One-body density matrix and static structure factor

Figure 6.2: Static structure factor $S(k)$ for a gas of HR at different values of the gas parameter $na_{1D}$ (symbols) and for a TG gas (dashed line).

Contrarily to the TG case, it is not possible to obtain analytical expressions for $g_1(z)$ and $S(k)$ in the HR problem. We have calculated them using configurations generated by a Monte Carlo simulation according to the exact probability distribution function $\vert\psi_{HR}\vert^2$. The results for the static structure factor are shown in Fig. 6.2. Compared to $S(k)$ in the TG regime, a clear peak is visible for values of $\k$ of the order of twice the Fermi wave vector $k_F=\pi n$ and the peak is more pronounced as $na_{1D}$ increases. The change of slope for small values of $\k$ reflects the increase of the speed of sound $c$ with $na_{1D}$. The long-range behavior of $g_1(z)$ can be obtained from the hydrodynamic theory of low-energy excitations [RC67,Sch77,Hal81]. For $\vert z\vert\gg\xi$, where $\xi=\hbar/(\sqrt{2}mc)$ is the healing length of the system, one finds the following power-law decay (1.199):

$$g_1(z)\propto1/\vert z\vert^\alpha,$$ (6.2)

where the exponent $\alpha$ is given by $\alpha=mc/(2\pi\hbar n)$. For a TG gas $mc=\pi\hbar n$, and thus $\alpha_{TG}=1/2$. For a HR gas one finds $\alpha=\alpha_{TG}/(1-na_{1D})^2$ and thus $\alpha>\alpha_{TG}$. This behavior at long range is clearly shown in Fig. 6.3 where we compare $g_1(z)$ of a gas of HR with $na_{1D}=0.1$, 0.2 and 0.3 to the result of a TG gas [JMMS80]. The long-range power-law decay of $g_1(z)$ is reflected in the infrared divergence of the momentum distribution $n(k)\propto 1/\vert k\vert^{1-\alpha}$ holding for $\vert k\vert\ll 1/\xi$. A gas of HR exhibits a weaker infrared divergence compared to a TG gas. The correlation functions of a HR gas at $na_{1D}=0.1$, 0.2 should accurately describe the physical situation of a Bose gas with large and negative $g_{1D}$. For $na_{1D}=0.3$ we expect already some deviations from the HR model, as it is evident from the equation of state in Fig. 6.1, which should broaden the peak in $S(k)$ and decrease the slope of the power-law decay in $g_1(z)$ at large distances. The analysis of correlation functions clearly shows that the super-Tonks regime corresponds to a Luttinger liquid where short range correlations are significantly stronger than in the TG gas.

Figure 6.3: One-body density matrix $g_1(z)$ for a gas of HR at different values of the gas parameter $na_{1D}$ (solid lines) and for a TG gas (dashed line). Higher values of density correspond to a faster decay of $g_1(z)$.