Multisite mean-field theory for cold bosonic atoms in optical lattices

We present a detailed derivation of a multisite mean-field theory (MSMFT) used to describe the Mott-insulator to superfluid transition of bosonic atoms in optical lattices. The approach is based on partitioning the lattice into small clusters which are decoupled by means of a mean-field approximation. This approximation invokes local superfluid order parameters defined for each of the boundary sites of the cluster. The resulting MSMFT grand potential has a nontrivial topology as a function of the various order parameters. An understanding of this topology provides two different criteria for the determination of the Mott insulator superfluid phase boundaries. We apply this formalism to $d$-dimensional hypercubic lattices in one, two, and three dimensions and demonstrate the improvement in the estimation of the phase boundaries when MSMFT is utilized for increasingly larger clusters, with the best quantitative agreement found for $d=3$. The MSMFT is then used to examine a linear dimer chain in which the onsite energies within the dimer have an energy separation of $\Delta$. This system has a complicated phase diagram within the parameter space of the model, with many distinct Mott phases separated by superfluid regions.

Figure 1

Partitioning of a one-dimensional chain into linear clusters of length $L$. Within our multisite mean-field formulation each cluster with open boundary conditions is treated exactly, and the intercluster couplings (generated by the hopping Hamiltonian) are treated in a mean-field decoupling approximation.

Figure 2

Partitioning of two-dimensional square lattice using various repeated clusters. Shown in panel (a) are single site monomers, dimers ($L=2$), and $L$-mers for $L=4$, 5, 6. Shown in panel (b) is a portion of a two-dimensional square lattice that is partitioned into $3\times 3$ clusters. The solid lines connect sites in a given cluster, dotted lines show the division of the lattice into such clusters, and the dashed lines show the intercluster couplings that are treated within mean-field theory. In each cluster there are three inequivalent sites: the central site $C$, which does not couple to neighboring clusters, and corner and edge sites, which respectively are labeled $A$ and $B$.

Figure 3

Partitioning of two-dimensional square lattice into $3\times 2$ clusters. The cluster consists of inequivalent corner ($A$) and edge ($B$) sites.

Figure 4

Nonzero eigenvalues of energy (top panel, in units of $U$) and stability (bottom panel) matrices for homogeneous two-dimensional square lattice using a $3\times 2$ cluster vs $J/U$. The chemical potential is $\mu /U=0.4$. One eigenvalue of the energy matrix crosses zero at $J/U=0.04815$, the same position at which one eigenvalue of the stability matrix first attains a magnitude of one (the solid black curve in each figure). The superfluid phase is the $T=0$ ground state for $J/U>0.04815$.

Figure 5

Normalized eigenvector components corresponding to energy (top) and stability (bottom) eigenvalues shown by solid black curves in Fig. 4 vs $J/U$. The bracketed numbers give the number of components having the indicated values and correspond to the corner sites (multiplicity 4) and edge sites (multiplicity 2) of the $3\times 2$ cluster. Note that at the transition these eigenvectors are the same, as required by Eq. (53).

Figure 6

Upper figure shows self-consistently determined order parameters corresponding to corner (${\psi }_A$) and edge (${\psi }_B$) sites of $3\times 2$ cluster as a function of $J/U$. The arrows indicate the values of ${\psi }_A$ at $J/U=0.049$ and 0.05. The lower figure shows the grand potential as a function of ${\psi }_A$ for the fixed ratio ${\psi }_B/{\psi }_A=0.9396$ found at the phase boundary at $J/U=0.04815$. The curves from top to bottom are for values of $J/U$ from 0.046 to 0.050 in steps of 0.001. In the SF phase, the grand potential shows minima at positions which are very close to the values of ${\psi }_A$ found from the self-consistent calculations (indicated by the arrows). These results were obtained for $\mu /U=0.4$.

Figure 7

MI-SF phase boundaries for linear chain predicted by MSMFT, compared to the DMRG results of Ref. [6] shown as circles. The phase boundary systematically approaches the exact result with increasing length $L$ of the cluster: $L=2$ (dash-dash-dotted red line), $L=4$ (dot-dashed green line), $L=8$ (dot-dot-dashed blue line). Also shown are the boundaries of the $N$-domain where the ground state of the full Hamiltonian for a finite chain of length $N_s$ has $N=N_s$ particles.

Figure 8

MI-SF phase boundaries for a two-dimensional square lattice, as calculated using SDMFT (solid black line) and MSMFT ($2\times 2$, dashed red line; $3\times 3$ dotted blue line). The MC data (solid black circles) are taken from Ref. [9]. Also shown (dot-dashed black line) is the $N$-domain ($N=9$) for the $3\times 3$ cluster determined by calculating the ground state of the full (non-MFT) Hamiltonian with periodic boundary conditions.

Figure 9

MI-SF phase boundaries for a three-dimensional cubic lattice, as calculated using SDMFT (solid black line) and MSMFT ($2\times 2\times 1$, dashed red line; $2\times 2\times 2$, dotted blue line). The MC data (solid black circles) are taken from Ref. [8].

Figure 10

Average correlation function of linear chain within MSMFT versus intersite spacing. Panel (a) shows results for $L=7$ in the MI phase: the six curves starting from 1 at $n=0$ correspond, from bottom to top, to $J/U=0.02$, 0.04, 0.06, 0.08, 0.10, and 0.12. The points are the calculated values of ${\overline{C}}_l\left(L\right)$ and the dashed curves are plots of the fitting function (69). The lowest curve in (a) shows results for $J/U=0.10$ for $L=$ 3, 4, 5, 6, and 7, all plotted on top of each other. This curve has been displaced from the others for clarity by multiplying the data by ${10}^{-4}$. Panel (b) shows results for $L=7$ in the SF phase: from top to bottom, $J/U=$ 0.14, 0.17, and 0.20.

Figure 11

Correlation length $\xi \left(L\right)$ vs $J/U$ for the linear chain of length $L=4$, 6, and 8 (bottom to top). The results were obtained by fitting the data to Eq. (69); the vertical bars indicate the estimated error.

Figure 12

Variation of averaged pair correlations with separation in units of lattice constant for two-dimensional $3\times 3$ cluster. The points are the result of the MSMFT calculation for $\mu /U=0.4$ and for $J/U=0.01$ (circles), 0.02 (squares), 0.03 (diamonds), 0.04 (triangles), and 0.049 (crosses). The values of ${\overline{C}}_{ij}$ as obtained from the fitting function in Eq. (71) are also plotted as solid points (red) which are joined by dotted lines as a guide to the eye. The MI-SF transition for this value of $\mu /U$ occurs close to $J/U=0.0492$.

Figure 13

Dependence of correlation length, $\xi$ (in units of lattice constant) on $J/U$ for the two-dimensional square lattice using a $3\times 3$ cluster. The results were obtained by fitting the data in Fig. 12 to Eq. (71).

Figure 14

Schematic of general dimer superlattice with alternating site energies ${\varepsilon }_A$ and ${\varepsilon }_B$. The level separation is $\Delta$. The different barrier heights along the chain lead to different intradimer ($J_1$) and interdimer ($J_2$) tunneling energies.

Figure 15

Portion of dimer superlattice used in analysis of MI-SF transition. The inversion symmetry about each lattice site results in equal intra- and interdimer tunneling energies $J$.

Figure 16

$N$-domains of single-dimer $L$-mer at $\tilde{J}=0$. The site occupations of the $A$ (lower level) and $B$ (upper level) sites is indicated by the number of black dots. As one crosses a downward (upward) sloping line with increasing $\mu /U$, the occupation of the lower (upper) level increases by one. This figure applies to any number of dimers in the $L$-mer being considered.

Figure 17

Phase diagram of a dimer chain within the MSMFT using a cluster consisting of a single dimer. The gray scale indicates the value of ${\tilde{J}}_{\mathrm{cr}}(\tilde{\mu },\tilde{\Delta })$ at which the MI-SF phase boundary occurs. Each region bounded by a black contour ($\tilde{J}=0$) corresponds to a different Mott lobe with an integer number of atoms $N$ in the dimer.

Figure 18

Slice through the phase diagram in Fig. 17 at $\tilde{\Delta }=0.75$. The average filling per site for each Mott lobe increases by $1/2$ with increasing $\mu /U$, starting with $n=0$ at the bottom of the figure. The system is a SF to the right of the phase boundary.

Figure 19

Phase diagram of dimer chain obtained for an $L$-mer consisting of single dimer at $J/\Delta =0.05$. The SF phase is indicated by the shaded region (the vertical black lines) which surround the various MI regions. The black lines are the boundaries of the $N$-domains within which Mott regions with $N$ particles per dimer are located. As discussed in the text, the $N$-domains are the regions where the ground state of ${\widehat{\mathcal{K}}}_L^0$ contains $N$ particles.

Figure 20

Phase diagram of dimer chain at $J/\Delta =0.05$ for $L$-mer containing two dimers. As compared to the single dimer phase diagram at in Fig. 19, we see that the Mott regions are larger, implying the increased stabilization of the Mott phase. This is particularly evident in the bottom-right corner of the figure which shows an additional Mott region with filling $n=3/2$.

Figure 21

Phase diagram of dimer chain at $J/\Delta =0.1$ for $L$-mer containing two dimers. The range of $\mu /U$ corresponds to that in Fig. 17 of Ref. [7] (note that their interaction parameter is $U^{\prime }=U/2$). The lowest unshaded region in the figure is a $n=0$ Mott insulator.

Figure 22

Plot of the onsite densities for terminating $A$ and $B$ sites of $L$-mer containing two dimers, as a function of $\tilde{\Delta }$, for the values of $J/\Delta =0.05$ and $\tilde{\mu }=0.5$. The densities $\langle {\widehat{n}}_A\rangle$ and $\langle {\widehat{n}}_B\rangle$ are obtained from the ground state of the full Hamiltonian (74); for comparison, we also show the densities ${\langle {\widehat{n}}_A\rangle }_0$ and ${\langle {\widehat{n}}_B\rangle }_0$ corresponding to the ground state of ${\widehat{\mathcal{K}}}_L^0$ in Eq. (75). The discontinuities in ${\langle {\widehat{n}}_A\rangle }_0$ and ${\langle {\widehat{n}}_B\rangle }_0$ occur at the boundaries of the $N$-domains. On the other hand, $\langle {\widehat{n}}_A\rangle$ and $\langle {\widehat{n}}_B\rangle$ vary continuously but show kinks at the boundaries between the MI and SF phases.

Figure 23

Plot of order parameters ${\psi }_A$ and ${\psi }_B$ as a function of $\tilde{\Delta }$ for $J/\Delta =0.05$ and $\tilde{\mu }=0.5$. The order parameters are shown for $L$-mers containing one and two dimers corresponding to the phase diagrams in Figs.  19 and  20.

Figure 24

Phase diagram for four-site superlattice which exhibits loopholes. The parameters defining the superlattice are given in the text. The shaded region corresponds to the SF phase as determined using MSMFT. The dashed lines show the phase boundaries determined using SDMFT. The numbers $n$ labeling each of the Mott domains are the average filling of the superlattice. We note that new Mott domains appear at $n=1$ and $n=2$ within MSMFT. The solid lines show the $N$-domains for the four-site cluster. Within SDMFT, the $N$-domain boundaries are straight lines, parallel to the $J/U$ axis, which start at the edges of the Mott domains along the $\mu /U$ axis.