Multireference symmetry-projected variational approximation for the ground state of the doped one-dimensional Hubbard model

The few determinant (FED) approximation introduced in our previous work [Phys. Rev. B 87, 235129 (2013)] is used to describe the ground state, characterized by well-defined spin and space group symmetry quantum numbers as well as doping fractions $N_e/N_{\mathrm{sites}}$, of one-dimensional Hubbard lattices with nearest-neighbor hopping and periodic boundary conditions. Within this multireference scheme, each ground state is expanded in a given number of nonorthogonal and variationally determined symmetry-projected configurations. The results obtained for the ground-state and correlation energies of half-filled and doped lattices with 30, 34, and 50 sites compare well with the exact Lieb-Wu solutions as well as with those obtained with other state-of-the-art approximations. The structure of the intrinsic symmetry-broken determinants resulting from the variational procedure is interpreted in terms of solitons whose translational and breathing motions can be regarded as basic units of quantum fluctuations. It is also shown that in the case of doped one-dimensional lattices, a part of such fluctuations can also be interpreted in terms of polarons. In addition to momentum distributions, both spin-spin and density-density correlation functions are studied as functions of doping. The spectral functions and density of states, computed with an ansatz whose quality can be well controlled by the number of symmetry-projected configurations used to approximate the $N_e\pm 1$ electron systems, display features beyond a simple quasiparticle distribution, as well as spin-charge separation trends.

Figure 1

The ratio of correlation energies $\kappa$ obtained with the FED approximation is plotted as a function of the inverse of the number of GHF transformations for a lattice with $N_{\mathrm{sites}}=30$ sites and $N_e=22$ electrons. Results are shown for onsite repulsions of $U=2t$, 4$t$, and 8$t$. For details, see the main text.

Figure 2

The quantities ${\xi }^i\left(j\right)$ [(a)–(d)] and ${\xi }_{cd}^i\left(j\right)$ [(e)–(h)] are plotted for a 1D lattice with $N_{\mathrm{sites}}=30$ sites and $N_e=18$ electrons at $U=2t$, as a function of lattice site $j$ for some typical symmetry-broken GHF determinants resulting from the FED VAP optimization. Results corresponding to the standard RHF approximation are plotted in red for comparison. For more details, see the main text.

Figure 3

Same as Fig. 2 but for $U=4t$. Results corresponding to the standard UHF approximation are plotted in red for comparison.

Figure 4

Same as Fig. 2 but for $U=8t$. Results corresponding to the standard GHF approximation are plotted in red for comparison.

Figure 5

The ground-state momentum distributions [Eq. (12)] for $N_{\mathrm{sites}}=30$ lattices with 14 (orange triangles), 18 (magenta triangles), 22 (cyan triangles), 26 (green triangles), and 30 (red triangles) electrons are shown for $U=2t$ (a), 4$t$ (b), and 8$t$ (c). DMRG results (open black circles) are also included in panels (a)–(c) for comparison. The FED values (filled triangles) are compared in panel (d) with a power-law [Eq. (13)] fitting (dashed lines) of the momentum distributions.

Figure 6

Fourier transforms of the ground-state spin-spin correlation functions in real space for $N_{\mathrm{sites}}=30$ lattices with 14 (orange diamonds), 18 (magenta diamonds), 22 (cyan diamonds), 26 (green diamonds), and 30 (red triangles) electrons are shown for $U=2t$ (a), 4$t$ (b), and 8$t$ (c). DMRG results (black triangles) are also included for comparison. Starting with 26 electrons, all the curves have been successively shifted by 0.3 to accommodate them in a single plot.

Figure 7

Maxima of Fourier transforms of the FED ground-state spin-spin correlation functions in real space [Eq. (14)] plotted as functions of ln$\delta$, with $\delta$ being the corresponding doping parameter. Results are shown for onsite repulsions $U=2t$ (red diamonds), 4$t$ (blue diamonds), and 8$t$ (green diamonds). DMRG values are plotted with open circles. A straight line has been fitted to guide the eye. For more details, see the main text.

Figure 8

Same as Fig. 6 but for the Fourier transforms of the ground-state density-density correlation functions in real space. Starting with 26 electrons, all the curves have been successively shifted by 0.2 to accommodate them in a single plot.

Figure 9

Inflection points at wave vector $q=2k_F$ of the FED Fourier-transformed density-density correlation functions in real space [Eq. (15)] are plotted as functions of ln$\delta$ for $U=2t$ (red diamonds), 4$t$ (blue diamonds), and 8$t$ (green diamonds). DMRG values are plotted with open circles. For more details, see the main text.

Figure 10

The hole (black) and particle (blue) SFs for a lattice with $N_{\mathrm{sites}}=30$ sites and $N_e=26$ electrons are plotted in panel (a) as functions of the excitation energy $\omega$ (in $t$ units). The DOS (black) $\mathcal{N}\left(\omega \right)$ is compared in panel (b) with the one (red) corresponding to half-filling. The latter has been shifted [i.e., $\mathcal{N}\left(\omega \right)+40$] for the sake of clarity. Results are shown for the onsite repulsion $U=4t$. A Lorentzian folding of width $\Gamma =0.05t$ has been used. For more details, see the main text.

Figure 11

The ground-state momentum distributions [Eq. (12)] for $N_{\mathrm{sites}}=34$ lattices with 18 (magenta triangles), 26 (green triangles), and 34 (red triangles) electrons are shown for $U=4t$. DMRG results (open black circles) are also included for comparison.