Mott transition in one dimension: Benchmarking dynamical cluster approaches

The variational cluster approach (VCA) is applied to the one-dimensional Hubbard model at zero temperature using clusters (chains) of up to ten sites with full diagonalization and the Lanczos method as cluster solver. Within the framework of the self-energy-functional theory (SFT), different cluster reference systems with and without bath degrees of freedom, in different topologies and with different sets of variational parameters, are considered. Static and one-particle dynamical quantities are calculated for half-filling as a function of $U$ as well as for fixed $U$ as a function of the chemical potential to study the interaction- and filling-dependent metal-insulator (Mott) transition. The recently developed $Q$-matrix technique is used to compute the SFT grand potential. For benchmarking purposes we compare the VCA results with exact results available from the Bethe ansatz, with essentially exact dynamical density-matrix renormalization-group data, with (cellular) dynamical mean-field theory and full diagonalization of isolated Hubbard chains. Several issues are discussed including convergence of the results with cluster size, the ability of cluster approaches to access the critical regime of the Mott transition, efficiency in the optimization of correlated-site vs bath-site parameters, and of multidimensional parameter optimization. We also study the role of bath sites for the description of excitation properties and as charge reservoirs for the description of filling dependencies. The VCA turns out to be a computationally cheap method which is competitive with established cluster approaches.

Figure 1

(Color online) The original system ($D=1$ Hubbard model, nearest-neighbor (NN) hopping) and different reference systems considered in this study together with the corresponding variational parameters being optimized. See text for discussion.

Figure 2

(Color online) $U$ dependence of the optimal intracluster hopping $t^{\prime }$ for different cluster sizes $L_c$ as indicated. VCA calculations for $\mu =U∕2$ (half-filling) using the reference system displayed in Fig. 1, A. The physical NN hopping $t=1$ sets the energy scale. Inset: optimal $t^{\prime }$ as function of $U^2$ for $U\to 0$.

Figure 3

(Color online) $U$ dependence of the VCA ground-state energy per site for cluster sizes $L_c=2$ and $L_c=10$ (see Fig. 1, A). The exact (Bethe ansatz) result of Eq. (2) is shown for comparison.

Figure 4

(Color online) SFT grand potential $\Omega \left(t^{\prime }\right)\equiv \Omega \left[\mathbf{\Sigma }\left(t^{\prime }\right)\right]$ (constantly shifted by $\mu ⟨N⟩$) per site and as a function of the intracluster nearest-neighbor hopping for $L_c=10$ and different $U$ $(\mu =U∕2)$. The arrows indicate the respective optimal $t^{\prime }$.

Figure 5

(Color online) VCA ground-state energy per site for $U=4$ (top) and $U=8$ (bottom) for different cluster sizes $L_c$ as a function of $1∕L_c$ compared to the exact (BA) result and the direct cluster approach.

Figure 6

(Color online) $U$ dependence of the insulating gap in the one-particle excitation spectrum as obtained from the VCA and from the direct cluster approach for $L_c=2$ and $L_c=10$ in comparison with the exact result of Eq. (3). VCA calculations using the reference system shown in Fig. 1, A.

Figure 7

(Color online) Optimized hopping parameters for reference systems shown in Fig. 1, A–C for $U=4$ and different cluster sizes ranging from $L_c=4$ to $L_c=10$ as indicated (physical hopping set to $t=1$). Black: hopping assumed to be uniform (A). Red: two hopping parameters varied independently, the hopping at the two cluster edges and the “bulk” hopping (B). Blue: hopping at the edges, next to the edges and bulk hopping varied. Green: four hopping parameters varied. Orange: five hopping parameters varied (for $L_c=10$ this corresponds to C).

Figure 8

(Color online) SFT grand potential per site $\Omega \left({\mathit{t}}^{\prime }\right)∕L\equiv \Omega \left[\mathit{\Sigma }\left({\mathit{t}}^{\prime }\right)\right]∕L$ as a function of the hybridization strength for $U=4$. Red lines: reference system Fig. 1, G; solid: $\Omega \left(V_1\right)$ at $V_2=\mathrm{opt.}$, i.e., varying $V_1$ of the edge bath sites while keeping the hybridization of the central bath sites $V_2$ at its optimal value. Dashed: $\Omega \left(V_2\right)$ at $V_1=\mathrm{opt.}$ The arrows indicate the respective minima. Blue line: reference system Fig. 1, H, i.e., $\Omega \left(V_1\right)$ at $V_2=0$.

Figure 9

(Color online) SFT grand potential per site as a function of the hybridization strength for $U=4$. Red lines: reference system Fig. 1, H for different cluster sizes ranging from $L_c=2$ to $L_c=8$. Green lines: reference system Fig. 1, I, i.e., with additional optimization of the nearest-neighbor hopping. The arrows indicate minima. Dashed black line: BA result for $E_0-\mu N$.

Figure 10

(Color online) Optimal values of the hybridization in Fig. 9 as a function of the inverse cluster size. The dashed lines show two possible extrapolations to the $L_c=\infty$ limit.

Figure 11

(Color online) Local density of states at a central site as obtained for different reference systems from the VCA Green’s function for $U=4$. Calculations are performed for two different Lorentzian broadening parameters $\eta$. Results for $\eta =0.005$ have been scaled by a common constant factor $1∕20$. The exact single-particle gap is indicated by dashed lines.

Figure 12

(Color online) Local Green’s function as a function of Matsubara frequency for $U=6$ using the $L_c=2$, $n_s=3$ and the $L_c=4$, $n_s=3$ reference systems (red solid lines) compared to dynamical DMRG data (black solid line), $L_c=2$ C-DMFT at finite (low) temperature $(\beta =20)$, and $L_c=2$ cluster dual-fermion theory (cluster DF, $\beta =20$) taken from Ref. 45.

Figure 13

(Color online) Electron density $n_c=⟨N⟩∕L_c=N_c∕L_c$ as a function of the chemical potential for an $L_c=10$ Hubbard chain (open boundary conditions) compared to the exact result for $L_c\to \infty$ from Ref. 29.

Figure 14

(Color online) Electron density $n$ within cluster-perturbation theory (CPT), calculated by integrating the CPT density of states up to zero excitation energy [red line, Eq. (12)] and calculated by differentiation of the CPT grand potential with respect to $\mu$ [blue line, Eq. (13)], as function of $\mu$ compared to the exact result (BA). Cluster size: $L_c=10$. Horizontal dashed lines: cluster fillings. Vertical dashed lines: critical chemical potentials at which the cluster ground state changes.

Figure 15

(Color online) The original system and different reference systems with corresponding variational parameters. See text for discussion.

Figure 16

(Color online) Electron density (filling $n$) as a function of the chemical potential close to half-filling for $U=4$ as obtained from the VCA with reference system A of Fig. 15 for different cluster sizes $L_c$. In all cases the cluster ground state is in the $N_c=L_c$ subspace. Inset: $n\left(\mu \right)$ for $L_c=4$ displayed for the entire $\mu$ range and with cluster ground state in the $N_c=L_c$ and in the $N_c=L_c∕2$ subspace.

Figure 17

(Color online) Filling $n$ as a function of $\mu$ close to half-filling for $U=4$ as obtained from the VCA with reference system B of Fig. 15 for $L_c=2$ and $L_c=4$. Exact and C-DMFT result from Ref. 31 are shown for comparison.

Figure 18

(Color online) Chemical-potential dependence of the optimal one-particle parameters for reference system C of Fig. 15 at $U=4$. $\Delta ϵ$ is an overall shift of all one-particle energies ($\epsilon$, ${\epsilon }_a$, and ${\epsilon }_b$).

Figure 19

(Color online) Filling versus chemical potential for $U=4$. VCA result using reference system C of Fig. 15 in comparison with the exact result (Bethe ansatz), with DMFT (from Ref. 31), and with the two-site dynamical impurity approximation (DIA).