Bose-glass, superfluid, and rung-Mott phases of hard-core bosons in disordered two-leg ladders

By means of Monte Carlo techniques, we study the role of disorder on a system of hard-core bosons in a two-leg ladder with both intrachain ($t$) and interchain ($t^{\prime }$) hoppings. We find that the phase diagram as a function of the boson density, disorder strength, and $t^{\prime }/t$ is far from being trivial. This contrasts with the case of spinless fermions where standard localization arguments apply and an Anderson-localized phase pervades the whole phase diagram. A compressible Bose-glass phase always intrudes between the Mott insulator with zero (or one) bosons per site and the superfluid that is stabilized for weak disorder. At half-filling, there is a direct transition between a (gapped) rung-Mott insulator and a Bose glass, which is driven by exponentially rare regions where disorder is suppressed. Finally, by doping the rung-Mott insulator, a direct transition to the superfluid is possible only in the clean system, whereas the Mott phase is always surrounded by the a Bose glass when disorder is present. The phase diagram based on our numerical evidence is finally reported.

Figure 1

Upper panel: Superfluid stiffness as function of the density $n$ at fixed $t^{\prime }/t=2$ on a lattice with $L=2\times 50$. Lower panel: clean phase diagram of (a) hard-core bosons and (b) spinless fermions on the two-leg ladder as function of $t^{\prime }/t$ and density $n$. The rung-Mott insulator is denoted by RMI, the superfluid by SF, and the band insulator by BI.

Figure 2

Upper left panel: superfluid stiffness ${\rho }_s$ as a function of the disorder strength $\Delta /t$ for different $n$ and $t^{\prime }/t=1$. Upper right panel: superfluid stiffness ${\rho }_s$ as a function of the density for different $\Delta /t$ and fixed $t^{\prime }/t=2$. Lower panel: low-density phase diagram of the hard-core bosonic model for $t^{\prime }/t=1$ and $t^{\prime }/t=2$. Calculations have been done on an $L=2\times 50$ system.

Figure 3

Superfluid stiffness ${\rho }_s$ as a function of $t^{\prime }/t$ for different $\Delta /t$ on a two-leg ladder with $L=2\times 50$ sites. The density has been fixed to $n=0.4$.

Figure 4

Distribution $P(E_g)$ of the gap as a function of disorder strength and fixed density $n=0.5$ and $t^{\prime }/t=2$ on a two-leg ladder with $L=2\times 50$ sites. The clean gap is shown for comparison in the upper-left box as a blue bar.

Figure 5

Left panel: Superfluid stiffness ${\rho }_s$ as a function of the density and several values of $\Delta /t$ and fixed $t^{\prime }/t=2$. The stiffness of the clean system is also shown for comparison. Right panel: Superfluid stiffness ${\rho }_s$ as a function of the disorder bound for several densities close to half-filling and fixed $L=2\times 50$ and $t^{\prime }/t=10$.

Figure 6

Results of the $2\times 50$ lattice for the phase diagram in the vicinity of the rung-Mott phase for $t^{\prime }/t=10$.

Figure 7

a) Tentative zero-temperature phase diagram of hard-core bosons on the two-leg ladder system. (b) Same phase diagram in (a) but for a larger value of disorder $\Delta /t$.