Quantum phases of canted dipolar bosons in a two-dimensional square optical lattice

We consider a minimal model to describe the quantum phases of ultracold dipolar bosons in two-dimensional square optical lattices. The model is a variation of the extended Bose-Hubbard model and apt to study the quantum phases arising from the variation in the tilt angle ${\displaystyle \theta }$ of the dipolar bosons. At low tilt angles, ${\displaystyle 0^{\circ }⩽\theta \lesssim {25}^{\circ }}$, the ground states of the system are phases with checkerboard order, which can be either checkerboard supersolids or checkerboard density waves. For high tilt angles, ${\displaystyle {35}^{\circ }\lesssim \theta \lesssim {55}^{\circ }}$, phases with striped order of the supersolid or density wave are preferred. In the intermediate domain, ${\displaystyle {25}^{\circ }\lesssim \theta \lesssim {35}^{\circ }}$, an emulsion or superfluid phase intervenes the transition between the checkerboard and the striped phases. The attractive interaction dominates at ${\displaystyle \theta \gtrsim {55}^{\circ }}$, which renders the system unstable, and there is a density collapse. For our studies we use Gutzwiller mean-field theory to obtain the quantum phases and the phase boundaries. In addition, we calculate the phase boundaries between an incompressible and a compressible phase of the system by considering second-order perturbation analysis of the mean-field theory. The analytical results, where applicable, are in excellent agreement with the numerical results. In our study, the incompressible phases have an average occupancy per site ${\displaystyle \rho \le 1}$, but the compressible phases can have ${\displaystyle \rho >1}$.

Figure 1

Schematics of dipolar bosons in a two-dimensional square optical lattice with dipolar interaction among the bosons at nearest-neighbor lattice sites. We consider the dipoles to be polarized in the y-z plane and the angle subtended by the direction of the dipole moments (polarization axis) with the $z$ axis is the tilt angle ${\displaystyle \theta }$. The tilt angle is illustrated by the orange-shaded sector. The angle between the polarization axis and the vector ${\displaystyle ({\vec{r}}_4-{\vec{r}}_1)}, {\displaystyle {\alpha }_{14}}$, is represented by the blue-shaded sector. The dipolar interaction between bosons at lattice sites ${\displaystyle (p,q)}$ and ${\displaystyle (p\pm 1,q)}$ is ${\displaystyle C_{\mathrm{dd}}}$, whereas the interaction between bosons at lattice sites ${\displaystyle (p,q)}$ and ${\displaystyle (p,q\pm 1)}$ is ${\displaystyle U_{\mathrm{dd}}\left(\theta \right)=C_{\mathrm{dd}}(1-3{sin}^2\theta )}$.

Figure 2

Density pattern of the system in distinct density-wave phases. Black squares represent lattice sites which are vacant, and white squares denote singly occupied lattice sites. The states are illustrated for fixed ${\displaystyle \mu /J=15}$ and ${\displaystyle C_{\mathrm{dd}}/U=0.8}$. CBDW (1,0) and SDW (1,0) states are obtained for ${\displaystyle J/U=0.033}$ at ${\displaystyle \theta =0^{\circ }}$ and ${\displaystyle {37}^{\circ }}$, respectively. The emulsion state is obtained for ${\displaystyle J/U=0.035}$ at ${\displaystyle \theta =31.5^{\circ }}$.

Figure 3

Phase diagrams in the ${\displaystyle J/U-C_{\mathrm{dd}}/U}$ plane for different values of the tilt angle ${\displaystyle \theta }$. The phase diagrams are obtained for ${\displaystyle \mu /J=15}$. Solid phase boundaries are obtained from the self-consistent numerical diagonalization of the mean-field Hamiltonian of the system, whereas filled circles represent the phase boundaries between an incompressible and a compressible phase of the system, which are obtained analytically considering perturbation analysis of the mean-field decoupling theory. In (c), the parameter region for the emulsion DW (1,0) state is shaded gold. In the emulsion state, the system simultaneously exhibits both orders, checkerboard and striped. The parameter region shaded pink in (f) represents the emulsion of SDW (1,0) and SDW (2,0) phases.

Figure 4

${\displaystyle C_{\mathrm{dd}}/U}$ value of the tip of the MI lobe as the tilt angle ${\displaystyle \theta }$ is changed. Filled green circles are obtained from Gutzwiller mean-field theory and the solid red line is obtained from Eq. (13).

Figure 5

Phase diagram in the ${\displaystyle J/U-\theta }$ plane for ${\displaystyle C_{\mathrm{dd}}/U=0.8}$ and ${\displaystyle \mu /J=15}$. Solid phase boundaries are obtained from the numerical computation of the Gutzwiller mean-field theory. Filled circles mark the phase boundaries between an incompressible and a compressible phase, which are calculated analytically by performing perturbation analysis of the mean-field decoupling theory. Parameter domains shaded silver and gold represent emulsion SS and emulsion DW (1,0) states, respectively. In these emulsion states, checkerboard and striped order coexist in the system.