Comparative study of the density matrix embedding theory for Hubbard models

We examine the performance of the density matrix embedding theory (DMET) recently proposed in Knizia and Chan [Phys. Rev. Lett. 109, 186404 (2012)]. The core of this method is to find a proper one-body potential that generates a good trial wave function for projecting a large-scale original Hamiltonian to a local subsystem with a small number of bases. The resultant ground state of the projected Hamiltonian can locally approximate the true ground state. However, the lack of the variational principle makes it difficult to judge the quality of the choice of the potential. Here we focus on the entanglement spectrum (ES) as a judging criterion; accurate evaluation of the ES guarantees that the corresponding reduced density matrix well reproduces all physical quantities on the local subsystem. We apply the DMET to the Hubbard model on the one-dimensional chain, zigzag chain, and triangular lattice, and test several variants of potentials and cost functions. It turns out that ES serves as a more sensitive quantity than the energy and double occupancy to probe the quality of the DMET outcomes. A symmetric potential reproduces the ES of the phase that continues from a noninteracting limit. The Mott transition as well as symmetry-breaking transitions can be detected by the singularities in the ES. However, the details of the ES in the strongly interacting parameter region depends much on these variants, meaning that the present DMET algorithm allowing for numerous variant is insufficient to fully characterize the particular phases that require characterization by the ES.

Figure 1

(a) Schematic illustration and (b) flowchart of the DET and DMET scheme. The original system is mapped to the impurity one by the projection operator, which is constructed from the Schmidt decomposition of the ground state of the reference system. The one-body potential in the reference system is optimized such that the one-body density or density matrix of the reference system matches that of the impurity one. (c) Energy per site as the function of $U$. The energy density of the entire system deviates from the DMRG result as increasing $U$ while that of subsystem A does not.

Figure 2

(a) Ground-state energy per site and (b) double occupancy of the 1D Hubbard chain at half-filling as the function of $U$ for $N=120$ and $N_{\mathrm{A}}=4$. The bottom panel shows the error with respect to the DMRG results for $N=40$ and $m=800$ under the antiperiodic boundary condition. All types reproduce the DMRG results with an insignificant error.

Figure 3

ES, EE, and the average of the one-body potential of the 1D Hubbard chain at half-filling as the function of $U$ for (a) DET-SU(2), (b) DET-Full, (c) DMET-SU(2), and (d) DMET-Full. Data points are the DMRG results. The solid and dotted lines in each bottom panel correspond to the SU(2) symmetric (${\overline{u}}^0$) and broken (${\sum }_{\mu =x,y,z}{\overline{u}}^{\mu }$) part of the one-body potential, respectively. The DET-SU(2), in which the one-body potential is exactly zero, well reproduces the overall structure of the exact low-energy ES and EE, while in other types the ES and EE deviate from the DMRG results as the one-body potential develops, which can be observed in the shaded region. This deviation starts to develop when the charge gap ${\mathrm{\Delta }}_c$ develops as well, which is shown in the bottom panel of (d). The charge gap ${\mathrm{\Delta }}_c$ is obtained by the Bethe ansatz.

Figure 4

(a) Ground-state energy per site and (b) double occupancy of the zigzag Hubbard chain at half filling as the function of $U$ for $N=120$ and $N_{\mathrm{A}}=4$. The bottom panel shows the error with respect to the DMRG results for $N=120$ and $m=1000$ under the open boundary condition. For a large $U$, the DET-SU(2) result significantly deviates from the DMRG results, while the other types reproduce them.

Figure 5

ES, EE, and the average of the one-body potential of the zigzag Hubbard chain at half-filling as the function of $U$ for (a) DET-SU(2), (b) DET-Full, (c) DMET-SU(2), and (d) DMET-Full. Data points are the DMRG results. The DET-SU(2) reproduces the low-energy ES and EE of DMRG for the low-$U$ region, while in the high-$U$ region, the ES starts to deviate from the data points. In other types, the ES also deviates from the DMRG results. The phase transition is observed as the onset of the ${\overline{u}}^{\mu }$ ($\mu =x,y,z$) in (b) DET-Full and (d) DMET-Full. This transition is also observed as the subtle anomaly of the EE in the inset of (d), which is not observed in the DET-SU(2) results as shown in the inset of (a). The charge gap ${\mathrm{\Delta }}_c$ in the bottom panel of (d) is extracted from Ref. [78].

Figure 6

Results of the triangular lattice Hubbard model at half filling. (a) Ground-state energy per site. The inset shows the energy near the first-order phase transition. (b) Double occupancy. We observe the first-order phase transition as the drop of the double occupancy, which is clearly seen in the inset. (c)–(f) ES, EE, and average of the one-body potential as the function of $U$. The structure of the ES change when crossing the phase transition point. Note that the one-body-potential optimization of the DMET-Full fails for large $U$. (g) Ground-state energy per site and double occupancy obtained by the DMET-SU(2) and DMET-SU(2)$+\pi$-flux. The drop of the double occupancy is observed at the different $U$. (h) ES, EE, and average of the one-body potential as the function of $U$ for the DMET-SU(2)$+\pi$-flux.