Ground-state energy

The energy of a dilute Bose gas in the presence of impurities is given by result (2.17). It is derived under the assumptions that the gas parameter is small $na^3 \ll 1$ and the external field is weak. Using the DMC algorithm we investigate the dependence of the ground-state energy both on the density $na^3$ and on the strength of the disorder $R = \chi (b/a)^2$. The main contribution to the energy comes from the mean-field term (see result (1.14) obtained from the Gross-Pitaevskii equation). In order to better understand the results for the energy it is useful to subtract the mean-field term $E_{MN}$ from the total energy $E$.

Figure 4.3: Beyond mean-field energy per particle $E - E_{MF}$ as a function of density $na^3$ for different strengths of disorder $R$. The solid lines correspond to the analytical prediction (2.17)

Let us first consider the energy dependence on $R$. For weak disorder ($R = 2$) the predictions well agree with the results of DMC simulations. By increasing $R$ while keeping $na^3$ fixed one sees deviation from analytical prediction. The Bogoliubov model is valid if the gas parameter $na^3$ is small. Fig. 4.3 shows that the values of the gas parameter where the theoretical prediction holds depends on the strength of disorder. For weak disorder ($R = 2$) agreement is found up to very high densities $na^3 \approx 10^{-2}$. By increasing the amount of disorder deviations appear for smaller values of $na^3$. For $R = 12.5$ numerical and analytical results coincide up to $na^3 \approx 5\cdot 10^{-4}$ and for very strong disorder $R = 100$ no agreement is found at densities $na^3 > 10^{-5}$.