In order to account for effects beyond local density approximation we have also
applied the DMC method to a system of particles interacting through the
Lieb-Liniger Hamiltonian in the presence of harmonic confinement
When the number of particles is large, the properties of the ground state of the
Hamiltonian (3.8) coincide with the ones obtained from the LL equation of
state within LDA. However, for small systems one expects deviations and the DMC
method provides us with a powerful tool. The relevant parameters are the same as for
the 3D simulation with the Hamiltonian (3.1): the number of particles , the
ratio
fixing the strength of the contact potential
and the
anisotropy parameter
fixing the strength of the longitudinal confinement
in units of
.
The importance sampling is realized through the Bijl-Jastrow trial wave function
(2.37). The one-body term is of gaussian form (2.41)
. The two-body term
is given by the formula
(2.51). The construction of the discussed in details in
Sec. 2.5.4.4. There are two variational parameters: gaussian width
and the matching distance
. We fix them by minimizing the variational energy.
We notice that our choice of the two-body Jastrow factor reproduces both the weakly
interacting and the Tonks-Girardeau regime. In fact, if is small,
and the contact potential is almost transparent. On the
other hand, if
approaches
,
and the
contact potential behaves as an impenetrable barrier.