We find that our data fits well to this expression truncated at t he quadratic order in Δ ( N ) µ , which means we have seven new parameters in our fit. However, most ofthese parameters are not de- termined reliably and contain large systematic errors. The only parameter that can be determined reliably is χ$_{0}$ and this is what we quote as a result from our analysis. We have performed extensive calculations at β = 0 . 43 , 0 . 50 and 0 . 20 . These values of β are chosen so that two of them are close to the critical coupling β$_{c}$ = 0 . 45421 on either side and one is far from it in the massive (disordered) phase. In this section we present fits of our results to the effective quantum mechanics description discussed above. As mentioned earlier, the effective description becomes useful only in the limit of small temperatures where excitations to higher energy levels can be neglected. Since the spacing between energy levels decreases with increase in volume, our approach works best on small spatial volumes. However, thanks to the efficient worm algorithm, we have been able to extract parameters of the effective quantum mechanics up to L = 16 at β = 0 . 43 and β = 0 . 5 . Although this lattice size is small compared to normal studies of bosonic lattice field theories, it still allows us to perfo rm a useful study of the L dependence of the physics and draw quantitative conclusions about the thermodynamic limit. At β = 0 . 2 we observe that the energy levels are more densely packed and we are able to compute quantities only up to L = 8 . We first consider L = 2 and vary L$_{t}$ in the range 40 ≤ L$_{t}$ ≤ 200 which is easy due to the small lattice size. In Fig. 2 we plot the behavior of our four observables as a function of the chemical potential at L$_{t}$ = 100 and β = 0 . 43 . Note that the particle number increases in steps of one at critical values of µ . This means energy per particle of the ground state in every particle number sector increases with the number of particles. In other words the particles repel each other. Thus, small systems containing particles of the non-linear sigma model will show the phenomena similar to Coulomb Blockade observed in nanoscale systems [30]. By fitting the data at L = 2 and β = 0 . 43 , 0 . 50 and 0 . 20 we have extracted the parameters µ ( N ) c , ρ$_{0}$ , ρ$_{1}$ and χ$_{0}$ for N = 0 , 1 , 2 and 3 . These are tabulated in Tab. I. In order to show the goodness of our fits, in Fig. 3 we plot the behavior of 〈 N 〉 and χ for values of µ close to the transition between the N = 0 and N = 1 sector for different values of L$_{t}$ at β = 0 . 43 . The solid lines represent the fit functions using the parameter values given in table I. Note that all the computed values of 〈 N 〉 shown in the left plot 11 V. RESULTS