Next: Results
Up: BEC-BCS crossover
Previous: Introduction
  Contents
The homogeneous two-component Fermi gas is described by the Hamiltonian
|
(9.2) |
where denotes the mass of the particles, and
label, respectively, spin-up and spin-down particles and
, being the total number of atoms. We model the
interspecies interatomic interactions using an attractive square-well potential:
for , and otherwise. In order to ensure that the mean
interparticle distance is much larger than the range of the potential we use
, where
is the gas number density. By varying the
depth of the potential one can change the value of the -wave scattering
length, which for this potential is given by
, where
. We vary is the range: . For
the potential does not support a two-body bound state and . For
, instead, the scattering length is positive, , and a molecular
state appears whose binding energy is determined by the trascendental
equation
, where
. The value
corresponds to
the unitary limit where and .
In the present study we resort to the Fixed Node Monte Carlo technique described in
Sec. 2.4. We make use of the following trial wave functions. A BCS wave
function
|
(9.3) |
and a Jastrow-Slater (JS) wave function
|
(9.4) |
where is the antisymmetrizer operator ensuring the correct antisymmetric
properties under particle exchange. In the JS wave function, Eq. (9.4), the
plane wave orbitals have wave vectors
, where is the size of the periodic cubic box fixed by ,
and are integer numbers. The correlation functions and
in Eqs. (9.3)-(9.4) are constructed from solutions of the two-body
Schrödinger equation with the square-well potential . In particular, in the
region we take for the function the bound-state solution
with energy and in the region the unbound-state
solution corresponding to zero scattering energy:
for and
for . In the unitary limit, ,
.
The JS wave function , Eq. (9.4), is used only in the region of
negative scattering length, , with a Jastrow factor
for . In order to reduce possible size effects due to the long range tail
of , we have used
for ,
with a matching point. The coefficients and are fixed by
the continuity condition for and its first derivative at ,
whereas the parameter is chosen in such a way that goes
rapidly to a constant. Residual size effects have been finally determined carrying
out calculations with an increasing number of particles , 38, and 66. In the
inset of Fig. 9.1 we show the dependence of the energy per particle
on in the unitary limit. Similar studies carried out in the BEC and BCS
regime show that the value is optimal since finite-size corrections in the
energy are below the reported statistical error in the whole BEC-BCS crossover. We
have also checked that effects due to the finite range of the potential are
negligible.
Figure:
Energy per particle in the BEC-BCS crossover. Solid symbols refer to
results obtained with the trial wave function , open symbols refer to
the ones obtained with . The red dot-dashed line is the expansion
(9.5) holding in the BCS region and the blue dotted line corresponds to the
binding energy . Inset: finite size effects in the unitary limit
.
|
Next: Results
Up: BEC-BCS crossover
Previous: Introduction
  Contents
G.E. Astrakharchik
15-th of December 2004