When β = 0 . 5 we are in the superfluid phase and the U (1) particle number symmetry is spontaneously broken. One then expects the low energy spectrum at finite volumes to be governed by O (2) chiral perturbation theory. Based on this we again expect µ ( N ) c ( L ) to be described by eq. (25) but with µ (0) c = 0 . While our data is again consistent with these expectations (see right plot of Fig. 8), without keeping higher order 1 /L corrections the fits again give a large χ $^{2}$/DOF . The solid lines in Fig. 8 describe eq. (25) with c = 0 . 33 , 0 . 22 , 0 . 2 , 0 . 2 for N = 0 , 1 , 2 , 3 . The phase diagram of the O (2) non-linear sigma model is an interesting research topic in itself. While the complete phase diagram requires more work, our results above allow us to compute the location of the transition line between the normal phase and the superfluid phase. In particular the value of µ (0) c as a function of β determines this line. Based on the evidence at β = 0 . 43 and 0 . 20 we predict that µ (0) c = M for all values of β < β$_{c}$ . The coordinates of the transition line are tabulated in Tab. V and shown on the phase diagram in Fig. 9. We expect this transition to be second order in the mean field universality class with loga rithmic corrections except at µ = 0 where it is governed by the 3 d XY universality class. Thus, when β < β$_{c}$ and ( β$_{c}$ − β ) /β$_{c}$ ≪ 1 we must have µ 0 c ∝ [( β$_{c}$ − β ) /β$_{c}$ ] ν where ν ≈ 0 . 671 [37]. VII. PHASE DIAGRAM In principle there could be other interesting phases at larger values of µ which we cannot rule out based on the current work. Since we have seen the particles have a repulsive interaction, an interesting possibility is the existence of a solid phase or a super-solid phase [38]. However, there are stringent constraints for super-solids to arise [39, 40] and we do not know if these rule out such a phase in the current model. In any case if a transition to a solid phase exists, it will most likely be first order similar to the solid-liquid phase transitions inmaterials and will occur at densities where the lattice structure may become important. These transitions can also be studied efficiently with the worm algorithm. We postpone these studies for the future, but have speculated the possibility of a solid phase in Fig. 9. TABLE V: The values of µ (0) c obtained by assuming that it is equal to the L → ∞ limit of M ( L ) at µ = 0 as discussed in the text. β 0.43 0.40 0.35 0.30 0.25 0.2 0.15 0.10 0.07 0.04 µ (0) c 0.29678(3) 0.530(4) 0.859(2) 1.172(7) 1.505(9) 1.85801(3) 2.267(3) 2.783(3) 3.210(3) 3.829(4) 23