The normal and superfluid fractions of a liquid can be obtained by
measuring the momenta of inertia of a rotating bucket. Consider a
liquid which is inserted between two cylindrical walls of radii
and . If then the system can be described as moving
between two planes. Let us denote by
the ground state
energy of the system in equilibrium with the walls which move with
velocity and the ground state energy of the system at
rest. The difference between the energies
and is due
to the superfluid component, which remains immobile in contrast to
the normal component which is carried along by the moving walls.
Thus, the superfluid fraction can be defined as
(3.48)
Let us introduce the wave-functions and
related to
the wave-functions of the system in the reference frames at rest
and in motion.
(3.49)
(3.50)
These wavefunctions satisfy the Schrödinger equation with the
following Hamiltonians
(3.51)
for the reference frame at rest and
(3.52)
for the reference frame at moving with velocity .
In the reference frame at rest one has
(3.53)
The Schrödinger equation in the moving frame is instead
(3.54)
Looking at (3.53) and (3.54) it is easy to write
the Bloch equations for the Green's functions in the rest frame
and in the moving frame
(3.55)
and
(3.56)
In general the wavefunction
of the system satisfies the
Schrödinger equation (3.2) and its
evolution in time is described by
(3.57)
The wavefunctions evolves in time as
(3.58)
so, substitution of (3.49) or (3.50) into (3.58)
gives
where the operator is defined as
.
Let us calculate the trace of the Green's function. From
(3.60) it follows that the trace of the Green's
function is equal to
(3.61)
Here it is possible to use the permutation property of the trace
(3.62)
This formula means that the trace of the Green's function is
unaffected by the presence of the trial wavefunction .
(3.63)
After long enough time of evolution the traces of the Green's function
is fixed by the ground state energy
Analogously, the trace of
is fixed
by the ground state energy
in the moving frame
(3.66)
The Green's function has to comply with periodic boundary conditions,
i.e. it must remain the same if one of the arguments is shifted
by the period
(3.67)
(3.68)
Let us define a new Green's function
in such a way that
(3.69)
The Green's function
satisfies the same
Bloch equation (3.55) as
, but the
boundary conditions differ from (3.67,
3.68) by the presence of a phase factor
(3.70)
Results (3.64) and (3.66) give the following
relation
(3.71)
By assuming that
one gets
(3.72)
The ratio of the traces is related to the energy difference
(3.73)
The Green's function coincides with apart when the
boundary conditions are invoked. Let us introduce the winding
number [25],
which counts how many times the boundary conditions
were used during the time evaluation
(3.74)
In the case of slowly moving walls, i.e. when
, the exponential
can be expanded in a Taylor series
(3.75)
Let us define through the distance the particles have gone
during the time
(3.76)
The average value of the linear term is equal to zero and the final
result is
(3.77)
An interpretation of this result is that the superfluid fraction is
equal to the ratio between the diffusion constant
of the
center of the mass of the system and the free diffusion constant
3.1
(3.78)
where the diffusion constants are defined as
(3.79)
(3.80)
where the center of the mass of the system is
(3.81)
By calculating the ratio
as a function of time one finds
that this ratio starts from at small time step, decreases and
finally reaches a constant plateau, In practice the best way of
finding the asymptotic value is to fit the ration
with
the function
, where , ,
are fitting parameters [26].
It is worth to remind that the calculation of is
independent of the choice of the trial wave-function and similarly
to the calculation of energy the superfluid fraction is a pure
estimator.
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