0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 5 f for attractive bosons PSfrag replacements f β k $^{2}$/ 2 m µ/T = log z$_{c}$ FIG. 14: The filling fraction f for attractive bosons for µ/T = − 2 , − 1 . 5 , − 1 . 08 = log z$_{c}$ . function c is shown in Figure 20. It has the limiting value lim µ/T → $_{0}$c = 0 . 303964 , (94) which is considerably less than the free field value c = 1 / 2. The scaling function b has the limiting values lim$_{T/µ}$$_{→}$$_{0}$ − b = 0, and lim$_{T/µ}$$_{→−∞}$ b = ∞ . Since the pressure is proportional to c , the same density occurs at two different pressures. One sees that for µ/T < − 0 . 56, the density increases with pressure as it should. For µ/T > − 0 . 56 the density decreases with increasing pressure, violating dp/dV < 0, and should thus be viewed as an unphysical region. We interpret this as a phase transition occuring at z$_{c}$ = e − 0 . 56 = 0 . 57, where the critical density At this critical point c ( − 0 . 56) = 0 . 24. This perhaps corresponds to a transition to a crystaline phase. We cannot prove this, since we have not calculated the shear n$_{c}$ = 0 . 25 mk$_{B}$T 2 π ℏ 2 ⇐⇒ T$_{c}$ ≈ 4 . 0 T$_{F}$ (95) 34