Universal Scaling of the Conductivity at the Superfluid-Insulator Phase Transition

The scaling of the conductivity at the superfluid-insulator quantum phase transition in two dimensions is studied by numerical simulations of the Bose-Hubbard model. In contrast to previous studies, we focus on properties of this model in the experimentally relevant thermodynamic limit at finite temperature $T$. We find clear evidence for deviations from ${\omega }_k$ scaling of the conductivity towards ${\omega }_k/T$ scaling at low Matsubara frequencies ${\omega }_k$. By careful analytic continuation using Padé approximants we show that this behavior carries over to the real frequency axis where the conductivity scales with $\omega /T$ at small frequencies and low temperatures. We estimate the universal dc conductivity to be ${\sigma }^{*}=0.45(5)Q^2/h$, distinct from previous estimates in the $T=0$, $\omega /T\gg 1$ limit.

Figure 1

Mean-field ground state phase diagram of the 2D Bose-Hubbard model. Shaded areas are the Mott-insulating phases. Dashed lines are constant density profiles in steps of 0.2. ● indicates the location of QCP at the tip of Mott lobe as determined by SSE simulations along the dotted line.

Figure 2

The compressibility (a) and SF density (b) vs $Zt/U$ at fixed $\mu /U=0.375$ and aspect ratio $\beta L^{-z}=0.5$. The dashed vertical line is drawn at $Z{t}_c/U=0.2385$. Error bars are displayed only if larger than the symbol size.

Figure 3

The conductivity $\sigma ({\omega }_k)$ in units of ${\sigma }_Q$ vs Matsubara frequency ${\omega }_k/{\omega }_c$ as obtained from ${\mathcal{H}}_{\mathrm{V}}$ (a). All results have been extrapolated to the thermodynamic limit $L\to \infty$ using the scaling form $f(L)=a+b\exp (-L/\xi )/\sqrt{L}$ [27] by calculating $\rho ({\omega }_k)$ at fixed $L_{\tau }$ using 9 lattice sizes from $L=L_{\tau }\dots 4L_{\tau }$ as shown in (b). $\sigma ({\omega }_k)$ in units of ${\sigma }_Q$ vs Matsubara frequency ${\omega }_k$ as obtained from SSE calculations of ${\mathcal{H}}_{\mathrm{BH}}$, with some typical error bars shown. All results have been extrapolated to the thermodynamic limit by calculating ${\Lambda }_{xx}({\omega }_k)$ for fixed $\beta$ using 5 lattice sizes $L=12,\dots ,30$ (c). Scaling plot of the conductivity data from (a) vs ${\omega }_k/T$. ● denotes extrapolations to $T\to 0$ ($L_{\tau }\to \infty$) at fixed ${\omega }_k/T$ using: $f(L_{\tau })=c+d\exp (-L_{\tau }/{\xi }_{\tau })/\sqrt{L_{\tau }}$ [27] (d).

Figure 4

The real part of the conductivity ${\sigma }^{\prime }$ at the critical coupling in units of ${\sigma }_Q$. The data marked $L_{\tau }$, plotted vs $\omega /{\omega }_c$, were obtained using ${\mathcal{H}}_V$, combined with the analytic continuation of $\rho (\omega /{\omega }_c)$ as explained in the text. Results for $\omega /{\omega }_c\gtrsim 1/2$ are denoted by dotted lines. The data marked SSE, plotted vs $\omega /10$, were obtained by direct SSE simulations of ${\mathcal{H}}_{\mathrm{BH}}$ with $L=20$, $\beta =10$ and subsequent maximum entropy analysis (a). Results as a function of $\omega /T$ (b).