Let us apply the theory developed in the previous section to a
dilute Bose system. As we will see, the system without disorder is
fully superfluid at zero temperature while the presence of
impurities create a depletion of the superfluid density.
According to (2.33) and (2.34) the normal component
is given by the limit of the transverse current-current response
function. Let us consider, for example,
, i.e.
response and in the direction (see
Fig. 2.2). First one has to take the limit and after and . This can be accomplished by
considering
and letting decrease toward
zero.
The -component of the current operator in second
quantization is given by the formula
(2.36)
where in the absence of disorder the particle creation and
annihilation operators can be expressed in terms of quasiparticle
operators by means of the Bogoliubov transformation (1.32).
Once the current (2.36) is calculated, the transverse response
function can be obtained by averaging the commutator
(2.37)
By taking the Fourier transform
(2.38)
the limiting procedure
(2.39)
yields the density of the normal fluid.
It is easy to check that in the absence of disorder
goes to zero as and that
the commutator (2.37) of goes to zero in the limit
. This corresponds to the fact that a homogeneous dilute
Bose gas is completely superfluid at .
A useful check consists in the calculation of the longitudinal
response (see Fig. 2.3). This means taking first
the limit , in
. We
consider
and then let . It is easy to
calculate in Bogloliubov approximation.
For the component of the current operator one has
(2.40)
Within this level of accuracy the limit (2.35) of the
longitudinal component is given by
(2.41)
Let us now study the system in the presence of the random external
potential.Starting from the transformation () between the particles operators
,
and the corresponding quasiparticle
operators
,
one can write
the contribution to the current proportional to the external
potential (there is no need to consider the contribution independent
of the external potential, because as calculated before it is equal to
zero).
(2.42)
Notice that in order to make calculations simpler we take
from the very begging. Result (2.34) is independent of the
order of the two limits.
The response function is then given by