Large system

Figs. 3.7-3.8 refer to a much larger system with $N=100$ and $a_{3D}/a_\perp =0.2$. In this case, we see a clear cross-over from 3D mean field, at large $\lambda$, to 1D LL at small $\lambda$. Important beyond mean-field effects become evident in the energy per particle as $N\lambda a_\perp^2/a^2\sim 1$, corresponding to $\lambda\sim 10^{-3}$. The Tonks-Girardeau regime would correspond to even smaller values of $\lambda$ which are difficult to obtain in our simulation. However, for the smallest values of $\lambda$ reported in Fig. 3.7 we find already very good agreement with the LL equation of state. One should notice that small deviations from mean field are also visible for $\lambda\sim 1$, and are due to high density corrections to the GP equation. The DMC results with the 1D Hamiltonian (3.8) follow exactly the LDA prediction showing that the deviations seen in Figs. 3.1-3.2 are due to finite size effects. In the cross-over region from the mean-field to the 1D LL regime, residual 3D effects are still present (see Fig. 3.9) and produce small deviations from the LL equation of state.

Figure 3.7: Energy per particle as a function of $\lambda$. Dashed line: GP equation (3.3), solid line: LL equation of states in LDA, dotted line: TG gas, dot-dashed line: non-interacting gas.

Figure 3.8: Mean square radius along $z$ as a function of $\lambda$. Dashed line: GP equation (3.3), solid line: LL equation of states in LDA, dotted line: TG gas, dot-dashed line: non-interacting gas.