MR 11/22 MR 13/26 MR 15/30 MR 10/20 MR 12/24 MR 14/28 0.04 0.02 0 0.04 0.02 0 0.04 0.02 0 0.04 0.02 0 0.04 0.02 0 0.04 0.02 0 0.2 0.1 0 0.2 0.1 0 0.2 0.1 0 0.2 0.1 0 0.2 0.1 0 0.2 0.1 0 11/21 13/25 15/29 10/19 12/23 14/27 13 FIG. 12: (Color online) Plots of ρ$_{TW}$ ( P ) for different system sizes (each panel is marked with the corresponding N/N$_{φ}$ ). To see ho w ρ$_{TW}$ ( P ) relates to the adiabatic curvature, for eac h size, we show | f ( P ) | on the left and ρ$_{TW}$ ( P ) on the right. computed the curvature ˆ F at each of these points, in a pre-chosen, arbitrary gauge. The corresponding f (see Eq. 51) vectors are gauge dependent. To compare the three vectors, we need to adiabatically transport them to the same point w . For this we compute the monodromies ˆ W$_{1}$ for the path w +Δ w$_{1}$ → w and ˆ W$_{2}$ for the path w +iΔ w$_{2}$ → w and then compute ˆ F$_{1}$ = ˆ W$_{1}$ ˆ F ( w +Δ w$_{1}$ ) ˆ W − 1 1 and ˆ F$_{2}$ = ˆ W$_{2}$ ˆ F ( w +iΔ w$_{2}$ ) ˆ W − 1 2 . By decomposing ˆ F$_{1}$$_{,}$$_{2}$ as in Eq. 51, we obtain the parallel transport of f ( w +Δ w$_{1}$ ) and f ( w + i Δ w$_{2}$ ) to w . We denote them by f$_{1}$ and f$_{2}$ , respectively. We now can ask if they are parallel. To quantify the answer to this question, we form the differ- ences δ$_{1}$ f = f$_{1}$ - f and δ$_{2}$ f = f$_{2}$ - f and define of the systems we attempted are still small, the trend seen in the sequence of plots shown in Fig. 10 suggests that the Non-Abelian curvature is localized in the vicinity of the fixed anyons. This implies a topological property for the Non-Abelian part of the mononodromies, i.e. their independence of the shape of braiding paths as long as the braiding path are far enough from the anyons. Fig. 12 demonstrates the existence of non-commutative monodromies. Indeed, if all the monodromies were com- muting with each other, that will imply the existence of a gauge in which the fiber-bundle of the zero modes becomes trivial. But this will imply ρ$_{TW}$ =0, which is contradicted by Fig. 12. Fig. 12 reveals much more, namely the splitting of the zero modes space as predicted by the fusion rules of the underlying CFT structure of the Moore-Read sequence. 13,14 For this, notice that the Twist density is practically zero when the mobile anyon comes near the fixed anyons and takes non-zero values only in between the fixed anyons. This implies that, although | f | takes ap- preciable values in the regions near the fixed anyons, the states are commutative. In other words, when the mobile anyon comes close to a fixed anyon, the two eigenmodes of the curvature matrix ˆ F ( P ) split the zero modes space into two sectors that don’t mix with each other during the braiding. If things happen as predicted by the fusion rules of the underlying CFT structure of the Moore-Read sequence, then we should be able to read from the classic This quantity is gauge invariant and measures how much are δ$_{1}$ f and δ$_{2}$ f deviating from being parallel. The two definitions of ρ$_{TW}$ given in Eqs. (55) and (56) coincide. Fig. 12 shows plots of the Twist density for different sys- tem sizes. We now can draw several conclusions regarding the non-commutative and topological properties of the Moore-Read states with four anyons. Although the size VIII. DISCUSSION ρ$_{TW}$ ≡ lim Δ w → 0 | δ$_{1}$ f × δ$_{2}$ f | Δ w$_{1}$ Δ w$_{2}$ . (56)