VI. THERMODYNAMIC LIMIT Using the results of the previous section it is tempting to extrapolate to the thermodynamic limit. However, in order to accomplish this task it is important to know how the effective quantum mechanical parameters depend on L . This dependence is non-universal in general and close to a critical point will depend on the nature of the phase transition. Assuming the phase transition is second order, close to the critical chemical potential where the density can be made arbitrarily small, we expect universal features to emerge. For example, when the particles have a purely repulsive interaction, the ground state energy of N particles is always less than the corresponding energy of N + 1 particles [31]. Based on the results of the previous section this scenario seems to be valid in the current model. Indeed the particle number always increases by one as we increase µ at every fixed value of L . The superfluid density ρ$_{s}$ also behaves like ρ . Thus, we conclude that at µ = µ (0) c in the thermodynamic limit, there is a second order transition to a superfluid phase. Based on this, below we discuss the extrapolations to the thermodynamic limit. First we consider β = 0 . 43 where the low energy physics contains massive bosons with repul- sive interactions. Then, the quantity µ (0) c is simply the mass of the particle M ( L ) at a finite L . This mass can be obtained by fitting the the temporal two-point cor relation function computed at µ = 0 , to the form G ( t ) ∼ exp( − M ( L ) t ) for values of t ≪ L$_{t}$/ 2 . In the definition of G ( t ) , y$_{⊥}$ and x$_{⊥}$ represent lattice sites at temporal slices 0 and t respectively. We have computed M ( L ) using this method and indeed we find excellent agreement with µ (0) c at all the three values of β . This means the true mass of the particle must be We can reverse this argument and obtain µ (0) c in the thermodynamic limit by simply measuring the mass of the particle at µ = 0 . Of course this result is not general and is valid only in the present study where there is clear evidence that the particles repel each other. In order to extract M in the massive phase ( β < β$_{c}$ ) we can use L¨uscher’s formula [32] extended to two spatial dimensions, At β = 0 . 43 we find that µ (0) c fits well to this form and gives M = 0 . 29680(4) , M$_{1}$ = 0 . 59(2) and ˜ m = 0 . 393(4) with a small χ $^{2}$/DOF . The data and the fit are shown in the left plot of Fig. 8. µ 0 c ≈ M + M$_{1}$ e − ˜ $^{mL}$. (24) M = lim L $_{→∞}$µ (0) c . (23) G ( t ) = ∑ x$_{⊥}$,y$_{⊥}$ 〈 e $^{iθ$_{x}$$_{⊥}$}$e − iθ$_{y}$$_{⊥}$ 〉 , (22) 21