Cluster-based mean-field and perturbative description of strongly correlated fermion systems: Application to the one- and two-dimensional Hubbard model

We introduce a mean-field and a perturbative approach, based on clusters, to describe the ground state of fermionic strongly correlated systems. In the cluster mean-field approach, the ground-state wave function is written as a simple tensor product over optimized cluster states. The optimization of the single-particle basis where the cluster mean field is expressed is crucial in order to obtain high-quality results. The mean-field nature of the Ansatz allows us to formulate a perturbative approach to account for intercluster correlations; other traditional many-body strategies can be easily devised in terms of the cluster states. We present benchmark calculations on the half-filled 1D and (square) 2D Hubbard model, as well as the lightly doped regime in 2D, using cluster mean-field and second-order perturbation theory. Our results indicate that, with sufficiently large clusters or to second-order in perturbation theory, a cluster-based approach can provide an accurate description of the Hubbard model in the considered regimes. Several avenues to improve upon the results presented in this work are discussed.

Figure 1

In the top schemes we show an initial guess (left) and optimized (right) spin orbitals that define a single cluster in an U-cMF calculation (12-site half-filled periodic 1D lattice, $U/t=4$) using clusters of two $\uparrow$ and two $\downarrow$ orbitals with one $\uparrow$ and one $\downarrow$ electrons. Orbitals are depicted (one per row) in the 12-site lattice (marked by small $+$ signs) using the following conventions: $\uparrow$ ($\downarrow$) orbitals are plotted in blue (red); filled (empty) circles indicate a positive (negative) orbital coefficient; the area enclosed by the circle is proportional to $\left|\varphi \right(j\left)\right|$. The guess orbitals (left) correspond to Boys-localized orbitals of the UHF solution (orbitals 1 and 2 are occupied; 3 and 4 are empty). The U-cMF optimized orbitals (right) displayed are those that diagonalize the $\uparrow$ and $\downarrow$ one-particle reduced density matrix, with orbitals 1 and 2 (3 and 4) having occupation of $0.9$ ($0.1$). Note that the orbitals remain fairly local in character within a subset of two lattice sites. Accordingly, the bottom scheme shows a simplified representation of the optimized solution expressed in the on-site basis. This is characterized by local dimer-like structures, which we connect by a solid blue line.

Figure 2

Energy per site obtained in cMF calculations on a $L=120$ half-filled periodic 1D lattice at $U/t=2$ (left) and $U/t=4$ (right) as a function of the inverse of the cluster size $l$. $l=1$ R-cMF and U-cMF results correspond to restricted HF and UHF, respectively. We consider cMF results in the on-site basis and with a fully-optimized single-particle basis (opt). They are compared with (exact) calculations on a single cluster (i.e., $L=l$) using open (OBC) and periodic (PBC) boundary conditions. The latter were computed by solving the corresponding Lieb-Wu [20] equations. The energies of $L=l$ calculations with OBC exactly match those of cMF calculations in the full $L=120$ lattice using the on-site basis.

Figure 3

Fraction of correlation (with respect to UHF) recovered in restricted (left) and unrestricted (right) cMF calculations in a $L=120$ periodic 1D lattice, as a function of $U/t$. (A log-2 scale is used in $U/t$ for clarity purposes; results are shown from $U/t=1$ to 16.) Cluster sizes from 2 to 12 were used. The inset in the right panel shows the total correlation energy per site, as a function of $U/t$.

Figure 4

Fraction of correlation (with respect to UHF) recovered in unrestricted cMF and cPT2 calculations in a $L=120$ periodic 1D lattice as a function of $U/t$. UMP2 and UCCSD results are provided for comparison purposes.

Figure 5

Spectrum of the cluster Hamiltonian ${\widehat{H}}_c^0$ in U-cMF calculations on a half-filled 1D lattice at $U/t=4$, using cluster sizes of 2, 4, and 6. All clusters in the $L=120$ lattice display an identical spectrum. The eigenvalues are classified according to the $n_{\uparrow }$ and $n_{\downarrow }$ quantum numbers within the cluster. Here, $(0,0)$ corresponds to a half-filled cluster, with $l/2 \uparrow$ and $\downarrow$ electrons. Only those Hilbert sectors relevant to the evaluation of the second-order energy are shown. [Certain Hilbert sectors are missing as their spectrum is identical to one that is displayed; e.g., the $(+1,0)$ and $(0,+1)$ spectra are equivalent, and so are the $(+1,-1)$ and $(-1,+1)$.]

Figure 6

Contributions to the second-order energy in U-cPT2 calculations ($L=120$ 1D periodic lattice) using a cluster size of 4 as a function of $U/t$. The notation used in the key takes the form “# clusters: interaction type.” Here, CT stands for charge transfer and SF refers to spin flip. The interaction channels are those from Table 1; two-electron charge transfer processes are of opposite spin due to the nature of the Hubbard interaction.

Figure 7

Real-space spin-spin correlations computed from unrestricted (left) and restricted (right) optimized cMF states for a $L=120$ periodic 1D lattice at $U/t=4$. The $l=1$ result in the left panel corresponds to UHF, while that in the right panel corresponds to restricted HF (i.e., a product of plane-wave states).

Figure 8

Fourier-transformed averaged spin-spin correlations ($L=120$ 1D periodic lattice at $U/t=4$) in U-cMF and R-cMF calculations as a function of the inverse of the cluster size. Two $q$ values are plotted, namely, 0 and $\pi$ (note that the $q=0$ curve is enhanced by a factor of 50 for clarity purposes). U-cMF and R-cMF calculations should tend to the same $q=\pi$ finite value as $l\to L$. The $l\to L$ limit at $q=0$ of U-cMF results should be 0 as the exact ground state corresponds to a true singlet state.

Figure 9

Tiling patterns adopted in U-cMF calculations on a square $12\times 12$ periodic 2D square lattice at half-filling. Sites within the same cluster are connected by solid lines; fully enclosed regions are shaded. A key is provided to the left of each structure. Two colors are used for clarity purposes in patterns where the clusters have different orientations.

Figure 10

(Left) Correlation energy (with respect to UHF), divided by the UHF energy, recovered in U-cMF calculations in a $12\times 12$ periodic 2D square lattice, as a function of $U/t$. The tiling patterns considered are those displayed in Fig. 9. (Right) Difference with respect to the double occupancy per site predicted by UHF in a variety of U-cMF calculations in a $12\times 12$ periodic square lattice. The double occupancy of UHF itself is shown in the inset.

Figure 11

Same as Fig. 10. Different tiling patterns with clusters of size 6 (cf. Figs. 9 and 12) are displayed.

Figure 12

Some additional tiling patterns (with clusters of size 6) adopted in U-cMF calculations on a square $12\times 12$ periodic 2D square lattice at half-filling.

Figure 13

Same as Fig. 10. U-cMF and U-cPT2 calculations are compared with UMP2, UCCSD, and AFQMC. AFQMC results, from Ref. [22], correspond to thermodynamic limit estimates.

Figure 14

Fourier-transformed averaged spin-spin correlations ($12\times 12$ half-filled periodic 2D square lattice at $U/t=8$) in U-cMF calculations as a function of the inverse of the cluster size. UHF ($l=1$) results are also displayed. Two $\mathbf{q}$ values are plotted, namely, $(0,0)$ and $(\pi ,\pi )$.

Figure 15

Hole and spin density profiles of UHF solutions ($10\times 10$ periodic lattice, $\langle n\rangle =0.8$) obtained at $U/t=8$ with linear (left) and diagonal (right) spin density wave character. The magnitude of the hole density is proportional to the area of the green circles. The magnitude of the spin density is proportional to the size of the arrows. Superimposed on them we display the tiling patterns adopted in U-cMF calculations. Clusters with six orbitals and four electrons (solid blue) have been used in the high hole density sectors, while the Néel regions are described in terms of staggered dimers (solid red); this leads to structures $\mathbf{1}l$ and $\mathbf{1}d$, respectively, for linear and diagonal order. We have additionally considered a tiling pattern where the staggered dimers are combined into size 4 clusters, following the dotted lines, leading to structures $\mathbf{2}l$ and $\mathbf{2}d$.

Figure 16

Hole and spin density profiles from U-cMF calculations ($10\times 10$ lattice, $\langle n\rangle =0.8$) at $U/t=8$ with linear (left, $\mathbf{1}l$) and diagonal (right, $\mathbf{1}d$) ordering. The magnitude of the hole density is proportional to the area of the red circles. The magnitude of the spin density is proportional to the size of the arrows.

Figure 17

Hole and spin density profiles of UHF solutions ($16\times 8$ periodic lattice, $\langle n\rangle =0.875$) obtained at $U/t=8$ with diagonal (top) and linear (bottom) spin density wave character. The magnitude of the hole density is proportional to the area of the green circles. The magnitude of the spin density is proportional to the size of the arrows. Superimposed on the bottom scheme we display the tiling pattern adopted in U-cMF calculations. The Néel regions are described in terms of staggered dimers (solid red), leading to structure $\mathbf{1}$, while regions of high hole density are described with slab-shaped clusters of size 6 (solid blue). We have also considered a pattern (structure $\mathbf{2}$) where the dimers are combined into half-filled clusters of size 6 and 4, following the dotted lines.

Figure 18

Fourier-transform $\overline{\mathcal{S}}(q_x,q_y)$ of the averaged spin-spin correlations obtained from U-cMF (structure $\mathbf{1}$) calculations in a $16\times 8$ ($\langle n\rangle =0.875$) lattice at $U/t=4$ (top) and $U/t=8$ (bottom).

Figure 19

Fourier-transform $\overline{\mathcal{N}}(q_x,q_y)$ of the averaged density-density correlations obtained from U-cMF (structure $\mathbf{1}$) calculations in a $16\times 8$ ($\langle n\rangle =0.875$) lattice at $U/t=4$ (top) and $U/t=8$ (bottom). The strong peak at $(0,0)$ ($=N^2/L$) has been removed for clarity purposes.

Figure 20

Convergence of the $\mathrm{UMP}n$ and $\text{U-cPT}n$ perturbation series for a half-filled $L=8$ periodic 1D lattice at $U/t=4$ (left) and $U/t=8$ (right). A cluster size of 2 has been used in $\text{U-cPT}n$ calculations. The exact ground-state energy is denoted by a dashed line. Due to the divergent nature of the $\text{U-cPT}n$ series at $U/t=8$, we have truncated at order $n=48$.

Figure 21

UHF, U-cMF, and U-cPT2 ground-state energies for a half-filled $L=8$ periodic 1D dimerized Hubbard lattice [cf. Eq. (B1)] as a function of the ratio $t_2/t_1$, with $U/t_1=4$. U-cMF and U-cPT2 calculations employ clusters of size 2.