Benchmarking the variational cluster approach by means of the one-dimensional Bose-Hubbard model

Convergence properties of the variational cluster approach with respect to the variational parameter space, cluster size, and boundary conditions of the reference system are investigated and discussed for bosonic many-body systems. Specifically, the variational cluster approach is applied to the one-dimensional Bose-Hubbard model, which exhibits a quantum phase transition from Mott to superfluid phase. In order to benchmark the variational cluster approach, results for the phase boundary delimiting the first Mott lobe are compared with essentially exact density matrix renormalization group data. Furthermore, static quantities, such as the ground state energy and the one-particle density matrix are compared with high-order strong coupling perturbation theory results. For reference systems with open boundary conditions the variational parameter space is extended by an additional variational parameter which allows for a more uniform particle density on the reference system and thus drastically improves the results. It turns out that the variational cluster approach yields accurate results with relatively low-computational effort for both the phase boundary as well as the static properties of the one-dimensional Bose-Hubbard model, even at the tip of the first Mott lobe where correlation effects are most pronounced.

Figure 1

(Color online) A system of two clusters. The intercluster hopping $\mathsf{T}$ is treated perturbatively.

Figure 2

(Color online) First Mott lobe of the 1D BH model obtained for reference systems with obc and the variational parameter sets (a) $\mathsf{x}=\left\{\mu \right\}$, (b) $\mathsf{x}=\left\{\mu ,t\right\}$, (c) $\mathsf{x}=\left\{\mu ,\delta \right\}$, and (d) $\mathsf{x}=\left\{\mu ,t,\delta \right\}$. The gray shaded area plotted in all subfigures indicates DMRG results obtained in Ref. 5.

Figure 3

(Color online) Spectral function $A\left(\mathbf{k},\omega \right)$ and density of states $N\left(\omega \right)$ for $t=0.15$, $\mu =0.35$, reference systems of $L=12$ sites with obc and the following sets of variational parameters (a) $\mathsf{x}=\left\{\mu \right\}$, (b) $\mathsf{x}=\left\{\mu ,t\right\}$, (c) $\mathsf{x}=\left\{\mu ,\delta \right\}$, and (d) $\mathsf{x}=\left\{\mu ,t,\delta \right\}$.

Figure 4

(Color online) Difference between the parameters of the reference system ${\widehat{H}}^{\prime }$ at the stationary point $\left\{{\mu }^{\prime },t^{\prime },{\delta }^{\prime }\right\}$ and the parameters $\left\{\mu ,t,\delta =0\right\}$ of the physical system $\widehat{H}$. The corresponding variational parameter sets are (a.∗) $\mathsf{x}=\left\{\mu \right\}$, (b.∗) $\mathsf{x}=\left\{\mu ,t\right\}$, (c.∗) $\mathsf{x}=\left\{\mu ,\delta \right\}$, and (d.∗) $\mathsf{x}=\left\{\mu ,t,\delta \right\}$.

Figure 5

(Color online) Difference between the parameters of the reference system ${\widehat{H}}^{\prime }$ at the stationary point $\left\{{\mu }^{\prime },t^{\prime },{\delta }^{\prime }\right\}$ and the parameters $\left\{\mu ,t=0.2,\delta =0\right\}$ of the physical system $\widehat{H}$ in dependence of the cluster size $L$. The corresponding variational parameter sets are (a) $\mathsf{x}=\left\{\mu \right\}$, (b) $\mathsf{x}=\left\{\mu ,t\right\}$, (c) $\mathsf{x}=\left\{\mu ,\delta \right\}$, and (d) $\mathsf{x}=\left\{\mu ,t,\delta \right\}$.

Figure 6

(Color online) Particle density $n^{\prime }\left(\alpha \right)$, first row, and $n\left(\alpha \right)$, second row, for reference systems with obc and size (a) $L=4$, (b) $L=8$ and (c) $L=12$. The parameters for the physical system are $t=0.15$ and $\mu =0.35$. The legend refers to the distinct sets of variational parameters.

Figure 7

(Color online) Phase boundaries of the first Mott lobe of the 1D BH model. The reference system consists of clusters with pbc. The variational parameters (a) $\mathsf{x}=\left\{\mu \right\}$ and (b) $\mathsf{x}=\left\{\mu ,t\right\}$ are used. The gray shaded area shows the DMRG results from Ref. 5.

Figure 8

(Color online) Spectral function $A\left(\mathbf{k},\omega \right)$ and density of states $N\left(\omega \right)$ for a 12 site cluster with pbc as reference system. The parameters of the physical system are $t=0.15$ and $\mu =0.35$. The corresponding variational parameter sets are (a) $\mathsf{x}=\left\{\mu \right\}$ and (b) $\mathsf{x}=\left\{\mu ,t\right\}$.

Figure 9

(Color online) Difference between the variational parameters at the stationary point of the grand potential and the parameters of the physical system for reference systems with pbc. The variational parameter sets (a.*) $\mathsf{x}=\left\{\mu \right\}$ and (b.*) $\mathsf{x}=\left\{\mu ,t\right\}$ are used.

Figure 10

(Color online) Difference between the variational parameters at the stationary point of the grand potential and the parameters of the physical system for reference systems with pbc for the hopping strength $t=0.2$ in dependence of the cluster size $L$. The variational parameter sets are (a) $\mathsf{x}=\left\{\mu \right\}$ and (b) $\mathsf{x}=\left\{\mu ,t\right\}$.

Figure 11

(Color online) Particle density $n\left(\alpha \right)$ obtained from $\mathsf{G}\left(\tilde{\mathbf{k}},\omega \right)$ for the parameters $t=0.15$ and $\mu =0.35$, and reference systems with pbc. The clusters are of size (a) $L=4$, (b) $L=8$, and (c) $L=12$. The legend refers to the considered sets of variational parameters.

Figure 12

(Color online) Deviations between the extrapolated gaps ${\Delta }^{\infty }$ (VCA) and the gap ${\Delta }^{DMRG}$ (DMRG), where (a) shows the deviations for reference systems with obc and (b) for reference systems with pbc. The dashed lines correspond to exact diagonalization results for the gap extrapolated to infinitely large systems.

Figure 13

(Color online) Small black dots indicate lattice sites and big red dots particles. (a) single-particle term of the Green’s function and (b) single-hole term of the Green’s function. In both situations the additional particle/hole can move freely on the lattice. (c) and (d), possible higher order excitations due to quantum fluctuations in the ground state of the single-particle term of the Green’s function.

Figure 14

(Color online) Spectral functions, first row, and density of states, second row, for (a) $t=0.01,\mu =0.5$, (b) $t=0.03,\mu =0.5$, and (c) $t=0.05,\mu =0.5$. For the visualization we used a threshold of 0.1. The reference system consists of an $L=12$ site cluster with obc.

Figure 15

(Color online) Comparison of the ground state energy $E_0$ for hopping parameters corresponding to the first Mott lobe obtained by means of VCA (dots connected by lines) and strong-coupling perturbation theory (solid line). The lines connecting the VCA results are guidance for the eyes. (a) directly compares the ground state energy obtained from the two approaches and (b) shows their difference.

Figure 16

(Color online) One-particle density matrix $C\left(\left|{\mathbf{r}}_{\mathbf{i}}-{\mathbf{r}}_{\mathbf{j}}\right|\right)$ for (a) nearest neighbors $C\left(1\right)$, (b) next nearest neighbors $C\left(2\right)$ and (c) next-next-nearest neighbors $C\left(3\right)$ obtained from VCA (dots connected by lines) and strong-coupling perturbation theory (solid line), respectively. (d) shows the dominating exponential drop-off $1/\xi$ of the one-particle density matrix for large enough distances in dependence of the hopping strength $t$ and the size of the reference system. The inset shows an extract of $1/\xi$ where the hopping strength is restricted to $0.1\le t\le 0.3$.