The problem of bosons in the presence of disorder has generated
much theoretical and experimental interest.
The superfluid fraction in liquid He has been measured for
different types of adsorbing porous media. In vycor, which has
small ( Å) pores and porosity of the superfluid
transition is considerably suppressed, but exhibits the same
critical exponent as in the bulk [7]. In contrast, in
aerogel, which is characterized by larger pores with a broad
distribution of sizes and porosity , the superfluid
transition is changed by only a few milli-kelvins while the
critical exponents are quite different from the bulk
[9,10].
Some experimental studies have also investigated, by measuring the
dynamic structure factor, the nature of the elementary excitations
in these systems
[28]
and the role played by the condensate fraction
[].
Theoretical studies of these effects have been proposed, mostly
concerning models on a lattice. Many of the theoretical works
address the problem of the superfluid-insulator transition and the
critical behavior near the phase transition [30,8,31]. The boson localization and the structure of the Bose-glass
phase have also been investigated.
Quantum Monte Carlo techniques have been used for numerical
simulations of the disordered Bose systems at low temperatures.
Most of them concern systems on a lattice using the Bose-Hubbard or
equivalent models. These studies have been carried out in 1D
[32], 2D [21,33,34]
and 3D [35,36] both at zero and finite temperatures. The structure of
the phase diagram has been investigated and the properties of the
superfluid, Mott insulator and Bose-glass phase have been
addressed.
There are very few simulations of disordered boson systems in the
continuum. In ref. [37] the effect of impurities on
the excitation spectrum in liqiud He is investigated using
PIMC. The same technique is applied to study the effect of disorder
on the superfluid transition in a Bose gas [38].
We apply DMC to study a hard-sphere Bose gas at zero temperature in
the presence of hard-sphere quenched impurities. Hard sphere
quenched impurities are easy are easy to implement in a numerical
simulation and provide a reasonable model for liquid He in the
porous media. Another possible physical realization of this model
is given by trapped gases in the presence of heavy impurities.
The free parameters in our simulations are:
the density of the particles , where is the
diameter of the hard-sphere particle scattering length,
the concentration of impurities
fixed
by the ratio of the number of impurities to the number of particles
used in the simulation,
the ratio , where
with
radius of the hard-sphere impurity.
The same parameters (2.8) were used in the
perturbative analysis discussed previously (see section ).
The goals of our study are:
recover the ``weak'' disorder regime
where the results of perturbation theory apply,
verify the scaling behavior of the effects due to
disorder in terms of the single parameter
as
predicted from the Bogoliubov model
understand if one can realize situations where the
superfluid density is smaller than the condensate fraction
investigate if within our model there exists a quantum
phase transition for strong disorder