0.75 1 1.25 1.5 1.75 2 0 1 2 3 4 5 6 < N > 0.8 1 1.2 1.4 1.6 1.8 0.8 0.9 1 1.1 1.2 1.3 L 2  ρ$^{s}$ 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.05 0.1 0.15 0.2 0.25 < N 2  > - < N >  2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 50 100 150 200 FIG. 2: The four observables as a function of µ up to four particle excitation. The data shown is for L = 2 , L$_{t}$ = 100 and β = 0 . 43 . The solid lines are fits to the effective quantum mechanicaldescription. of Fig. 3 can be fit with just one parameter namely µ ( N ) c . We have repeated the above analysis at larger values of L . We find the physics remains qualita- tively similar to the L = 2 case. In particular the average particle number jumps by one at critical values of µ . In Fig. 4 we show the average particle number as a function of µ at different values of L at β = 0 . 43 and β = 0 . 50 . In Fig.5 we plot the average particle number at L = 8 for all three values of β at different values of L$_{t}$ close to µ ( N ) c for N = 1 , 2 , 3 . The effective quantum mechanics description continues to fit all our data well as lo ng as µ is close to the critical values 12