Scaling behaviour

The strength of disorder is described by two independent parameters: the particle-impurity scattering length $b$ and the concentration $\chi = N^{imp}/N$. One of the important results of the Bogoliubov model is that a single parameter $R = \chi (b/a)^2$ is sufficient to describe the effect of disorder (see eqs. (2.17), (2.22), (2.65)). We have calculated the condensate and superfluid fraction by changing both $b$ and $\chi$ while keeping $R = \chi (b/a)^2$ constant. The results are shown in Fig. 4.7 for $na^3 = 10^{-2}$ and $R=2,~4$ and in Fig. 4.8 for $na^3 = 10^{-4}$ and $R=25,~100$.

Figure 4.7: Condensate fraction $N_0/N$ and superfluid fraction $\rho _0/\rho$ as functions of impurity size $b$ for two values of the scaling parameter $R = 2$ and $R = 4$, and density $na^3 = 10^{-2}$

Figure 4.8: Condensate fraction $N_0/N$ and superfluid fraction as functions of impurity size $b$ for fixed value of the scaling parameter $R = 25$ and $R = 100$, and density $na^3 = 10^{-4}$

Both at low and high density we see that the scaling behavior for $N_0/N$ and $\rho_s\rho$ is well satisfied for the smallest values of $R$ ($R = 25$ for $na^3 = 10^{-4}$ and $R = 2$ for $na^3 = 10^{-2}$). It is worth noticing that these values of $R$ and $na^3$ correspond to regime where the results of first order perturbation theory do not apply (see Figs. 4.5, 4.6). This means that the scaling behavior in the parameter $R$ is valid also beyond the region of applicability of the perturbation expansion.