Equilibrium phases of two-dimensional bosons in quasiperiodic lattices

We report on results of quantum Monte Carlo simulations for bosons in a two-dimensional quasiperiodic optical lattice. We study the ground state phase diagram at unity filling and confirm the existence of three phases: superfluid, Mott insulator, and Bose glass. At lower interaction strength, we find that sizable disorder strength is needed in order to destroy superfluidity in favor of the Bose glass. On the other hand, at large enough interaction superfluidity is completely destroyed in favor of the Mott insulator (at lower disorder strength) or the Bose glass (at larger disorder strength). At intermediate interactions, the system undergoes an insulator to superfluid transition upon increasing the disorder, while a further increase of disorder strength drives the superfluid to Bose glass phase transition. While we are not able to discern between the Mott insulator and the Bose glass at intermediate interactions, we study the transition between these two phases at larger interaction strength and find no evidence of a Mott-glass-like behavior.

Figure 1

Ground state phase diagram of the system described by Eq. (1) at filling factor $n=1$. The horizontal and vertical axes are the onsite interaction strength $U/J$ and disorder strength $\Delta /J$, respectively. Using these two parameters as tuning knobs, the system can form a Mott-Insulator (MI), a superfluid (SF), and a Bose glass (BG). Simulations results for the SF-insulator phase boundary are shown using solid orange circles (the solid orange line is a guide to the eye), while solid purple squares (the dashed line is a guide to the eye) correspond to the phase boundary between the MI and BG phases. At lower disorder and intermediate interactions we are unable to distinguish between the MI and the BG (dashed blue region). The shaded, light blue region to the left of the dot-dashed line corresponds to $U/J\le 10$ which has not been explored (see text).

Figure 2

Main plot: Scaled superfluid stiffness ${\rho }_s{L}^{-(d+z-2)}$ with $z=2$, as a function of $\Delta /J$ for $U/J=22$ and $L=21$, 34, 55, and 89 using red circles, blue squares, empty black squares, and black diamonds, respectively. In these simulations we have used $\beta ={(L/2)}^2$ to scale the imaginary time dimension $L_{\tau }$. Inset: Data collapse using $\nu =0.67,a=-9.4,\omega =-0.9$, and ${\overline{\Delta }}_c=10.21$ corresponding to the critical point extracted from the main plot. Here $\overline{\Delta }=\Delta /J$. The symbols are the same as those used for the main plot.

Figure 3

Main plot: $\kappa$ vs $\Delta /J$ for $U/J=45,L=21$, and $\beta =L/2$. The compressibility becomes finite at $\Delta /J\sim 23.5$ and plateaus at $\Delta /J\sim 24.5$. Inset: The top and bottom panels show $\kappa L^{d-z}$ versus $\Delta /J$ for $z=0.75$ and $z=1$, respectively, at $U/J=45$. The scaled compressibility is shown for $L=21,$ 34, 55, and 89 using black squares, red circles, blue triangles, and green diamonds, respectively. Our data indicates that $0.75\lesssim z\lesssim 1.25$.

Figure 4

The plot shows $log\kappa$ as a function of $\beta$, at $U/J=45$ and $L=21$ for $\Delta /J=22$, 23.5, 23.6, 23.7, 23.8, 23.9, and 26. Below the critical point $\Delta /J=23.76\pm 0.05$ (see inset of Fig. 3) the behavior is consistent with that of a MI. Above the transition, the behavior is consistent with that of the BG phase.