Cluster size convergence of the density matrix embedding theory and its dynamical cluster formulation: A study with an auxiliary-field quantum Monte Carlo solver

We investigate the cluster size convergence of the energy and observables using two forms of density matrix embedding theory (DMET): the original cluster form (CDMET) and a new formulation motivated by the dynamical cluster approximation (DCA-DMET). Both methods are applied to the half-filled one- and two-dimensional Hubbard models using a sign-problem free auxiliary-field quantum Monte Carlo impurity solver, which allows for the treatment of large impurity clusters of up to 100 sites. While CDMET is more accurate at smaller impurity cluster sizes, DCA-DMET exhibits faster asymptotic convergence towards the thermodynamic limit. We use our two formulations to produce new accurate estimates for the energy and local moment of the two-dimensional Hubbard model for $U/t=2,4,6$. These results compare favorably with the best data available in the literature, and help resolve earlier uncertainties in the moment for $U/t=2$.

Figure 1

Translational symmetry in DMET. (a) The original lattice with translational symmetry, divided into three supercells. (b) The CDMET impurity cluster with broken intracluster translational symmetry, between the center site and the edge sites. (c) The DCA-DMET impurity cluster restores the intracluster translational symmetry through a basis transformation and interaction coarse graining.

Figure 2

Definition of the real (left) and reciprocal (right) lattice vectors for the DCA transformation for a “hypercubic” cluster with $L_c=2$. The intercluster component of the real lattice vector $\tilde{\mathbf{r}}$ labels the origin of the cluster, and the intracluster component $\mathbf{R}$ labels the site within the cluster. The reciprocal space of $\tilde{\mathbf{r}}$ and $\mathbf{R}$ are labeled by $\tilde{\mathbf{k}}$ and $\mathbf{K}$, respectively.

Figure 3

Sum-of-square of the one-body impurity-environment coupling Hamiltonian $|h_c{|}^2={\sum }_{i\in C=0,j\in C^{\prime }\ne 0}{\left|h_{ij}\right|}^2$ for the CDMET and DCA formulations, in one dimension. The fittings follow constant (CDMET) and $1/L_c$ (DCA) scalings, respectively.

Figure 4

Energy per site $e$ for the half-filled 1D Hubbard model versus inverse impurity size, $1/L_c$, from CDMET (blue) and DCA-DMET (red). For comparison, we also plot the same numbers from AFQMC with PBC (purple) and TABC (orange) for $U/t=4$. The extrapolations use $e=a+b{L}_c^{-1}+c{L}_c^{-2}$ for CDMET and $e=a+b{L}_c^{-2}+c{L}_c^{-3}$ for DCA-DMET. (a) $U/t=4$, (b) $U/t=8$.

Figure 5

Spin order in the 1D Hubbard model. (a) Local spin moments $m$ from CDMET (blue) and DCA-DMET (red) in finite impurity cluster calculations at $U/t$=4. $x$ is the site index scaled to the interval $[0,1]$ for the CDMET results. (b) and (c) CDMET AF order parameters $m\left(L_c\right)$ divided by spin correlation function $S{(L_c/2)}^{1/2}$, versus inverse impurity cluster size $1/L_c$ for $U/t=4$ and $U/t=8$ (blue, center average; green, entire cluster average). The extrapolation uses the form $m\left(L_c\right)/S{(L_c/2)}^{1/2}=a+b{L}_c^{-1}+c{L}_c^{-2}$; see Eq. (24) for details. (d) and (e) DCA-DMET and CDMET (center average) AF order parameters $m\left(L_c\right)$ versus inverse impurity cluster size $1/L_c$ for $U/t=4$ and $U/t=8$. The extrapolation for DCA-DMET values uses the form $m\left(L_c\right)=a+b{L}_c^{-1}+c{L}_c^{-2}$; see Eq. (26) for details.

Figure 6

Energy per site $e$ versus $1/L_c$ in the 2D Hubbard model from CDMET (blue), DCA-DMET (red), and finite system AFQMC (orange, TABC; purple, PBC; brown, APBC for $y$ direction and PBC for $x$ direction) (from Ref. [18]). The consensus range illustrated by the gray-shaded region represents the TDL results of AFQMC, DMRG, and iPEPS calculations in Refs. [14, 17, 18, 41]. (a) $U/t=2$. (b) $U/t=4$. (c) $U/t=6$.

Figure 7

Antiferromagnetic order parameter $m$ versus $1/L$ in the 2D Hubbard model from CDMET (blue), DCA-DMET (red), and finite system AFQMC using TABC [18] (orange) and modified boundary conditions [17] (cyan). The DMET results extrapolate to the TDL uses the form $m\left(L\right)=a+b{L}_c^{-1}+c{L}_c^{-2}$. (Insets) CDMET and DCA-DMET TDL estimates with error bars including fitting and AFQMC statistical uncertainties, compared to the determinantal Monte Carlo simulations by Scalettar and coworkers [27], pinning field QMC simulations by Wu and coworkers [28], AFQMC with TABC by Qin et al. [18] and the modified boundary conditions by Sorella [17]. (a) $U/t=2$. (b) $U/t=4$. (c) $U/t=6$.