In this section the superfluid fraction will be
obtained directly from Gross-Pitaevskii equation in a perturbative
manner. This derivation is new and the result coincides with the one
obtained from the Bogoliubov model presented in the previous
section.
The Gross-Pitaevskii equation (1.11) for the
condensate wavefunction in the absence of external field takes the form
(2.46)
Let us add a moving impurity that creates an external field
and let us
treat it as a perturbation to the solution
of
the equation (2.46), i.e.
The perturbation follows the moving impurity, so
is a function of .
Let us introduce the new variable
. It means that the coordinate derivative can be related to time derivative
(from now on, the subscript over will be dropped)
Taking the Fourier transform of this equation and treating
as a real constant (i.e. the solution for the uniform case
)
one obtains
(2.52)
where we used the property of Fourier components
.
The substitution of in the equation complex
conjugate of (2.52) gives
(2.53)
The solution for the system of linear equations
(2.52 - 2.53) is given by
(2.54)
The energy has the minimum at fixed for the
ground state function . It means that does not have
terms linear in and
, so
.
Here the last term comes from the linear expansion of the energy
. The term
being quadratic in and
satisfies the Euler identity:
(2.55)
which using the variational equation
(2.56)
can be rewritten as
(2.57)
To start with, let us Fourier transform the first term. Exchanging
time derivatives with gradients by the rule (2.49) one
obtains
(2.58)
The terms of interest are the ones that are quadratic in the
velocity . It means that the term
in
the denominator of (2.54) can be neglected and
turns out to be
(2.59)
The energy does not have terms linear in , because all
terms independent of in (2.59) are even in , so multiplied by and integrated over momentum space
they provide zero contribution to the energy. The only term that is
left is the following
(2.60)
For the calculation of one should consider
taking into account that
and then
integrate it over momentum space
(2.61)
The integral of the second term over momentum space is equal to
zero. The third term is diverging and needs the renormalization of
(as discussed in sections 1.3.1 and 2.2.2) in
order to be calculated correctly. However, its correction does not
depend on and will be omitted.
The energy is defined by the following integral
(2.62)
In the integral
can be replaced by
due to the equivalence of different directions.
Then the integral can be easily calculated if one recall the
following integral identity
(2.63)
The result is the following
(2.64)
where
The term in front of in (2.64) can be interpreted as an
effective mass of the particles which follow the external
perturbation, i.e. the normal (and not superfluid) component of the
fluid. Then normal fraction can then be easily obtained which, as
anticipated, coincides with result (2.45).
(2.65)
Let us compare the results for the superfluid density (2.65)
and the condensate fraction (2.22). It is
interesting to note that in both formulae the effect of disorder
enters as , where
is the universal
scaling parameter which already entered the result (2.17) for
the energy. This means that systems with different disorder
concentration and size of the impurities , but same
experience the same effect due to disorder.
Another interesting result is that disorder is more efficient (by a
factor ) in depleting the superfluid density than the
condensate. Taking into account that even pure systems ()
exhibit a nonzero quantum depletion due to particle interactions,
one infers that at the critical amount of disorder
the depletion of the superfluid density becomes
larger than the depletion of the condensate fraction.
Huang and Meng [16], who first derived results
(2.22) and (2.45) even if for
a different model of disorder, have used them at to predict
two distinct transitions as a function of the amount of disorder :
first a superfluid-insulator transition where followed
by a Bose-Einstein transition where . These authors also
argue that the intermediate phase corresponding to and
should be identified with a Bose-glass phase. However,
in [19] it is stressed that results () and (2.45) are valid in the weak
disorder regime and cannot be applied if the depletion due to
disorder is large.
The range of validity of results (2.22) and
(2.65) will be investigated in detail in section
4.4.2 using Monte Carlo techniques.
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