Cluster Gutzwiller method for bosonic lattice systems

A versatile and numerically inexpensive method is presented allowing the accurate calculation of phase diagrams for bosonic lattice models. By treating clusters within the Gutzwiller theory, a surprisingly good description of quantum fluctuations beyond the mean-field theory is achieved approaching quantum Monte Carlo predictions for large clusters. Applying this powerful method to the Bose-Hubbard model, we demonstrate that it yields precise results for the superfluid to Mott-insulator transition in square, honeycomb, and cubic lattices. Due to the exact treatment within a cluster, the method can be effortlessly adapted to more complicated Hamiltonians in the fast progressing field of optical lattice experiments. This includes state- and site-dependent superlattices, large confined atomic systems, and disordered potentials, as well as various types of extended Hubbard models. Furthermore, the approach allows an excellent treatment of systems with arbitrary filling factors. We discuss the perspectives that allow for the computation of large, spatially varying lattices, low-lying excitations, and time evolution.

Figure 1

(a) Within the Gutzwiller theory a single lattice site is coupled to the mean field $\langle {\widehat{b}}_{j^{\prime }}^{white†black}\rangle$ of its nearest neighbors (illustrated for a square lattice). The lattice sites are expanded in Fock states $|n\rangle$ with $n=0,1,2,\cdots$ particles. (b) In the supercell method, the single site is replaced by a cluster of lattice sites and is expanded in the many-site Fock basis $|N\rangle$.

Figure 2

(a) The cluster Gutzwiller method applied to the Bose-Hubbard model of a square two-dimensional (2D) lattice. The gray lobe corresponds to the single-site Gutzwiller method and the blue lines to supercells with $s=$ 4, 9, and 16 sites (from left to right). Using periodic boundary conditions for $s=12$ and 16 sites sites (dashed green lines) improve the results significantly. The vertical lines depict the critical ratio $J/U$ obtained by mean-field theory (MFT) [3, 4, 6] and quantum Monte Carlo (QMC). The QMC results (open circles) are taken from Ref. [35]. The cluster calculation takes $f=7$ fluctuations into account (see text), where the red error bar corresponds to $f=\infty$ ($f=8$ for $s\ge 16$) and the black error bar to $f=5$. (b) Results for the honeycomb (HC) lattice with three nearest neighbors for $s=$ 4, 12, and 18 sites (blue) as well as $s=18$ and 12 sites with periodic boundary conditions (dashed green). The vertical line corresponds to the process chain approach (PCA) where the shaded area is the estimated error [36]. (c), (d) Finite-size scaling of the critical points for different cluster sizes, where circles (diamonds) indicate results without (with) periodic boundary conditions. The scaling parameter $\lambda$ is zero for a single site and one for an infinite lattice (see text). For (c) 2D and (d) HC lattices, the predicted critical points for $\lambda =1$ match with the QMC and the PCA predictions, respectively, within the error bounds.

Figure 3

(a) Results for three-dimensional cubic lattices (3D) for supercells with $s=$ 4, 12, 18 sites (blue lines) and supercells with 12 and 18 sites using periodic boundaries (dashed green lines). The blue and green lines for 18 and 12 sites, respectively, nearly coincide. In three dimensions, periodic boundary conditions can be applied in two directions for the $3\times 3\times 2$ cluster (dotted-dashed green line). The QMC results (open circles) are taken from Ref. [37]. See Fig. 2 for further details. (b) The finite-size scaling for clusters without (circles) and with periodic boundary conditions in one (diamonds) and two directions (squares).

Figure 4

Mott insulator phase boundaries for filling factors $n=1$–3 obtained by the supercell approach, which allows for the calculation of arbitrary filling factors. The results for the cubic (3D), the square (2D), and the honeycomb lattice (HC) are plotted for the largest cell sizes with periodic boundary condition along one direction as a function of $zJ/U$. In this unit the mean-field boundary (gray) is independent of the lattice geometry.