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1D system: beyond local density approximation

In order to account for effects beyond local density approximation we have also applied the DMC method to a system of $N$ particles interacting through the Lieb-Liniger Hamiltonian in the presence of harmonic confinement

$\displaystyle \hat H^{trap}_{LL}= N\hbar\omega_\perp - \frac{\hbar^2}{2m}\sum_{...
...}
+g_{1D}\sum_{i<j}\delta(z_i-z_j)+\sum_{i=1}^N \frac{m\omega_z^2 z_i^2}{2} \;.$     (3.8)

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 $N$, the ratio $a_{3D}/a_\perp $ fixing the strength of the contact potential $g_{1D}$ and the anisotropy parameter $\lambda $ fixing the strength of the longitudinal confinement in units of $\hbar\omega_\perp$.

The importance sampling is realized through the Bijl-Jastrow trial wave function (2.37). The one-body term is of gaussian form (2.41) $f_1(z)=\exp\{-\alpha_z z^2\}$. The two-body term $f_2(z)$ 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 $\alpha _z$ and the matching distance $R_{m}$. 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 $kR_{m}$ is small, $f_2(z)\simeq 1, z=0$ and the contact potential is almost transparent. On the other hand, if $kR_{m}$ approaches $\pi/2$, $f_2(z)\to 0,z=0$ and the contact potential behaves as an impenetrable barrier.


next up previous contents
Next: Results Up: Theory Previous: 1D system: local density   Contents
G.E. Astrakharchik 15-th of December 2004