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ference between the two, plotted in the lower-right panel. 

From the adiabatic curvature we compute the Berry phases accumulated (i.e. the exponent in Eq. 48) as we move one anyon along different paths of the form θ =ct., while keeping the other anyon fixed at the North Pole of the sphere. Fig. 8 shows the Berry phase as func- tion of the area enclosed by the path. We use this figure to draw several important conclusions. First, we point out that the Berry phase plotted in this figure includes also the Aharonov-Bohm phase due to the magnetic flux φ$_{B}$ : ψ$_{AB}$ = e $^{∗}$φ$_{B}$ . Thus it is expected that the total Berry phase, as a function of the enclosed area, to go asymptot- ically to a linear curve, a feature that is obviously present in Fig. 8. The slope of the asymptotic part of the curve is equal to the charge e $^{∗}$= e/ 4 of the anyons. 

Excepting the length of the lines, the graphs for dif- ferent system sizes look very similar, implying that the thermodynamic limit is achieved very fast. We expect the graphs for the larger systems to represent the thermody- namic limit with a high degree of accuracy. Looking at the largest systems, we can see that, for large enclosed ar- eas, the curves representing the Berry phase for systems with even number of electrons are shifted upward by π relative to the curves representing the Berry phase for systems with odd number of electrons. This is a topo- logical effect, since the shift is independent of the area enclosed by the braiding loop or of the shape of the loop, 

as long as the loop is large enough. The finding is in total agreement with the SO spinorial representation of the braid group derived in Ref. 19 on the basis of under- ling CFT structure of the Moore-Read state. This effect was also considered the most direct signature of the Non- Abelian statistics for the Moore-Read state . 21,32 

We conclude this subsection with the observation that the coefficient F depends on how we compute the area enclosed by Γ in Eq. 48, more precisely on what metric tensor is used when computing the area enclosed by the loops. The plots shown in Fig. 7 were obtained with the standard metric of the sphere. We have repeated the calculations using the quantum metric tensor Eq. (38) instead, in which case we have to re-scale the coefficient: 

According to the our previous analytic prediction (see Eq. 40), F $^{q}$( P ) should be identically 1. We numerically checked this prediction in the following way. The quan- tity that is readily available in the numerical calcualtions is the quantum distance. We can compute the determi- nant of the quantum metric tensor by considering a circle of radius ρ (in the standard metric of the sphere) centered at P and calculate the quantum distance between P and the points of this circle. If d q M and d q m denote the max- imum, respectively minimum quantum distance to the 

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