Quantum phases of dipolar bosons in a multilayer optical lattice

We consider a minimal model to investigate the quantum phases of hardcore, polarized dipolar atoms confined in multilayer optical lattices. The model is a variant of the extended Bose-Hubbard model, which incorporates intralayer repulsion and interlayer attraction between the atoms in nearest-neighbor sites. We study the phases of this model emerging from the competition between the attractive interlayer interaction and the interlayer hopping. Our results from the analytical and cluster-Gutzwiller mean-field theories reveal that multimer formation occurs in the regime of weak intra- and interlayer hopping due to the attractive interaction. In addition, intralayer isotropic repulsive interaction results in the checkerboard ordering of the multimers. This leads to an incompressible checkerboard multimer phase at half-filling. At higher interlayer hopping, the multimers are destabilized to form resonating valence-bond-like states. Furthermore, we discuss the effects of thermal fluctuations on the quantum phases of the system.

Figure 1

Schematic representation of the dipolar bosonic atoms in a bilayer optical lattice. The lattice spacing within a layer is $a$, and interlayer spacing is $a_z$. The red spheres represent atoms, and the arrows indicate the orientation of the dipole. The dotted arrows indicate possible extension to multilayered optical lattice.

Figure 2

Schematic representation of the incompressible phases. The red (gray) spheres represent particles or atoms (holes). (a), (b), and (c) correspond to the DCBS, VCBS $\rho =1/4$ and VCBS $\rho =3/4$ phases, respectively. The blue shading across the interlayer bonds in (b) and (c) denote the entangled state, as given by Eq. (6).

Figure 3

Phase diagrams in the $J/V-\mu /V$ plane for $J^{\prime }/V=0$ and 0.5, respectively for $V^{\prime }=-1.0V$. The black lines represent the phase boundaries between the incompressible and compressible phases, obtained numerically with $2\times 2\times 2$ clusters in the CGMF method. Metastable quantum phases with intermediate values of $\rho$ are present in the red shaded region in the phase diagram. And, the filled black circles denote the analytical phase boundaries obtained from the site-decoupled mean-field method, as given in Eqs. (30, 31, 32).

Figure 4

Phase diagrams in the $J/V-\mu /V$ plane for $J^{\prime }/V=0.8$, 1.0, 1.2 and 1.5 for $V^{\prime }/V=-1.0$ are shown in (a), (b), (c), and (d), respectively. The black lines indicate the phase boundaries between the incompressible and compressible phases, and the blue lines represent the SS-to-SF phase boundary, which are computed numerically with $2\times 2\times 2$ clusters in the CGMF method. Filled black circles are the phase boundaries obtained from the perturbative analysis of the mean-field Hamiltonian, as given in the Eqs. (30), (31), (32), (39), and (40). The red shaded region in the (d) corresponds to an incompressible phase with no dimer structure.

Figure 5

Phase diagrams of the bilayer optical lattice in the $J/V-\mu /V$ plane for $J^{\prime }/V=0$ and $V^{\prime }/V=-1.0$ obtained from the clusters of different sizes: $1\times 1\times 1$ (magenta), $1\times 1\times 2$ (gold), $2\times 1\times 2$ (green), $4\times 1\times 2$ (red), and $2\times 2\times 2$ (black). The filled black circles represent the analytically obtained phase boundary obtained from the Eqs. (30, 31, 32).

Figure 6

The incompressible to compressible phase boundaries in the $J/V-\mu /V$ plane for $J^{\prime }/V=1.0$ and $V^{\prime }/V=-1.0$ obtained from clusters of different sizes: $1\times 1\times 1$ (magenta), $1\times 1\times 2$ (gold), green for $2\times 1\times 2$ (green), $4\times 1\times 2$ (red), and $2\times 2\times 2$ (black). The filled black circles indicate the analytical phase boundaries obtained from Eqs. (30), (39), and (40).

Figure 7

Phase diagram in the $J/V\text{−}\mu /V$ plane for $V^{\prime }/V=-0.25$ (top) and $V^{\prime }/V=-2.0$ (bottom). The subplots (a) and (b) correspond to $J^{\prime }/V=0.25$ and 0.5, respectively. The subplots (c) and (d) correspond to $J^{\prime }/V=1.0$ and 1.5, respectively. The black solid lines represent the phase boundaries between incompressible and compressible phases, and the blue lines indicate the phase boundaries between SS and SF phases. These phase boundaries are obtained numerically using $2\times 2\times 2$ clusters in the CGMF method. The black circles represent the analytically obtained phase boundary points obtained from the Eqs. (30), (31), (32), (39), and (40).

Figure 8

Finite temperature phase diagram for the bilayer optical lattice in the $J/V\text{−}\mu /V$ plane for $J^{\prime }/V=1.0$, $V^{\prime }/V=-1.0$, and $k_{B}T=0.05V$. The phase boundaries are obtained numerically with $2\times 2\times 2$ clusters in the CGMF method. The green lines represent the phase boundary between the NF and the compressible phase, while black lines indicate the phase boundaries between incompressible, unmelted phases and compressible phases.

Figure 9

Plot of the $\kappa$ and $S(\pi ,\pi )$ as a function of temperature. The blue solid (dashed) line indicates the $S(\pi ,\pi )$ for DCBS $\rho =1/2$ (VCBS $\rho =1/4$) state at low temperatures. The red solid (dashed) line indicates the $\kappa$ for DCBS $\rho =1/2$ (VCBS $\rho =1/4$) state. Here the system parameters $J^{\prime }/V=1.0$ and $V^{\prime }/V=-1.0$ are considered. The value of $J/V$ is fixed at 0.08, and the values of the chemical potential $\mu /V$ are $-0.5$ and 1.5 for VCBS ($\rho =1/4$) and DCBS phases, respectively.

Figure 10

Phase diagram for trilayer optical lattice in the $J/V\text{−}\mu /V$ plane for $J^{\prime }/V=1$ and $V^{\prime }/V=-1$. The black solid lines represent the phase boundaries between incompressible and compressible phases, and the blue lines indicate the phase boundaries between SS and SF phase, obtained numerically with $2\times 1\times 3$ clusters in the CGMF method and with periodic boundary condition along $z$ direction. The black circles represent the analytically obtained phase boundary points from the Eqs. (41, 42, 43, 44, 45).

Figure 11

Phase diagram for trilayer optical lattice in the $J/V\text{−}\mu /V$ plane for $J^{\prime }/V=1$ and $V^{\prime }/V=-1$. The black solid lines represent the phase boundaries between incompressible and compressible phases, and the blue lines indicate the phase boundaries between SS and SF phase, obtained numerically with $2\times 1\times 3$ clusters in the CGMF method and with open boundary condition along $z$ direction.