next up previous contents
Next: Scaling behaviour Up: Results Previous: Ground-state energy   Contents


Superfluid density and condensate fraction

The predictions of Bogoliubov theory for the condensate and superfluid fraction are given by (2.22) and (2.65). As already mentioned in section 2.3.3 a very interesting consequence of these results is that for any value of $na^3$ and $R>5.1$ the superfluid fraction $\rho _s/\rho $ is less than the condensate fraction $N_0/N$. We have investigated the dependence of the superfluid and condensate fraction on the density $na^3$ and the strength of disorder $R$. The results of the DMC simulations are presented in Fig. 4.4. At low density the DMC results always confirm the predictions of Bogoliubov model, but the region of validity of the model depends on the strength of disorder. If disorder is weak ($R = 2$) the superfluid fraction is described correctly up to density $na^3 \approx 5\cdot 10^{-3}$ while the condensate fraction starts to deviate much earlier. By increasing disorder we find agreement over a smaller range in density. For $R = 12.5$ the superfluid fraction agrees with the Bogoliubov prediction only up to $na^3 \approx
5\cdot 10^{-4}$ and the condensate fraction only up to $na^3 \approx 2\cdot 10^{-4}$. The strength of disorder is here larger than the critical value $R_c=5.1$ and Bogoliubov model predicts $\rho_s/\rho < N_0/N$. Our results show that the condensate fraction decreases faster than predicted and we do not see this phenomenon. In the presence of very strong disorder $R = 100$ we find no quantitative agreement for $na^3 > 10^{-5}$, at large densities, however, we find $\rho_s/\rho < N_0/N$.

Figure 4.4: Condensate fraction $N_0/N$ (open symbols) and superfluid fraction $\rho _s/\rho $ (solid symbols) as a functions of density $na^3$, for $R = 2$, $R = 12.5$, $R = 100$. The solid curve is the Bogoliubov prediction for the superfluid fraction [Eq.(2.65)], the dashed curve for the condensate fraction [Eq.(2.22)].
\includegraphics[width=\textwidth]{Figure5.eps}

Let us now fix $na^3 = 10^{-4}$ and study the dependence of $\rho _s/\rho $ and $N_0/N$ on the strength of disorder From Fig. 4.5 one sees that for very weak disorder (i.e. small $R$) Bogoliubov results are valid. For larger disorder we find deviations. Bogoliubov model predicts a linear dependence on $R$, with a different slope for $\rho _s/\rho $ and $N_0/N$. We find instead that the two decrease togrther up to the strong disorder regime where $\rho_s/\rho < N_0/N$ as in Fig. 4.4.

Figure 4.5: Condensate fraction $N_0/N$ and superfluid fraction $\rho _s/\rho $ as a function of $R = \chi (b/a)^2$. Here $na^3 = 10^{-4}$ and $b/a = 5$. The dashed and solid lines show Bogoliubov predictions for $N_0/N$ and $\rho _s/\rho $ respectively.
\includegraphics[width=\textwidth]{Figure1.eps}

A different behavior exhibited at the larger density $na^3 = 10^{-2}$ as shown in Fig.4.6. Even in the pure case ($R = 0$) the condensate fraction does not agree with Bogoliubov prediction and by increasing disorder deviations are more evident. On the contrary, the superfluid density well agrees with Bogoliubov prediction. For small values of $R$, analogously to the case $na^3 = 10^{-4}$, $\rho _s/\rho $ and $N_0/N$ decrease linearly with a similar slope.

Figure 4.6: Condensate fraction $N_0/N$ and superfluid fraction $\rho _s/\rho $ as a function of $R = \chi (b/a)^2$. Here $na^3 = 10^{-2}$ and $b/a = 2$. The dashed and solid lines show Bogoliubov predictions for $N_0/N$ and $\rho _s/\rho $ respectively.
\includegraphics[width=\textwidth]{Figure2.eps}

Let us comment on the result $\rho_s/\rho < N_0/N$ which we find for large values of $R$ (see Fig. 4.4 and 4.5). This result is highly unusual since in general4.1 $\rho_s/\rho < N_0/N$. For example, in liquid $^4$He at low temperatures only $10\%$ of the particles are in the condensate, although the system is entirely superfluid. An extreme example is provided by two-dimensional Bose systems at $T\ne 0$ which do not exhibit Bose-Einstein condensation (Hohenberg theorem), but do exhibit superfluidity below the Kosterlitz-Thoules transition temperature. It is interesting to understand how it is possible to realize a system with $\rho_s/\rho < N_0/N$, or even to realize a normal system (i.e. $\rho_s = 0$) with nonzero condensate. A possible answer is by realizing an adsorbing medium with isolated cavities of typical size larger than the healing length. The gas gets trapped in the cavities and a condensate can still exist in each of the cavities, while the overall conductivity is absent. Let us estimate what is the critical value for the excluded volume at which the gas the critical parameters when the fluid can not flow from one side of the box to another (i.e. the percolation threshold). The relative excluded volume for the impurities can be estimated as
\begin{displaymath}
p = \frac{V^{excl}}{V} =
\frac{\frac{4}{3}\pi b^3N_{imp}}{V}
=\frac{4\pi}{3}\chi\left(\frac{b}{a}\right)^3 na^3
\end{displaymath} (4.9)

The percolation threshold is given by $p_c=0.70$ [39]. In the case of the results of Fig. 4.5 (i.e. $na^3 = 10^{-4}$ and $b/a = 5$) the percolation threshold corresponds to $R_c = 350$. This means that the system is approaching the percolation limit and we can expect that $\rho_s/\rho < N_0/N$.
next up previous contents
Next: Scaling behaviour Up: Results Previous: Ground-state energy   Contents