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
and the superfluid fraction is less
than the condensate fraction .
We have investigated the dependence of the superfluid and
condensate fraction on the density and the strength of
disorder . 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
() the superfluid fraction is described correctly up to
density
while the condensate fraction
starts to deviate much earlier. By increasing disorder we find
agreement over a smaller range in density. For the
superfluid fraction agrees with the Bogoliubov prediction only up
to
and the condensate fraction only
up to
. The strength of disorder is
here larger than the critical value and Bogoliubov model
predicts
. 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 we find no quantitative agreement for , at large
densities, however, we find
.
Figure 4.4:
Condensate fraction (open symbols) and superfluid fraction
(solid symbols) as a functions of density , for , , . The solid curve is the Bogoliubov
prediction for the superfluid fraction [Eq.(2.65)], the dashed curve for the
condensate fraction [Eq.(2.22)].
Let us now fix
and study the dependence of
and on the strength of disorder From
Fig. 4.5 one sees that for very weak disorder (i.e. small
) Bogoliubov results are valid. For larger disorder we find
deviations. Bogoliubov model predicts a linear dependence on ,
with a different slope for and . We find
instead that the two decrease togrther up to the strong disorder
regime where
as in Fig. 4.4.
Figure 4.5:
Condensate fraction and superfluid fraction
as a function of
. Here
and .
The dashed and solid lines show Bogoliubov predictions for
and respectively.
A different behavior exhibited at the larger density
as shown in Fig.4.6. Even in the pure case () 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 , analogously to the case ,
and decrease linearly with a similar slope.
Figure 4.6:
Condensate fraction and superfluid fraction
as a function of
. Here
and .
The dashed and solid lines show Bogoliubov predictions for
and respectively.
Let us comment on the result
which we find
for large values of (see Fig. 4.4 and 4.5).
This result is highly unusual since in general4.1
.
For example, in liquid He at
low temperatures only of the particles are
in the condensate, although the system is entirely superfluid.
An extreme example is provided by two-dimensional Bose systems at
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
, or even to realize a normal system
(i.e. ) 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
(4.9)
The percolation threshold is given by [39].
In the case of the results of Fig. 4.5
(i.e. and ) the percolation threshold
corresponds to .
This means that the system is approaching the percolation
limit and we can expect that
.
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