Quantum phase transition

By adding disorder to the system the condensate and superfluid density are depleted. One expects that at some critical amount of disorder superfluidity vanishes and the system becomes normal. We want to investigate the quantum phase transition of the hard-sphere gas in the presence of hard-sphere impurities at $T = 0$. In the vicinity of the phase transition the correlation length becomes large. This means that in the MC simulation one has finite size errors (see 3.4.3) and it is necessary to carry out calculations with systems of different size and finally extrapolate to the thermodynamic limit. We calculate the superfluid density as a function of the concentration of impurities $\chi$ while keeping the size of the impurity constant. The simulation is carried out for systems of 16, 32 and 64 particles with periodic boundary conditions. The results for the low density $na^3 = 10^{-4}$ is presented in Fig. 4.10 and for the density $na^3 = 10^{-2}$ in Fig. 4.11.

Figure 4.10: Superfluid density measured as a function of $\chi$ at density $na^3 = 10^{-4}$ and $b/a = 5$ with 16, 32 and 64 particles in the simulation box

Figure 4.11: Superfluid density measured as a function of $\chi$ at density $na^3 = 10^{-2}$ and $b/a = 2$ with 16, 32 and 64 particles in the simulation box

The figures show that there is no significant finite size effect present which means that we are still far from the critical region. The further increase of the strength of disorder in Fig. 4.10 and Fig. 4.10 is impossible, because of the constraint of non-overlapping impurities. We conclude that within our model of non-overlapping impurities the superfluid-insulator quantum transition is absent.