Superfluid-insulator transition in the disordered two-dimensional Bose-Hubbard model

We investigate the superfluid-insulator transition in the disordered two-dimensional Bose-Hubbard model through quantum Monte Carlo simulations. The Bose-Hubbard model is studied in the presence of site disorder, and the quantum critical point between the Bose glass and superfluid is determined in the grand canonical ensemble at $\mu /U=0$ (close to $\rho =0.5$), $\mu /U=0.375$ (close to $\rho =1$), and $\mu /U=1$ as well as in the canonical ensemble at $\rho =0.5$ and 1. Particular attention is paid to disorder averaging, and it is shown that a large number of disorder realizations are needed in order to obtain reliable results. Typically, more than $100000$ disorder realizations were used. In the grand canonical ensemble, we find $Z{t}_c/U=0.112\left(1\right)$ with $\mu /U=0.375$, significantly different from previous studies. When compared to the critical point in the absence of disorder ($Z{t}_c/U=0.2385$), this result confirms previous findings showing that disorder enlarges the superfluid region. At the critical point, we then study the dynamic conductivity.

Figure 1

The phase diagram of the 2D Bose-Hubbard model. Shaded areas are the Mott insulating phases for zero disorder as determined from strong-coupling expansions in Refs. 37 and38. The mean-field phase boundaries and constant density profiles for zero disorder are shown as red dotted lines. The dashed line indicates the constant chemical potential $\mu /U=0.375$. The solid triangle indicates the location of the transition to the Mott phase in the absence of disorder as determined by SSE simulations along the dashed line from Ref. 39. The three solid squares from bottom up are for the locations of superfluid to Bose-glass transitions in the presence of disorder at $\mu /U=0,0.375,1$, respectively, as determined from SSE simulations in this work.

Figure 2

A sketch of the conductivity with the two regimes $\hslash \omega /\left(k_{B}T\right)\gg 1$ and $\hslash \omega /\left(k_{B}T\right)\ll 1$ clearly apparent. Also shown is the difference between the universal dc conductivity ${\sigma }^{*}$ and the conductivity ${\overline{\sigma }}^{*}$ obtained from extrapolating the high-frequency numerical data. An approximate position of the first nonzero Matsubara frequency is also shown.

Figure 3

The energy difference between two replicas $E_{\alpha }-E_{\gamma }$ as a function of MCS averaged over 100 disorder samples. Results are shown for $L=16$, $t=0.5$, $U=11$, and $\beta =12.8$. We start recording the energy difference after 50 MCS.

Figure 4

The autocorrelation function $C_{W^2}\left(t_a\right)$ for $W^2$ averaged over 100 disorder samples for $L=16$, $t=0.5$, $U=11$, and $\beta =12.8$. Note that the $y$ axis is on a $\log$ scale.

Figure 5

Convergence of the superfluid density for $t=0.5$ and $U=11$ as a function of the number of disorder samples $N_{\mathrm{sample}}$ for various lattice sizes $L=6,8,10,$ and 16.

Figure 6

Superfluid density scaling from canonical simulations, i.e., fixed particle number with density $\rho =0.5$, for $\Delta =12$ and $t=0.5$, for various lattice sizes, $L=6,8,10,$ and 16.

Figure 7

Dynamic conductivity scaling plot $\sigma \left({\omega }_k\right)/{\sigma }_Q$ vs Matsubara frequency ${\omega }_k$ for $Zt/U_c=0.182$ and $t=0.5$.

Figure 8

(a) Scaling curves of the superfluid density from grand canonical simulations at $\mu /U=0.375$. The inset shows the slope of these curves at the QCP on a log-log scale. (b) The compressibility $\kappa$ in the same region.

Figure 9

Dynamic conductivity $\sigma \left({\omega }_k\right)/{\sigma }_Q$ vs Matsubara frequency ${\omega }_k$ for $Z{t}_c/U=0.112$ in a disordered grand canonical system at $\mu /U=0.375$.