Quantum phases of dipolar soft-core bosons

We study the phase diagram of a system of soft-core dipolar bosons confined to a two-dimensional optical lattice layer. We assume that dipoles are aligned perpendicular to the layer such that the dipolar interactions are purely repulsive and isotropic. We consider the full dipolar interaction and perform path-integral quantum Monte Carlo simulations using the worm algorithm. Besides a superfluid phase, we find various solid and supersolid phases. We show that, unlike what was found previously for the case of nearest-neighbor interaction, supersolid phases are stabilized by doping the solids not only with particles but with holes as well. We further study the stability of these quantum phases against thermal fluctuations. Finally, we discuss pair formation and the stability of the pair checkerboard phase formed in a bilayer geometry, and we suggest experimental conditions under which the pair checkerboard phase can be observed.

Figure 1

Phase diagram of Hamiltonian (1) as a function of $V_{\mathrm{dd}}/J$ and particle density $n$, computed via QMC simulations, at $U/t=20$ (see text). CDW I, CDW II: Charge-density waves at $n=0.5$ and $n=1$, and MI: Mott insulator. The shaded region corresponds to solids stabilized at different rational fillings.

Figure 2

Configurations corresponding to the phases stabilized by model Eq. (1) as shown in Fig. 1. The radius of each sphere is proportional to the density at a given site for a specific Monte Carlo configuration.

Figure 3

Critical temperatures for the disappearance of off-diagonal long-range order (solid black line) and diagonal long-range order (dotted blue line) for $V_{\mathrm{dd}}/t=6$. The SS phase disappears via a two-step transition (see text for details).

Figure 4

The superfluid stiffness ${\rho }_s$ vs temperature $T/t$ at $V/t=6$ and $n=0.469$ for $L=$ 8, 12, 16, 20, and 24 shown using open squares, filled circles, filled squares, open circles, and filled stars, respectively. The supersolid phase melts in two stages. Inset: The Kosterlitz-Thouless scaling described in [20] used to find the critical temperature for the first transition. At $T_{\mathrm{KT},\mathrm{SS}}\approx 0.20\pm 0.01$ the SS melts into a liquid-crystal-like phase. The critical temperature is given by the intersection of the dashed line of best fit with the vertical axis.

Figure 5

Plot of the minimum value $V_{\mathrm{dd}}/t$ needed to stabilize the PCB phase in the case of a bilayer geometry as a function of the distance between the layers $d_z/a$. Once the layers are separated by $d_z/a>2.8$ they behave as independent layers; that is, the CB solids on the two layers are not correlated and the minimum $V_{\mathrm{dd}}/t$ has saturated to the case of a single layer. Note that these results where obtained using a cutoff of three nearest neighbors. This results in a shift of $\sim 7\%$ of the minimum $V_{\mathrm{dd}}/t$ for the single layer.