Density matrix embedding from broken symmetry lattice mean fields

Several variants of the recently proposed density matrix embedding theory (DMET) [G. Knizia and G. K-L. Chan, Phys. Rev. Lett. 109, 186404 (2012)] are formulated and tested. We show that spin symmetry breaking of the lattice mean-field allows precise control of the lattice and fragment filling while providing very good agreement between predicted properties and exact results. We present a rigorous proof that at convergence this method is guaranteed to preserve lattice and fragment filling. Differences arising from fitting the fragment one-particle density matrix alone versus fitting fragment plus bath are scrutinized. We argue that it is important to restrict the density matrix fitting to solely the fragment. Furthermore, in the proposed broken symmetry formalism, it is possible to substantially simplify the embedding procedure without sacrificing its accuracy by resorting to density instead of density matrix fitting. This simplified density embedding theory (DET) greatly improves the convergence properties of the algorithm.

Figure 1

Energy per site of the 1D Hubbard model at half-filling evaluated with various embedding approximations with a fragment of two (top) and four sites (bottom). Bethe ansatz (BA) and Hartree-Fock (HF) results are added for comparison. The error with respect to BA is plotted in the bottom panel. The HF wave function is not constrained to preserve ${\widehat{S}}^2$ symmetry.

Figure 2

Comparison of double occupancy for the 1D Hubbard model at half-filling evaluated with various embedding approximations with a fragment of two (top) and four sites (bottom). Bethe ansatz (BA) results are added for comparison. The error with respect to BA is plotted in the bottom panel.

Figure 3

Energy density of the 1D Hubbard model as a function of hole doping evaluated with various embedding approximations for $U=4t$ (top), $6t$ (middle), and $8t$ (bottom). Bethe ansatz (BA) results are added for comparison. The error with respect to BA is plotted in the bottom panel.

Figure 4

Lattice filling as a function of chemical potential evaluated with various embedding approximations. The embedded fragment consists of two sites in each case. Exact Bethe ansatz (BA) values are shown for comparison.

Figure 5

Comparison of lattice filling as a function of chemical potential between the DET embedding scheme with a fragment composed of four sites and the exact Bethe ansatz (BA) results. The data corresponds to $U=4t$.

Figure 6

Spin-spin correlation function evaluated for a Hubbard ring of eight sites with DMET and DET embedding. Computations are performed with a fragment of four sites. DMRG results (deemed exact) are provided for comparison. For clarity, the results for $U=8t$ are shifted down by 0.2.

Figure 7

DET and DMET spin-spin correlation functions (SSF) evaluated for a ring of 30 sites with 30, 26, and 22 electrons and $U=4t$. Calculations are performed with a fragment of two sites. DMRG results (deemed exact) are included for reference. For clarity, data for 26 and 22 electrons are shifted by $-$0.2 and $-$0.4, respectively. The bottom panel presents the basis set coverage (see text for details).