FIG. 8: (Color online.) The Berry phase accumulated by an anyon when moved along a path θ =const, with the other anyon fixed at the North pole. Different lines refer to differ- ent system sizes that are marked in the diagram with their corresponding N/N$_{φ}$ numbers. A point on a particular line represents one braiding path, which is labeled by the area enclosed by the path, shown on the horizontal axis. FIG. 7: (Color online.) The adiabatic curvature F ( P ) ob- tained by moving one anyon while keeping the other fixed. Left-upper panel shows the results for odd number of elec- trons: N/N$_{φ}$ =9/16, 11/20, 13/24, 15/28 and the right- upper panel shows the results for even number of electrons: N/N$_{φ}$ =10/18, 12/22, 14/26, 16/30. The lower-left and lower- right panels show the sum and the difference between the odd and even results, respectively. For example, we added and subtracted the result for N/N$_{φ}$ =10/18 and N/N$_{φ}$ =9/16, and then the results for N/N$_{φ}$ =12/22 and N/N$_{φ}$ =11/20, etc.. Next we use the fact that ˜ Ψ$_{i}$ are analytic functions of w , i.e. they are functions of the single variable w rather than of two independent variables w$_{1}$ and w$_{2}$ , in which case: where ˆ F is a D × D matrix. At P , in the same system of coordinates, we can prove the following general fact: where g 0 is the standard metric of the sphere. This shows that the quantum metric and the standard metric of the sphere are related by a conformal transformation, which involves the trace of the curvature tensor. We start the proof of Eq. (40). The 2 n − 1 many body wavefunctions Ψ$_{i}$ of the zero modes depend parametri- cally on the position w of the mobile anyon and take the following general form: 23 The scalar product is real, which allows us to easily com- pute the imaginary part, resulting in: where the crucial fact is that the vectors ˜ Ψ$_{i}$( w ) are all analytic of w . A$_{i}$ ( w ) are normalization factors that are not analytic of w . We have: If one repeats exactly the same arguments for the quan- tum metric tensor given in Eq. 38, the conclusion will be that: Due to the presence of 1 − P$_{w}$ , we can ignore the action of the derivatives on the normalization constants A$_{i}$ ( w ), which then can be pulled out of the scalar product as below: Since w$_{1}$ and w$_{2}$ coincide with the geodesic coordinates of the sphere at P , i.e. the coordinates in which the metric tensor becomes the identity matrix when evaluated at P : g $^{0}$( P )$_{µν}$ = δ$_{µν}$ , the proof of Eq. 40 is completed. Tr ˆ F ( w ) = 2 ∑ i |A$_{i}$ ( w ) | $^{2}$× Im {〈 ∂$_{w}$ ˜ Ψ$_{i}$( w ) , [1 − P$_{w}$ ] ∂$_{w}$ ˜ Ψ$_{i}$( w ) 〉 ( ∂$_{1}$w ) $^{∗}$∂$_{2}$ w } . (44) g q $_{µν}$( P ) = Tr { ˆ F ( P ) } g $^{0}$( P )$_{µν}$ , (40) Ψ$_{i}$( w ) = A$_{i}$ ( w ) ˜ Ψ$_{i}$( w ) , i = 1 , . . . , 2 n − 1 (41) Tr ˆ F ( w ) = 2 ∑ i |A$_{i}$ ( w ) | $^{2}$× 〈 ∂$_{w}$ ˜ Ψ$_{i}$( w ) , [1 − P$_{w}$ ] ∂$_{w}$ ˜ Ψ$_{i}$( w ) 〉 . (45) g q $_{µν}$( w ) = 2 ∑ i |A$_{i}$ ( w ) | 2 ×〈 ∂$_{w}$ ˜ Ψ$_{i}$( w ) , [1 − P$_{w}$ ] ∂$_{w}$ ˜ Ψ$_{i}$( w ) 〉 δ$_{µν}$. (46) Tr { ˆ F ( w ) } = 2Im ∑ i 〈 ∂$_{1}$ Ψ$_{i}$( w ) , [1 − P$_{w}$ ] ∂$_{2}$ Ψ$_{i}$( w ) 〉 . (42) Tr ˆ F ( w ) = 2 ∑ i |A$_{i}$ ( w ) | $^{2}$× Im 〈 ∂$_{1}$ ˜ Ψ$_{i}$( w ) , [1 − P$_{w}$ ] ∂$_{2}$ ˜ Ψ$_{i}$( w ) 〉 . (43) } Berr y Phase (units of  ! ) Area 11/20 12/22 13/24 14/26 16/30 15/28 ! Even N$_{el}$ Odd N$_{el}$ Distance Distance Curvature Odd Even Distance Curvature Dif ference Sum Difference 0 2 4 6 8 0 2 4 6 8 0.3 0.4 0.5 0.6 0.1 0.15 0.2 0.25 0.3 0.35 Distance Cur vatur e 9