiator approximation. Near-optimal correlators can be obtained using a simple projective equation, thus obviating the need for a numerical optimization of the correlator. The FCIQMC method, as a projective technique, is well suited for such non-Hermitian problems, and its stochastic nature can handle the three-body interactions exactly without undue increase in computational cost. The highly compact nature of the right eigenvectors means that the initiator approximation in FCIQMC is not severe and that large lattices can be simulated, well beyond the reach of the method applied to the original Hubbard Hamiltonian. Results are provided in lattice sizes up to 50 sites and compared to auxiliary-field QMC. New benchmark results are provided in the off-half-filling regime, with no severe sign problem being encountered. In addition, we show that methodology can be used to calculate excited states of the Hubbard model and lay the groundwork for the calculation of observables other than the energy.

Figure 1

Top: $L^2$ norm of the HF state $c_{HF}^{2(L/R)}$ and within the HF determinant and double excitations, $L_{(0,2)}^{(L/R)}$, for a half-filled 6-site Hubbard chain with periodic boundary conditions at $U/t=4$ and $\mathbf{k}=0$ for the left $|{\mathrm{\Phi }}_0^L\rangle$ and right $|{\mathrm{\Phi }}_0^R\rangle$ ground-state eigenvector of $\overline{H}$ as a function of $-J$. Bottom: The Hartree-Fock energy $E_{HF}\left(J\right)=\langle {\mathrm{\Phi }}_{HF}|\overline{H}\left(J\right)|{\mathrm{\Phi }}_{HF}\rangle$ as a function of $-J$. The dash-dotted line indicates the exact ground-state energy $E_{ex}$ for this system, and since $\overline{H}$ is not Hermitian $E_{HF}$ can drop below the exact energy. Also indicated are the results of minimizing the variance of the energy of the similarity-transformed Hamiltonian $E_{\mathrm{var}}$ by Tsuneyuki [72], the result of solving Eq. (22) $E_{HF}\left(J_{\mathrm{opt}}\right)$, and the result from a VMC optimization $E_{\mathrm{VMC}}$ [96]. The vertical dashed line indicates $J_{\max }$, where $L_{(0,2)}^2$ of $|{\mathrm{\Phi }}_0^R\rangle$ is maximal. All energies are in units of $t$ and the two panels share the $x$ axis.

Figure 2

Histogram of $|{\overline{H}}_{ij}|/p_{ij}$ ratios for the half-filled, 50-site, $U/t=4$ Hubbard model with periodic boundary conditions for uniform, weighted, and mixed generation probabilities.

Figure 3

The error of the energy per site $|e_J-e_{ex}|$ for the half-filled, 18-site Hubbard model for the original $J=0$ and different strengths of the correlation parameter $J$ at $\mathbf{k}=0$ for $U/t=2$ (left) and $U/t=4$ (right) versus the walker number $N_w$. The dashed lines indicate the statistical errors of the $N_w={10}^6$ results with $n_{\mathrm{init}}=1.2$ for $U/t=2$ and of the $N_w=5\times {10}^7$ with $n_{\mathrm{init}}=2.0$ for $U/t=4$. The exact reference results were obtained by Lanczos diagonalization [99].

Figure 4

$L^2$ norm captured within the HF determinant $c_{HF}^{2(L/R)}$ and additionally double excitations, $L_{(0,2)}^2$, for the half-filled, 18-site Hubbard model at $U/t=4$ for the left and right ground-state eigenvectors of the non-Hermitian similarity-transformed Hamiltonian (19) as a function of $-J$. The results were obtained by a noninitiator ST-FCIQMC calculation. $|{\mathrm{\Phi }}_L\rangle$ was sampled by running the simulation with positive $J$, which corresponds to conjugating $\overline{H}$. $J_{\max }$ indicates the position of the maximum of $L_{(0,2)}^2$ and $J_{\mathrm{opt}}$ is the result of solving Eq. (22).

Figure 5

The $L^2$ norm contained in each excitation level relative to the HF determinant, indicated by excitation level 0, for the half-filled, 18-site, $U/t=4,\mathbf{k}=0$ Hubbard model for different values of $J$. The inset shows the tail of the same data on a logarithmic scale.

Figure 6

Error of the energy per site versus the excitation level truncation $n_{\mathrm{trunc}}$ in the half-filled, 18-site, $U/t=4,\mathbf{k}=0$ Hubbard model for different values of $J$. The inset shows the absolute error on a logarithmic scale. The dashed lines in the inset indicate the statistical error for the $n_{\mathrm{trunc}}=8$ results for each value of $J$.

Figure 7

Energy per site error compared to exact Lanczos results [99] for the $14e^{-}$ in the 18-site, $U/t=4,\mathbf{k}=0$ Hubbard model for the (a) ground, (b) first excited, and (c) second excited state as a function of walker number $N_w$. All three panels share the $x$ and $y$ axes. The dashed lines indicate the statistical errors of the $N_w={10}^7$ for each value of $J$.

Figure 8

The three different square lattices studied in this paper. The 18- and 50-site lattices have tilted periodic boundary conditions with lattice vectors ${\mathbf{R}}_1=(3,3),{\mathbf{R}}_2=(3,-3)$ and ${\mathbf{R}}_1=(5,5),{\mathbf{R}}_2=(5,-5)$, respectively. The 36-site lattice is studied with periodic and mixed, periodic along the $x$ axis and antiperiodic boundary conditions along the $y$ axis. The filled red circles in the 18-site lattice indicate the doubly occupied states and the half-filled green circles the singly occupied states in the subspace studied in Sec. 4a2.

Figure 9

The energy per site versus the excitation level truncation $n_{\mathrm{trunc}}$ in the half-filled, 50-site, $U/t=4,\mathbf{k}=0$ Hubbard model. AFQMC reference [108] and nontruncated i-ST-FCIQMC results are also shown. Similarly to the 18-site system at half-filling (Fig. 6), the energies are well converged at 6-fold excitations.

Figure 10

Error of the first 10 eigenstate energies obtained by the projected energy $e_p$ and shift energy $e_s$ for the $6e^{-}$ in the 6-site 1D periodic Hubbard model at $U/t=4$ and $\mathbf{k}=0$ compared to exact diagonalization results. The horizontal dashed lines indicate the averaged statistical errors. The green pluses show the difference of the exact overlaps from the overlaps obtained within FCIQMC; see Eq. (C1). A correlation parameter of $J=-0.1$, initiator threshold of $n_{\mathrm{init}}=1.2$, and maximum walker number of $N_w={10}^5$ were used. (a) Exact overlap is ill defined due to degeneracy of states 6 and 7.