Symmetry-projected variational approach to the one-dimensional Hubbard model

We apply a variational method devised for the nuclear many-body problem to the one-dimensional Hubbard model with nearest neighbor hopping and periodic boundary conditions. The test wave function consist for each state out of a single Hartree–Fock determinant mixing all the sites (or momenta) as well as the spin projections of the electrons. Total spin and linear momentum are restored by projection methods before the variation. It is demonstrated that this approach reproduces the results of exact diagonalizations for half-filled $N=12$ and $N=14$ lattices not only for the energies and occupation numbers of the ground but also of the lowest excited states rather well. Furthermore, a system of ten electrons in an $N=12$ lattice is investigated and, finally, an $N=30$ lattice is studied. In addition to energies and occupation numbers we present the spectral functions computed with the help of the symmetry-projected wave functions as well.

Figure 1

Energy spectrum of the half-filled $N=4$-lattice as obtained with interaction strength $U∕t=4$ either by complete diagonalization or by the LMSPHF-approach out of Sec. 2B. In the latter case each of the states is represented by a single linear momentum- and spin-projected Slater determinant.

Figure 2

Energies for some states of the half-filled $N=12$ lattice as obtained using the interaction strength $U∕t=4$ by various approaches. The different columns refer to a simple unprojected HF calculation (HF), to HF with only linear momentum (LMPHF), or only spin projection (SPHF) before the variation and to HF with restoration of the total linear momentum and only the three-component of the total spin $\left({\mathrm{LMS}}_z\mathrm{PHF}\right)$. Finally, the energies of the lowest few states obtained with linear momentum- and full spin-projection before the variation as described in Sec. 2B (LMSPHF) are compared with those resulting from a complete diagonalization (EXACT).

Figure 3

Again for the half-filled $N=12$-lattice and $U∕t=4$, the energy spectra for the lowest $S=0$, $S=1$, and $S=2$ states as obtained with the LMSPHF method for the various linear momentum quantum numbers $\xi$ are displayed. Because of the degeneracy of the $\xi =12-i$ with the $\xi =i$ spectra for $i=1\text{–}5$, only the spectra for $\xi =0\text{–}6$ are presented.

Figure 4

For the half-filled $N=12$ lattice and $U∕t=4$ the occupation numbers [Eq. (66)] of the various basis states in the ($S=0$, $\xi =6$)-LMSPHF-ground state are displayed for various interaction strengths $U∕t$. Since in all cases the exact and LMSPHF results cannot be distinguished, only the latter are displayed.

Figure 5

Again for the half-filled $N=12$ lattice and $U∕t=4$, the hole-[Eq. (65)] and particle-[Eq. (66)] spectral functions are plotted for all possible values of ${\xi }_{\pm 1}$ as functions of the energy $\omega$ ($={ϵ}_{\tilde{h}}$ and $={ϵ}_{\tilde{p}}$, respectively). The upper part of the figure has been obtained by using only the ground state determinant $\mid D^1⟩$ in the calculation for the spectral functions $(m=1)$. Using instead all the $m=5$ determinants corresponding to the lowest five states out of Fig. 2 in these calculations, one obtains the results displayed in the lower part of the figure.

Figure 6

Again for the half-filled $N=12$ lattice and $U∕t=4$, the hole spectral functions for $\xi =0\text{–}3$ as obtained with $m=5$ determinants are plotted vs the excitation energy and compared with the “exact” results computed with the Lanczos approach.

Figure 7

Same as in Fig. 3, but for only ten electrons in the $N=12$ lattice.

Figure 8

Same as in Fig. 5, but for only ten electrons in the $N=12$ lattice. Here only the results obtained with one determinant $(m=1)$ are presented.

Figure 9

Same as in Fig. 2, but for the half-filled $N=14$ lattice.

Figure 10

Same as in Fig. 3, but for the half-filled $N=14$ lattice.

Figure 11

(Color online) For the half-filled $N=14$ lattice, the hole- and the particle-spectral functions are displayed as functions of the excitation energy $\omega$ (in units of $t$) and the linear momentum $k$ in units of $(2\pi ∕N)$. The spectral functions for momenta identical to the Fermi momentum are displayed in red color. The spectral functions have been obtained by using the $m=5$ determinants corresponding to the five lowest LMSPHF solutions in Fig. 9. In order to obtain continuous functions for each state a Lorentz shape with constant width of $0.05t$ has been used.

Figure 12

Same as in Fig. 2, but for the half-filled $N=30$ lattice, also assuming $U∕t=4$. The exact result for the energy of the ground state has been determined using the Bethe–Ansatz.

Figure 13

(Color online) Same as in Fig. 11, but for the half-filled $N=30$ lattice. Only the hole part of the spectral distribution is displayed.