Coupled-cluster impurity solvers for dynamical mean-field theory

We describe the use of coupled-cluster theory as an impurity solver in dynamical mean-field theory (DMFT) and its cluster extensions. We present numerical results at the level of coupled-cluster theory with single and double excitations (CCSD) for the density of states and self-energies of cluster impurity problems in the one- and two-dimensional Hubbard models. Comparison to exact diagonalization shows that CCSD produces accurate density of states and self-energies at a variety of values of $U/t$ and filling fractions. However, the low cost allows for the use of many bath sites, which we define by a discretization of the hybridization directly on the real frequency axis. We observe convergence of dynamical quantities using approximately 30 bath sites per impurity site, with our largest 4-site cluster DMFT calculation using 120 bath sites. We suggest that coupled-cluster impurity solvers will be attractive in ab initio formulations of dynamical mean-field theory.

Figure 1

Single-site DMFT and two-site CDMFT results for the 1D Hubbard model, comparing the use of the CC and exact diagonalization (ED) impurity solvers. Numbers in parentheses indicate the number of impurity sites and the total number of bath sites. Results are shown at half filling with weak interactions ($U/t=2$, left) and strong interactions ($U/t=6$, right). The chemical potential is fixed at $\mu =U/2$ and the DOS is plotted with a broadening of $\eta =0.5t$ for clarity.

Figure 2

Single-site DMFT results for the 1D Hubbard at half filling with weak interactions ($U/t=2$, left) and strong interactions ($U/t=6$, right), showing convergence with respect to the number of bath sites $N_{\omega }$. The chemical potential is fixed at $\mu =U/2$ and the DOS is plotted with a broadening of $\eta /t=0.5$ for clarity.

Figure 3

Two-site CDMFT results as a function of doping. The DOS obtained using $N_{\omega }=30$ (60 bath sites in total) is shown in the top two panels, where the chemical potential is indicated by a vertical line. The site occupancy as a function of the chemical potential using $N_{\omega }=40$ (open symbols) and $N_{\omega }=60,80$ (filled symbols) is shown in the bottom panel, along with the numerically exact result from Bethe ansatz [61] and the mean-field result.

Figure 4

Single-site DMFT and four-site ($2\times 2$) CDMFT results for the 2D Hubbard at half filling with weak interactions ($U/t=2$, left) and strong interactions ($U/t=8$, right). We used 30 bath sites for single-site DMFT and 120 bath sites for the four-site CDMFT. The chemical potential is fixed at $\mu =U/2$ and the DOS is plotted with a broadening of $\eta /t=0.5$ for clarity.

Figure 5

Impurity self-energy on the real frequency axis from a two-site CDMFT calculation for the 1D Hubbard at half filling with $U/t=2$. RCCSD is used as the impurity solver and two bath sites are coupled to each impurity site. A small broadening of $\eta =0.001t$ is used to show the noncausal behavior.