FIG. 1: Plot of ρ as a function of µ for β = 0 . 43 , L = 12 and L$_{t}$ = 144 . The data shows clear “wiggles” at small values of ρ which disappears for larger values. The left inset magnifiesthe region of the “wiggles”. The right inset shows the plot of ρ as function of L at µ = 0 . 32 assuming L$_{t}$ = L 2 which also shows clear non-monotonic behavior. temperatures (large L$_{t}$ ) in a finite spatial volume (fixed L ). While these singularities are smoothed out at finite L$_{t}$ , they leave behind non-trivial finite size effects observedabove. While it may still be possible to develop practically useful finite size scalin g relations using L$_{t}$ = L $^{2}$, we find it natural to consider a finite size scaling theory for quantiti es as a function of L$_{t}$ and µ for a fixed value of L close to the critical values of µ where energy levels cross each other. As we discuss below this leads to an effective quantum mechanics problem. The finite size effects studied here have been observed earlier in the context of quantum spin-systems in a magnetic field [29], but they were not analyzed using the techniques we introduce below. 7 0.3 0.4 0.5 0.6 0.7 0 0.025 0.05 0.075 0.27 0.3 0.33 0.36 0 0.004 0.008 0.012 8 12 16 20 24 L 0.004 0.006 0.008