Homogeneous system

We calculated the pair distribution matrix for a number of densities ranging from very small value of the gas parameter $n_{1D}a_{1D}\ll 1$ (TG regime) up large densities $n_{1D}a_{1D}\gg 1$ (GP regime). The results are presented in the Fig. 5.3. In the GP regime the correlations between particles are very weak and $g_2(z)$ arrives very quickly at the bulk value. Decreasing the $\vert a_{1D}\vert$ (thus making the coupling constant $g_{1D}$ larger) we enhance beyond-mean field effects and enforce correlations. For the smallest considered value of the gas parameter $n_{1D}a_{1D} = 10^{-3}$ we see oscillations in the pair distribution function, which, in this sense becomes more similar to the one of a liquid, rather than of a gas. At the same density we compare the pair distribution function with the one of the TG gas and find no visible difference.

Figure 5.3: Pair distribution function function for different values of the gas parameter. In ascending order of the value at zero $n_{1D}\vert a_{1D}\vert = 10^{-3}, 0.3, 1, 30, 10^3$. Arrows indicate the value of $g_2(0)$ as obtained from Eq. 5.10.

On the same Figure we plot predictions for the value of the pair distribution function at zero and find perfect agreement with the analytical prediction. In the TG regime particles never meet each other and consequently $g_2(0)=0$. Making the interaction between the particles weaker we find finite probability of two particles coming close to each other according to the Eq. 5.10. As we go further into the direction of the GP the interaction potential becomes more transparent and we approach the ideal gas limit $g_2(0) = 1$.

In the Fig. 5.4 we present the static structure factor obtained from $g_2(z)$ according to relation (1.31). At the smallest density $n_{1D}\vert a_{1D}\vert = 10^{-3}$ our points lie exactly on the top of the $S(k)$ of the TG gas (Eq. 5.5). For all densities the small-momenta part of the structure factor comes from generation of a phonon. We compare DMC results with the Feynman prediction (Eq. 5.9). We see that in the strongly correlated regime phononic description works well even to values of the momenta of the order of inverse density $n_{1D}^{-1}$, although in the MF regime the healing length becomes significantly larger than the mean interparticle distance leading to earlier deviations.

Figure 5.4: Static structure factor for the same values of $n_{1D}\vert a_{1D}\vert$ as in Fig. 5.3 (solid lines). The dashed lines are the corresponding long-wavelength asymptotics from Eq. 5.9.

We calculate the value at zero of the three-body correlation function (1.22) over a large range of densities. At large density $n_{1D}\vert a_{1D}\vert = 10^4$ the probability of three-body collisions is high. Making the density smaller we find decrease in the value of $g_3(0)$. Close to the MF limit the result of the Bogoliubov theory (Eq. 5.13) provides fairly good description of $g_3(0)$ (see. Fig. 5.5). Further decrease in the density leads to fast decay of three-body collision rate and it becomes vanishing at values of the gas parameter smaller than one. In order to resolve the law of the decay we plot same data on the log-log scale (Fig. 5.6) and show that the decay goes with the forth power of the gas parameter in agreement with Eq. 5.12. Numerical estimation of $g_3(0)$ at smaller densities becomes very difficult due to very small value of the measured quantity itself. It is interesting to note that $g_2^3(0)$ follows closely to $g_3(0)$.

We compare the result of an experiment done in NIST [TOH+04] with the theoretical prediction of the Lieb-Liniger model, see Figs. 5.5,5.6. In this experiment the three body recombination rate was measured and the value of $g_3(0)$ was extracted. We find an agreement between experiment and theory. The result of DMC calculation is slightly closer to the experimental data point than the estimation $g_2^3(0)$, although the error bars of the experimental measurement cover both values.

Figure 5.5: Value at zero of the three-body correlation function $g_3(0)$ (squares), Bogoliubov limit Eq. 5.13 (dashed line), $g_2^3(0)$ Eq. 5.10 (solid line), experimental result of [TOH+04] (diamond).

Figure 5.6: Value at zero of the three-body correlation function, log-log scale, $g_3(0)$ (squares), TG limit Eq. 5.12 (dashed line), Bogoliubov limit Eq. 5.13 (short-dashed line), $g_2^3(0)$ Eq. 5.10 (solid line), experimental result of [TOH+04] (diamond).

We calculated the spatial dependence of the one-body density matrix for different densities. At small distances we compare the function with the short range expansion (5.14) and find an agreement for $zn_{1D}\ll 1$ (see Fig. 5.7). For distances larger than the healing length we expect the hydrodynamic theory to provide a correct description. Indeed, we see that the long-range decay has a power-law form in agreement with the prediction Eq. 5.7 (see Fig. 5.8). We fix the coefficient of proportionally in Eq. 5.7 by fitting the data. By doing it we conclude complete description of the one-body density matrix starting from small distances up to large ones. The deviations on Fig. 5.8 from power law-decay are at largest distances ($z\approx L/2$) are due to finite size effects.

We derive a highly accurate expression for the coefficient $C_{asympt}$ from a hydrodynamic approach considering weak interactions and low density fluctuations. In terms of Euler's constant $\gamma=0.577$ we have (1.199):

$$C_{asympt} = \left(\frac{e^{1-\gamma}}{8\pi\alpha}\right)^\alpha(1+\alpha)$$ (5.15)

Although this constant is formally derived in the limit $\alpha\ll 1$ (i.e. limit of weak interaction $n_{1D}a_{1D}\gg 1$) it provides a very good description in the whole range of densities. Indeed, the coefficient defined by fitting $g_{1}$ as shown in Fig. 5.8 is always in agreement with prediction (5.15) within the uncertainty errorbars we get from our DMC calculation. Further, in the most strongly interacting TG regime we compare (5.15) with the exact result provided by formula (5.6) and find only $0.3\%$ difference. A different expression was obtained by Popov [Pop80] (and later recovered in [MC02]) $C^{Popov}_{asympt} = \left(\frac{e^{2-\gamma}}{8\pi\alpha}\right)^\alpha$. Both expressions coincide for small values of $\alpha$, but Popov's coefficient lead up to larger $10\%$ maximal error, as it was pointed out in [Caz04]. The comparison of different coefficients is presented in Table 5.2.

Table: Coefficient of the long-range decay of the one-body density matrix defined as in (5.7). First column is the one-dimensional gas parameter, second column is the fitting coefficient extracted from Eq. 5.8, third column is Popov's prediction, fourth column is formula 5.15. Density $n_{1D}a_{1D} = 0.001$ is deeply in the TG regime and here one can apply Eq. 5.6 leading to $C^{TG}_{asympt} = 0.5214$

$n_{1D}a_{1D}$	$C^{DMC}_{asympt}$	$C^{Popov}_{asympt}$	$C_{asympt}$
1000	1.02	1.0226	1.0226
30	1.06	1.0588	1.0579
1	0.951	0.9646	0.9480
0.3	0.760	0.8145	0.7814
0.001	0.530	0.5746	0.5227

Figure 5.7: Short range behavior of the one-body density matrix for different values of the gas parameter $n_{1D}\vert a_{1D}\vert = 10^{-3}, 0.3, 1, 30, 10^3$ (the lowest curve corresponds to $n_{1D}\vert a_{1D}\vert = 10^{-3}$, the uppermost to $n_{1D}\vert a_{1D}\vert = 10^{3}$, $g_1(z)$ (solid lines), series expansion at zero (eq. 5.14) (dashed lines).

Figure: Large range behavior of the one-body density matrix (solid lines), fits to the long-wavelength asymptotics from eq. 5.7 (dashed lines). Values of the density are same as in Fig. 5.8. The arrows indicate the value of $\xi n$: the leftmost corresponds to $n_{1D}\vert a_{1D}\vert = 10^{-3}$, the rightmost to $n_{1D}\vert a_{1D}\vert = 10^3$

We obtain the momentum distribution from the Fourier transformation (see Eq. 1.26) of the one-body density matrix at short distances and the fit power-law decay at large distances. In an infinite homogeneous system the momentum distribution has an infrared divergence (Eq. 5.8). In order to present the momentum distribution in the most efficient way we plot in Fig. 5.9 a combination $k n(k)$, where this divergence is absent. We notice that the infrared asymptotic behavior is recovered for values of $\k$ considerably smaller than the inverse healing length $1/\xi$. At large $\k$ the numerical noise of our results is too large to extract evidences of $1/k^4$ behavior predicted in [OD03].

Figure 5.9: Momentum distribution for the same values of $n_{1D}\vert a_{1D}\vert$ as in Fig. 5.8 (solid lines). The dashed lines correspond to the infrared behavior of Eq. 5.8. The arrows indicate the value of $1/\xi n_{1D}$: the rightmost corresponds to $n_{1D}\vert a_{1D}\vert = 10^{-3}$, the leftmost to $n_{1D}\vert a_{1D}\vert = 10^3$.