The energy of a dilute Bose gas in the presence of impurities
is given by result (2.17). It is derived under the assumptions that
the gas parameter is small and the external field is
weak.
Using the DMC algorithm we investigate the dependence of the
ground-state energy both on the density and on the strength
of the disorder
.
The main
contribution to the energy comes from the mean-field term
(see result (1.14) obtained from the Gross-Pitaevskii equation).
In order to better understand the results for the energy
it is useful to subtract the mean-field term
from the total energy .
Figure 4.3:
Beyond mean-field energy per particle as
a function of density for different strengths of disorder
. The solid lines correspond to the analytical prediction
(2.17)
Let us first consider the energy dependence on . For weak
disorder () the predictions well agree with the results of
DMC simulations. By increasing while keeping fixed one
sees deviation from analytical prediction.
The Bogoliubov model is valid if the gas parameter is small.
Fig. 4.3 shows that the values of the gas parameter where
the theoretical prediction holds depends on the strength of
disorder. For weak disorder () agreement is found up to very
high densities
. By increasing the amount of
disorder deviations appear for smaller values of . For numerical and analytical results coincide up to
and for very strong disorder no agreement
is found at densities
.
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