Symmetry-projected variational approach for ground and excited states of the two-dimensional Hubbard model

We present a symmetry-projected configuration mixing scheme to describe ground and excited states, with well defined quantum numbers, of the two-dimensional Hubbard model with nearest-neighbor hopping and periodic boundary conditions. Results for the half-filled $2\times 4$, $4\times 4$, and $6\times 6$ lattices as well as doped $4\times 4$ systems, compare well with available results, both exact and from other state-of-the-art approximations. We report spectral functions and density of states obtained from a well-controlled ansatz for the $(N_e\pm 1)$-electron system. Symmetry projected methods have been widely used for the many-body nuclear physics problem but have received little attention in the solid state community. Given their relatively low (mean-field) computational cost and the high quality of results here reported, we believe that they deserve further scrutiny.

Figure 1

The energy spectrum, obtained via Eq. (21) for the half-filled $2\times 4$ lattice at $U=4t$ is shown in (a). This spectrum can be hardly distinguished from the one obtained using an exact diagonalization (ED). Therefore, in (b), the absolute errors are plotted for each of the predicted 120 solutions. For more details, see the main text.

Figure 2

The DOS $\mathcal{N}\left(\omega \right)$ [see Eq. (30)] for the half-filled $2\times 4$ lattice at $U=4t$ is plotted in (a) as a function of the shifted excitation energy $\omega -U/2$ (in $t$ units). Results have been obtained by approximating the ($N_e\pm 1$)-electron systems [see Eqs. (23) and (27)] with $n_T=1$ (red) and $n_T=5$ (blue) Slater determinants out of Sec. 2a. As can be observed from (b) the DOS obtained with exact diagonalization (ED) and the one obtained using $n_T=5$ HF transformations can hardly be distinguished. The hole (blue) and particle (black) spectral functions, computed with $n_T=5$ HF transformations, are plotted in (c). A Lorentzian folding of width $\Gamma =0.05t$ has been used.

Figure 3

Occupation numbers [see Eq. (26)] of the basis states in the ground state of the half-filled $2\times 4$ lattice are plotted for various $U$ strengths.

Figure 4

The energy spectrum, obtained via Eq. (21) for the half-filled $4\times 4$ lattice at $U=4t$ is shown in (a). In (b), the excitation energies from the ground state to the lowest-lying $S=1$ and $S=2$ states from (a) are plotted as functions of the linear momentum quantum numbers $\Gamma =(0,0)$, $R_1=(1,0)$, $P=(2,0)$, $R_2=(2,1)$, $Q=(2,2)$, and $R_3=(1,1)$, respectively. In addition to $U=4t$ (blue boxes), results for $U=0t$ (red diamonds) are also included for comparison.

Figure 5

The DOS $\mathcal{N}\left(\omega \right)$ [see Eq. (30)] for the half-filled $4\times 4$ lattice at $U=4t$ is plotted in (a) as a function of the shifted excitation energy $\omega -U/2$ (in $t$ units). Results have been obtained by approximating the ($N_e\pm 1$)-electron systems [see Eqs. (23) and (27)] with $n_T=5$ HF determinants. Hole (blue) and particle (black) spectral functions are displayed in (b). A Lorentzian folding of width $\Gamma =0.2t$ has been used.

Figure 6

The same as Fig. 5 but for $U=8t$. The shapes of the DOS as well as the spectral functions for momenta $(\pi ,0)$, $(\pi /2,0)$, and $(0,0)$ are qualitatively similar to the ones obtained using Lanczos calculations.[57]

Figure 7

Energy spectrum, obtained via Eq. (21), for the $4\times 4$ lattice with $N_e=15$ electrons at $U=4t$.

Figure 8

Ground-state energy of the $4\times 4$ lattice with $N_e=15$ electrons at $U=4t$ computed with various approaches. The different columns refer to the unprojected Hartree-Fock (HF) calculation, HF with linear momentum projection (LM), HF with projection of linear momentum and only the $z$ component of the total spin (LM + $S$${}_z$) and HF with projection of linear momentum and full spin projection before the variation (LM + $S$). For all these methods, we have used the approximation discussed in Sec. 2b with five transformations. Note that the LM + $S$ method corresponds to the symmetry-projected configuration mixing approach used throughout the paper. The predicted energies are compared with the exact (EXACT) one.[17] For details, see the main text.

Figure 9

The energy spectrum, obtained via Eq. (21) for the $4\times 4$ lattice with $N_e=14$ electrons at $U=4t$ is shown in (a). In (b), the excitation energies from the ground state to the lowest-lying $S=0$, 1, and $S=2$ states from (a) are plotted as functions of the linear momentum quantum numbers $\Gamma =(0,0)$, $R_1=(1,0)$, $P=(2,0)$, $R_2=(2,1)$, $Q=(2,2)$, and $R_3=(1,1)$, respectively. In addition to $U=4t$ (blue boxes), results for $U=0t$ (red diamonds) are also included for comparison.

Figure 10

The same as Fig. 5 but for the $4\times 4$ lattice with $N_e=14$ electrons at $U=4t$.

Figure 11

The same as Fig. 5 but for the $4\times 4$ lattice with $N_e=14$ electrons at $U=8t$. The shapes of the DOS as well as the spectral functions are qualitatively similar to the ones obtained using Lanczos calculations.[57]

Figure 12

The same as Fig. 4 but for the half-filled $6\times 6$ lattice at $U=4t$.

Figure 13

The same as Fig. 5 but for the half-filled $6\times 6$ lattice at $U=4t$.