Equilibrium phases of dipolar lattice bosons in the presence of random diagonal disorder

Ultracold gases offer an unprecedented opportunity to engineer disorder and interactions in a controlled manner. In an effort to understand the interplay between disorder, dipolar interactions, and quantum degeneracy, we study two-dimensional hard-core dipolar lattice bosons in the presence of on-site bound disorder. Our results are based on large-scale path-integral quantum Monte Carlo simulations by the worm algorithm. We study the ground-state phase diagram at a fixed half-integer filling factor for which the clean system is either a superfluid at a lower dipolar interaction strength or a checkerboard solid at a larger dipolar interaction strength. We find that, even for weak dipolar interactions, superfluidity is destroyed in favor of a Bose glass at a relatively low disorder strength. Interestingly, in the presence of disorder, superfluidity persists for values of the dipolar interaction strength for which the clean system is a checkerboard solid. At a fixed disorder strength, as the dipolar interaction is increased, superfluidity is destroyed in favor of a Bose glass. As the interaction is further increased, the system eventually develops extended checkerboard patterns in the density distribution. Due to the presence of disorder, though, grain boundaries and defects, responsible for a finite residual compressibility, are present in the density distribution. Finally, we study the robustness of the superfluid phase against thermal fluctuations.

Figure 1

Phase diagram of the system described by Eq. (1) at filling factor $n=0.5$. The horizontal and vertical axes are the dipole-dipole repulsive interaction strength $V/J$ and the disorder strength $\mathrm{\Delta }/J$, respectively. Solid red circles represent superfluid-to-insulator transition points as determined by standard finite-size scaling. The solid line is a guide to the eye. The blue circle is the interaction strength at which superfluidity disappears in favor of a checkerboard solid in the clean system. The inset shows finite-size scaling of the superfluid stiffness ${\rho }_s$. We plot ${\rho }_{s}L$ vs $V/J$ at $\mathrm{\Delta }/J=7$ for system sizes $L=12$, 16, 20, and 24 (red circles, blue squares, green up triangles, and purple down triangles, respectively). The transition point corresponds to the value of $V/J$ where curves referring to different system sizes cross. Here, $V_c/J=3.68\pm 0.25$. Error bars are within the symbols if not visible in the plots.

Figure 2

Imaginary-time average of the density distribution for a given Monte Carlo configuration and a specific disorder realization at (a) $V/J=4.2, \mathrm{\Delta }/J=2$, and (b) $V/J=4.0, \mathrm{\Delta }/J=7$.

Figure 3

Compressibility $\kappa$ as a function of $V/J$ for system sizes $L=12$, 16, 20, 24, and 28 (red squares, blue up triangles, green down triangles, orange diamonds, and purple stars, respectively) at fixed disorder strength (a) $\mathrm{\Delta }/J=4$, (b) 5, and (c) 7. Compressibility remains finite beyond the superfluid-to-insulator transition, confirming that the superfluid phase disappears in favor of a Bose glass. At a large enough interaction, $\kappa$ seems to plateau. Upon approaching the plateau, extended CB patterns appear in the density distribution (see text for details). Error bars are within the symbols if not shown in the figures.

Figure 4

Imaginary-time average of the density distribution for a given Monte Carlo configuration and a specific disorder realization at (a) $V/J=10.0, \mathrm{\Delta }/J=7.0$, (b) $V/J=4.5, \mathrm{\Delta }/J=7.0$, and (c) $V/J=1.0, \mathrm{\Delta }/J=9.5$. The radius of each circle is proportional to the density at that site. Defects and grain boundaries within the checkerboard order are present in (a). They contribute to a small residual compressibility.

Figure 5

All plots refer to $V/J=2$. (a) Main plot: Critical temperature $T_c/J$ for disappearance of superfluidity via a KT transition as a function of $\mathrm{\Delta }/J$. Bottom-left-hand inset: Superfluid stiffness ${\rho }_s$ as a function of $\mathrm{\Delta }/J$ for system size $L=24$. Top-right-hand inset: Compressibility $\kappa$ as a function of $\mathrm{\Delta }/J$ for system size $L=24$. (b) Superfluidity ${\rho }_s$ as a function of $T/J$ for $L=12$, 16, 20, 24, 36, and 60 (blue squares, green up triangles, purple down triangles, orange diamonds, pink stars, and yellow asterisks, respectively), at $\mathrm{\Delta }/J=4$. The dotted line corresponds to ${\rho }_s=T/\pi$. Its intersection points with each ${\rho }_s$ curve give “critical” temperatures $T_c\left(L\right)/J$ for a finite system. (c) $T_c\left(L\right)/J$ vs $1/{ln}^{2}L$. Error bars are within symbol size if not visible in the plots.