Characterization of the Bose-glass phase in low-dimensional lattices

We study by numerical simulation a disordered Bose-Hubbard model in low-dimensional lattices. We show that a proper characterization of the phase diagram on finite disordered clusters requires the knowledge of probability distributions of physical quantities rather than their averages. This holds in particular for determining the stability region of the Bose-glass phase, the compressible but not superfluid phase that exists whenever disorder is present. This result suggests that a similar statistical analysis should be performed also to interpret experiments on cold gases trapped in disordered lattices, limited as they are to finite sizes.

Figure 1

(Color online) Upper panel: superfluid stiffness ${\rho }_s$ as a function of the disorder strength $\Delta /t$ for different densities of hard-core bosons. Lower panel: low-density phase diagram of the hard-core bosonic model. Calculations have been done on a $2\times 50$ ladder system.

Figure 2

(Color online) Distribution $P\left(E_g\right)$ of the gap in the 1D Bose-Hubbard model for different values of $U/t$ and $L=60$ sites.

Figure 3

(Color online) Superfluid stiffness ${\rho }_s$ (solid lines on the left) and the minimum gap $E_g^{\text{min}}$ (dashed lines on the right) for different clusters. Two-leg (with $2\times 40$ sites) and three-leg (with $3\times 50$) ladders are shown; the 2D case with a $12\times 12$ cluster is also reported for comparison. In all cases, the disorder strength is $\Delta /t=5$. Arrows indicate the opening of the charge gap according to $\Delta =E_g^{\text{clean}}/2$.

Figure 4

(Color online) Distribution $P\left(E_g\right)$ of the gap for the three-leg ladder $3\times 50$ (upper raw) and 2D $12\times 12$ (lower raw) for different values of $U/t$. In all cases, the disorder strength is $\Delta /t=5$.

Figure 5

(Color online) Local density for one disorder realization of the $2\times 40$ ladder. Darker (brighter) spots indicate lower (higher) densities. The values of the stiffness ${\rho }_s$ for the same disorder realization are also reported.

Figure 6

(Color online) Upper panel: variational results for the excitation gap for a $2\times 40$ ladder. Lower panels: momentum distribution $n_k$ for the same cluster.