The FN-DMC energies for atoms and the potential
with
are shown in Fig. 9.1 and in Table 31 as a function of the
interaction parameter
. The numerical simulations are carried out both with
the BCS wave function, Eq. (9.3), and with the JS wave function, Eq.
(9.4). For
we find that
gives lower energies,
whereas for smaller values of
, including the unitary limit and the BEC
region, the function
is preferable. This behavior reflects the level of
accuracy of the variational ansatz for the nodal structure of the trial wave
function. We believe that in the intersection region,
, both wave
functions
and
give a poorer description of the exact nodal
structure of the state, resulting in a less accurate estimate of the energy. In the
BCS region,
, our results for
are in agreement with the
perturbation expansion of a weakly attractive Fermi gas9.1 [HY57,LY57]
In the unitary limit we find
, with
. This result
is compatible with the findings of Refs. [CCPS03,CPCS04] obtained using a
different trial wave function which includes both Jastrow and BCS correlations. The
value of the parameter
has been measured in experiments with trapped
Fermi gases [OHG+02,BAR+04b,BKC+04], but the precision is too low
to make stringent comparisons with theoretical predictions. In the region of
positive scattering length
decreases by decreasing
. At approximately
, the energy becomes negative, and by further decreasing
it rapidly approaches the binding energy per particle
indicating the
formation of bound molecules [CPCS04]. The results with the binding energy
subtracted from
are shown in Fig. 9.2. In the BEC region,
, we find that the FN-DMC energies agree with the equation of state of a
repulsive gas of molecules
![]() |
![]() |
In Fig. 9.4 we show the results for the pair correlation function of
parallel,
, and antiparallel spins,
. For parallel spins,
must
vanish at short distances due to the Pauli principle. In the BCS regime the effect
of pairing is negligible and
coincides with the
prediction of a noninteracting Fermi gas
. This result
continues to hold in the case
, where it is consistent with the picture
of a gas in the unitary regime as a noninteracting Fermi gas with effective mass
. In the BEC regime the static structure factor
of composite
bosons can be estimated using the Bogoliubov result:
, where
is the Bogoliubov dispersion
relation for particles with mass
, density
and coupling constant
. The pair distribution function
of composite bosons,
obtained through
using
the value
, is shown in Fig. 9.4 for
and
compared with the FN-DMC result. For large distances
, where Bogoliubov
approximation is expected to hold, we find a remarkable agreement. This result is
consistent with the equation of state in the BEC regime and shows that structural
properties of the ground state of composite bosons are described correctly in our
approach. For antiparallel spins,
exhibits a large
peak at short distances due to the attractive interaction. In the BEC regime the
short range behavior is well described by the exponential decay
fixed by
the molecular wave function
. In the unitary regime correlations extend
over a considerably larger range compared to the tightly bound BEC regime. In the
BCS regime the range of
is much larger than
and is determined by the coherence length
, where
is the gap parameter. In this regime the wave function we use does not
account for pairing and is inadequate to investigate the behavior of
.
![]() |