4 8 12 16 20 L 0.2 0.4 0.6 0.8 1 c N = 3 N = 2 N = 1 N = 0 $_{c}$ = 0.2968 4 8 12 16 L 0 0.2 0.4 0.6 0.8 c N = 3 N = 2 N = 1 N = 0 FIG. 8: The finite size scaling of the N-particle energy level s as a function of the spatial lattice size L at β = 0 . 43 (left, massive phase) and β = 0 . 50 (right,superfluid phase). The spatial size dependence of the energy of N particles in three spatial dimensions has been calculated using models of quantum mechanics [33, 34]. Recently, this dependence was also computed using effective field theory [35]. In the special ca se of two particles the problem was also solved in a general massive quantum field theory in threespatial dimensions [36]. All these studies indicate that the ground state energy of N particles satisfies the relation E ( N ) 0 − E (0) 0 ∝ N ( N + 1) /L $^{3}$. For N ≥ 1 , remember that µ ( N ) c is the difference in the ground state energies of N + 1 particles and N particles. Extending the known results to two spatial dimensions and making the assumption that the particle density in the thermodynamic limit close to the critical point is of the form ρ ∼ c ( µ − µ (0) c ) at leading order in the superfluid phase, we expect for sufficiently large L and N . Figure 8 shows that our data is described reasonably well by this equation. In the left plot of Fig. 8 we show the values of µ ( N ) c ( L ) obtained from table III. The solid lines show the dependence of µ ( N ) c ( L ) on L as described by eq. (25) with c ≈ 0 . 18 , 0 . 16 and 0 . 15 at N = 1 , 2 and 3 respectively. Clearly, for large values of L the solid lines pass through the data. The value of c changes slightly since N is small. Unfortunately a fit of our data to eq. (25) yields a large χ $^{2}$/DOF . We believe this is due to the fact that our data has very small errors and hence is sensitive to higher order corrections which we do not know analytically at the moment in two spatial dimensions. 22 µ ( N ) c ( L ) = µ (0) c + ( N + 1) cL 2 (25)