Shape of the one-body density matrix

The shape of the one-body density matrix (1.45) has been calculated within the Bogoliubov model and is given by (1.47) in the pure system and by (2.24) in the system with disorder. The derivation of these results is valid for dilute systems in the presence of weak disorder, so one expects to find agreement with the DMC results for small values of $na^3$ and $R$. Since the Bogoliubov model neglects short-range correlation, we also expect that result (2.24) is valid for large values of $r$ ($r\gg a$, $r\gg b$). The DMC simulation gives the mixed estimator for the OBDM and the VMC gives the variational estimate. By calculating these results using the extrapolation technique (3.91), we can estimate the pure OBDM. We have calculated the OBDM at different densities $na^3$ and for a fixed strength of the disorder $R = 100$. The results are shown in Fig. 4.9. Although the strength of the disorder is large we find good agreement at small densities while by increasing in the density we see significant differences.

Figure 4.9: One-body density matrix $\rho _s(r)/\rho (0)$ at different densities $na^3 = 10^{-5}, 2\cdot 10^{-5}, 5\cdot 10^{-5}, 2\cdot 10^{-4}$, and $R = \chi (b/a)^2 = 100$ with $b/a = 5$. Solid lines correspond to result (2.24).